Section 1.1 Arithmetic Sequences Math 20 – 1

Many patterns and designs linked to mathematics are found in nature and the human body. Certain patterns occur more often than others. Logistic spirals, such as the Golden Mean Spiral, are based on the Fibonacci number sequence. The Fibonacci sequence is often called Nature’s Numbers.

1, 1, 2, 3, 5, 8, 13……

A sequence is an ordered list of numbers usually separated by commas. It contains elements or TERMS that follow a pattern or rule to determine the next term in the sequence.

The numbers in a sequence are called TERMS.

An ARITHMETIC SEQUENCE is an ordered list of terms in which the difference between consecutive terms is a CONSTANT. The value added to each term to create the next term is the COMMON DIFFERENCE.

Given the sequence -5 , -1, 3 …..

a) What is the value of t1? t3? t4?

b) Determine the value of the common difference.

c) What could you do to find the value of t10?

Typically you will be asked to find 2 things. First you will be asked to find the GENERAL EXPRESSION.

Secondly, you will be asked to find the value of a specific term:

For the arithmetic sequence -3, 2, 7, 12, …..

a) Determine t20.

b) Which term in the sequence is 212?

c) What is the formula for the general term?

Which of the following sequences are arithmetic? Find the value of d.

In an arithmetic sequence t4 = 24, t10 = 66, determine the equation of the general term and determine t1.

Homework Pg. 8 #4 – 21 (omit 9)

Section 1.3 Geometric Sequence Math 20 – 1

A GEOMETRIC SEQUENCE is an ordered list of terms in which the difference between consecutive terms is a RATIO. The value multiplied to each term to create the next term is the COMMON RATIO and is represented by r.

Given the sequence 4, 8, 16 …..

a) What is the value of t1? t3? t4?

b) Determine the value of the common ratio.

c) What could you do to find the value of t10?

Typically you will be asked to find 2 things. First you will be asked to find the GENERAL EXPRESSION. With the sequence 4, 8, 16….

Secondly, you will be asked to find the value of a specific term:

Homework Pg. 35#3 - 19

1 + 2 + 3 + . . . 98 + 99 + 100.

Section 1.2 Arithmetic Series Math 20 – 1

Without a calculator, can you please find the sum of the first 100 numbers?

Determine the SUM of the series 17 + 12 + 7 + … -38

The sum of the first 2 terms of an arithmetic series is 19 and the sum of the first 4 terms is 50.

a) What are the first 4 terms of the series?

b) What is the sum of the first 20 terms?

Homework Pg. 19 #6 – 10, 15, 18

Section 1.4 Finite Geometric Series Math 20 – 1

Calculate the sum of the series 5 + 15 + 45 + … + 10935

Algebraically determine the sum of the first 7 terms of the series 27 + 9 + 3 + …

How many terms in the series 2 – 4 + 8 – 16 + … will yield a sum of 342?

The common ratio of a geometric series is ¼ and the sum of the first 4 terms is 1275. What is the value of the first term?

Homework Pg. 48 #4 – 7, 9 – 11, 13

Section 1.6 Infinite Geometric Series Math 20 – 1

A geometric series with an infinite number of terms, in which the sequence of partial sums continue to grow are considered to be DIVERGENT, and as such you cannot determine the sum.

A geometric series with an infinite number of terms, in which the sequence of partial sums actually decrease and approach a finite number are considered to be CONVERTENT, and as such you can determine the sum.

Math 20-1 Chapter 1 Arithmetic and Geometric Sequences and Series Review

Key Ideas Description or Example

Sequences ● An ordered list of numbers where a mathematical pattern can be used to determine the next terms.

● Example: 1, 5, 9, 13, 17... or 1000, 100, 10, 1...● n is the term position or the number of terms, n must be a natural number

Series ● The sum of all the terms of a finite sequence● Example: 5 + 10 + 15 + 20

1 + 0.5 + 0.25 + 0.125...

ArithmeticSequence ● A sequence that has a common difference,

● Example: 2, 4, 6, 8, 10, 12, 14,... where d = 2

Graph of an Arithmetic Sequence

● Always discrete since the n values or the term position must be natural numbers.

● Related to a linear function y = mx + b where m = d and .● The slope of the graph represents the common difference of the general

term of the sequence.● t1 = b + m , add the y-intercept to the slope to get the value of the first term

of the sequence.

ArithmeticSeries

● The sum of an arithmetic sequence.

Use when you know the first term, last term, and the number of terms.

Use when you know the first term, the common difference, and the number of terms.

n

tn

You may need to determine the number of terms by using .

GeometricSequence ● A sequence that has a common ratio.

● Example: 3, 9, 27, 82, 243, 729, 2187... where r = 3

Graph is discrete, not continuous, and not linear.

Problem

Finite Geometric Series

A finite geometric series is the expression for the sum of the terms of a finite geometric sequence.The General formula for the Sum of the first n terms

Known Values are: Known Values are:t1, r and n t1, r and tn

Problem

Infinite Geometric Series

A geometric series that does not end or have a final term.

It may be convergent (sum approaches a value, there is a formula for this) or divergent (sum gets infinitely larger).

The series is convergent if the absolute value of r is a fraction or decimal less than one: |r|<1 , -1 < r < 1The series is divergent if the absolute value of r is greater than one |r|>1, -1 > r > 1

Sum of an Infinite Geometric Series, only if it converges

You must know the value of the first term and the common ratio.

General or explicit formula

● The unique parameters are substituted into the formula.

● The parameters are and r for a geometric sequence or d for arithmetic sequence.

● Example: or

Vocabulary Definition

Common Difference occurs in an arithmeticsequence or series

● The difference between successive terms in an arithmetic sequence, which may be positive or negative.

● Formula:

Common Ratio occurs in a geometric sequence or series

● The ratio of successive terms in a geometric sequence, which may be positive or negative.

● Formula:

Finite Sequence ● A sequence that ends and has a final term.

Divergent ● Where the sum of the infinite geometric series continues to grow and not approach a finite number. There is no sum.

● When |r|>1,

Convergent ● Where the sum of the infinite geometric series approaches a finite number. There is a sum.

● When |r|<1,

Common Errors Description

Formulas ● Using the wrong formula such as using the general term formula instead of the sum formula for a series.

Sequences and Series ● Confusing sequences with series. Sequences are the list of all the terms where series are the sum of all the terms.

Divergent or Convergent

● Must know the difference

Discrete or Continuous

The n value refers to the number of terms or a specific term. The value of n must be a natural number making the graph of the sequence discrete.

2.1 Absolute Value Math 20-1

Putting the absolute values in order from least to greatest

In order to do this, simply determine the actual value of each element, and then visualize placing that on the number line.

|-5|, |-7.8|, |3.11|, |0|, |613 |

First, determine the value of each of the above:

Secondly, place the absolute value of each element on the number line.

Evaluating Absolute Value expressions:

What is the distance between -5.9 and -10.2 on a number line?

What is the distance between 2.75 and -3.5 on a number line?

Determine the solution to √(3−5)2 **the principle root is always Positive. We will discuss this further in Sec 2.2

What about an actual expression?

|100 – 32| – 2|5 – 6| * treat |a| like brackets. Do first

|5x2 + 3x – 4|, when x = 3

**hmwk Pg. 89 #3 – 17

2.2 Radicals Math 20-1

√100=10 √5=√5 – is the simplest form of writing the answer.

***When taking the square root of a PERFECT SQUARE the result will be a rational number. When taking the square root of a NON-PERFECT SQUARE, the result will be an irrational number.

With CUBED roots, there is only 1 possible root, but because you will be dealing with an ODD index, the potential for taking the root of a negative number now becomes a possibility.

3√1000=10 3√−27=−3

3√49=3√49 – is the simplest form of writing the answer.

***When taking the cubed root of a PERFECT CUBE the result will be a rational number. When taking the cubed root of a NON-PERFECT CUBE, the result will be an irrational number.

Working with Radicals:

Product Property: √ab=√ (a )(b)=(√a ) (√b ) Where a,b ≥ 0

What does this mean? We use this rule to help simplify an entire radical into a mixed radical (a mixed radical is the simplest form of an entire radical)

Step 1 – determine the largest perfect square (or perfect cube) factor of the radicand.

Step 2 – break the radicand up into its factors with the perfect square (or cube) first

Step 3 – convert the perfect square (or cube) to its rational value

√125 √300 √ x5

= √(25)(5) = √(100)(3) =√(x2)(x2)(x) = √25√5 = √100√3 =√ x2√ x2√ x = 5√5 = 10√3 =(x)(x)√ x

= x2√ x

You Try: √40 −√108 √75 x3

What about Cubes?

3√48 3√−250 =3√(8)(6) =3√(−125)(2)

= 3√8 3√6 = 3√−125 3√2 = 23√6 = −5 3√2

You Try: 3√6000 x5 2 3√−40 5√486

Division Property: n√ ab=

n√an√b

where n€N, a,b€R, b≠0

Use all the same steps as the product property to simplify these expressions.

3√−4081

4√ 32243

Converting from MIXED radical to an ENTIRE radical

Just try to remember the steps from above, and work backwards.

3√712

3√160

You Try: 3 3√5

Applications

Consider the line segment to the right. How long is the line segment?

We need to use Pythagorean Theorem to determine this. Make a right triangle with the points and then use algebra.

c2 = a2 + b2

Restrictions on the Variable

Anytime you have a variable in an expression, you are required to define the values that are applicable to that variable. This simply means that you have to tell the variable what it is allowed to be so that it follows all the rules of math. The 2 rules that we will need to pay attention to are the following:

a. You cannot take the square root, fourth root, sixth root etc. of a negative number.

b. You cannot divide by zero.

Taking the example √54 x3, we know that because this is a square root situation, the radicand; 54x3, must be greater than or equal to 0. In order for this to be accomplished, each part of the radicand must be ≥ 0. There are 2 parts to the radicand: 54, which is already ≥ 0, and the x3. The x3 can be anything because we have no idea what x will be. But in order for the original expression to follow the rules of math, we must make sure that x3 ≥ 0. The restriction for the entire expression then is x ≥ 0.

What about: 3√12 x5 4√−12x3

**hmwk Pg. 100 #3 – 16 (****13, 15, 16)

2.3 +/- Radicals Math 20-1

Adding and subtracting radicals does not work the same way as multiplying and dividing. In order to Add or Subtract them, you MUST have LIKE TERMS.

Examples: 3√7+2√7 5√11−2√11 5√2+3√5

You Try:

8√7−3√7+15√7 185√10+12 5√10−7 5√10 5√ x−4 √x *define x

8 3√2 x+7√2x−5 3√2 x+√2 x

What happens if the radicals are not in “simplified” form? (aka mixed radicals)

Before you can combine radicals, you need to make sure they are in simplified form. Once simplified, you then combine like radicals.

Examples: √2+√8 5√12+6√48 7√27−3√75+2√147

5√8 x3+4 y √75 y3−2√27 y5−3 x√50 x 4√81 p3q5−2 4√ p3 q5

Word Problems:

Calculate the length of x.

Hmwk: Pg.114 #4 – 12(***9 – 12)

2.1 – 2.3 Check Point for Quiz Math 20 – 1

For the quiz, you need to make sure you understand the following:

a. Determine the absolute value of a number.b. Compare and order absolute values.c. Relate the absolute value on a number line. Section 2.1d. Determine the absolute value in an expression.e. Compare and order radicals.f. Convert radicals from Mixed to Entire and Entire to Mixed.

Section 2.2g. Identify the values of a variable for which the radical is defined.h. Identify like radicals.i. + and – to simplify radicals. Section 2.3j. Identify the values for which the radical is identified.

To help get yourself prepared for this quiz, you may wish to look at the following pages in your workbook:

**Pg. 108 – 109, 136 #1 – 3

Your quiz will be ALL LONG ANSWER. In order to get full marks, you will need to make sure you have complete sentences, and show all work requested and necessary. Format matters far more than the correct answer, so please be thorough.

2.4 x/÷ Radicals Math 20-1

Multiplying and dividing radicals work the same as any other multiplying. The main difference is to remember to only combine like elements, and to simplify at the end when necessary.

HMWK Pg. 126 #3 – 5, 7, 8

There are situations where the denominator is a radical that does not divide out. This means that it remains in the denominator. Radicals are not allowed to be in a simplified radical. In order to remove this, you need to RATIONALIZE the denominator. This is accomplished by multiplying both top and bottom by the actual radical. Let’s see what happens…

5√7+3√7

9√2−3√5√12

3√2−63−2√6

HMWK Pg. 127 #6, 9, 10, 12 – 16

2.5 Solving Radical Equations Math 20-1

**said to have 1 real root

You Try:

3√ x=5 4 √x+1−5 = 3

Your Turn: √2x+3−√x+2=2

Word Problems:

The formula d = √13 h is used to estimate the distanced of the horizon from the observer at a height above sea level. (*d is in km, and h is in m) If an observation platform is built on a tower that is 100m above sea level, how high would the platform be for the observer to be 50Km from the horizon?

d = 50h = 100 + t

You Try: The period, P, in seconds, of a pendulum is the time to complete one full swing. The period can be determined by:

P=2 π √ L9.8

Where L is the length of the pendulum in meters. Approximately how long should a pendulum be to complete 1 full swing in 5s.

HMWK Pg. 145 #4 – 16

Chapter 2 Absolute Value and Radicals Review Math 20 - 1

Key Ideas Description or Example

Absolute value Represents the distance from zero on a number line, regardless of direction. Absolute value is written with a vertical bar around a number or expression. It represents a positive value.

Example: |24| = 24 |-2| = 12

The absolute value of a positive number is positive, The absolute value of a negative number is positive, and the absolute value of zero is zero.

Piecewise Definition of an absolute value function

Radical means root. The index determines which root you are looking for.

Principle Square Root

Negative Square Root

Equations with even indices have roots.

Equations with odd indices have one root.

the root is positive

The radical is negative, the root will be negative

Perfect Squares

Perfect Cubes

Perfect Fourth Roots

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169…x2

1, 27, 64, 125, … x3

1, 16, 81,… x4

Perfect Square Roots

Perfect Cubes

Perfect Fourth Roots

Convert entire radicals to mixed radicals

Remember

Entire

Mixed Radical . The radicand is in lowest terms.

not -3

Convert mixed radicals to entire radicals

Apply the index as an exponent to the base under the radicand.

A negative does not enter the radicand, the entire radicand stays negative.

Comparing and ordering radical expressions Write as entire radicals to compare, apply the proper index.

Write as a decimal and compare.

Identifying restrictions on the values for a variable in a radical expression.

The radicand must be greater or equal to zero.

For the restriction is

For the restriction is

Simplifying radical expressions using addition or subtraction

Radicals must have the same index and the same radicand to be added or subtracted. Only add or subtract the coefficients.

Do not add radicands.

Multiply radicals Radicals must have the same index.

Multiply coefficients, multiply radicands, simplify.

Divide radicals Radicals must have the same index.

Divide coefficients, divide radicands, simplify.

,

Rationalize the denominator

Monomial denominator

Binomial denominator

Solve an equation with one radical term.

State restrictions on the variable in the radicand. Check for extraneous roots.

Isolate the radical term on one side of the equation and then apply the Power Rule with squares. Verify by substitution.

Solve an equation with two radical terms

State restrictions on the variable in the radicand. Check for extraneous roots.

Separate the radicals, one on each side of the equal sign.

Square both sides of equation, not individual terms.

Vocabulary Definition

Absolute Value Distance from zero on a number line.

Piecewise definition

Entire Radical

The coefficient is one.

Mixed Radical The radicand is in simplest form.

Radicand The quantity under the radical sign.

Index The small number in a radical that indicates which root to take.

Rationalize the denominator A procedure for converting a denominator containing a radical into a rational number. The value of the expression does not change.

Conjugates Two binomial factors whose product is a difference of squares.

The conjugate of is .

Restrictions on the variableFor even index .

For odd index, the radicand may be any real numb.

Review: Page 156 – 164

3.1 Factoring Polynomial Expressions Math 20 - 1

Factoring trinomials require an understanding of how each term is created. If you know the pattern, then it is easy to factor.

Part A: Expand and Simplify

Part B: Factor

HMWK Pg. 176 #1 – 22

3.2 Solving Quadratic Equations by Factoring Math 20 - 1

Remember, SOLVE means find the zeros. It’s finding where the parabola crosses the x axis.

To solve quadratic equations by factoring, apply theZero Product Property which states that, if the productof two real numbers is zero, then one or both of the numbersmust be zero.Thus, if ab = 0, then either a = 0, or b = 0 or both equal 0.

x(x + 1) = 0 (x – 5)(x + 1) = 0

x = 0 OR x + 1 = 0 x – 5 = 0 x + 1 = 0 -1 = -1 +5 = +5 - 1 = - 1

x = -1 x = 5 x = -1

Solve the following by Factoring then VERIFY

x2 – 2x – 8 = 0 (2x – 3)(x + 1) = 3

2x2 + 18x = 12x 2x2 = 4x

What if you have a radical in the equation?

√ x+3−1=x **remember to check for extraneous roots

Determine the roots of the following:

-16 = x2 – 10x 3x2 + 19x – 14 = 0 x2 + 8x = 0

(a + 2)2

+ 3(a + 2) + 2 = 0

x2 – 36 = 0 (x – 2)2 – 25 = 0 x2 + 4x = -4

21 3 04

x x

What happens if you are given the roots and asked to find the quadratic equation? Just remember that the roots were determined by factoring the equation. This means if you want to get back to the equation all you need to do is multiply the “zero” factors.

If the zeros are -6 and 3

16x2

– 121 = 0

Now is this the only possible equation? What others would also result in zeros of -6 and 3?Putting it all together:

Find the length of the two unknown sides of a triangle if the hypotenuse is 15cm long and the sum of the other two legs is 21cm

A rectangular garden has the dimensions of 5m by 7m. When both dimensions are increased by the same length, the area of the garden increases by 45m2. Determine the dimensions of the larger garden.

Hmwk Pg. 189 #1 – 22

3.3 Solving Quadratic Equations Using Square Roots Math 20 – 1

A ball is thrown up in the air. Three different forms for the height of the ball, in feet, as a function of time, x in seconds, are:

Standard Form: y = -16x2 + 32x + 48

Vertex Form y = -16(x – 1)2 + 64

Factored Form: y = -16(x – 3)(x + 1)

How could you show that the three forms are equivalent?

What characteristic of the graph of the function does each form reveal?

Two ways to solve the same equation:

Solve using the square root method:

(x + 1)2 = 25 2(x – 1)2 – 8 = 0

Solving by COMPLETING THE SQUARE

You try:

x2 – 10x + 3 = 0 2x2 – 14x + 8 = 0

-½x2 + 6x – 1 = 0 ½x2 + 3x – 9/2 = 0

The sum of the squares of three negative consecutive integers is 77. Write a quadratic equation to represent the sum of the integers. Determine the value of the integers.

HMWK Pg. 206 #1 – 17

3.4 Solving using the Quadratic Formula and the Discriminates Math 20 – 1

Remember that the general form of the quadratic equation is:

f(x) = ax2 + bx + c

When the equation is in this form, you are able to use the quadratic formula to calculate the zeros. This method works all the time, even if there are no zeros.

x=−b±√b2−4ac2a

Try this : 6x2 – 3 = 7x

Step 1 – rearrange the equation into standard formStep 2 – determine values for A, B, and CStep 3 – Write formula and substituteStep 4 – determine both zeros, state as exact and as an estimateStep 5 – verify by graphing in calculator.

Try this: 2x2 + 8x – 5 = 0

Try this: 5x2 – 10x + 3 = 0

Not only is it important for you to be able to determine the type of roots when we have the full formula, it is important that you are able to work backwards to determine a missing element, knowing what type of roots you would like.

Hmwk: Pg. 217 #1 – 22Pg. 232 #1 – 20

4.1 Properties of Quadratic Functions Math 20-1

Every quadratic function can be graphed, and it takes the shape of a parabola. It either has a maximum or a minimum point called the vertex. Every parabola is symmetrical about a vertical line, called the axis of symmetry that passes through the vertex.

Listing the properties of a quadratic function from a graph

From a graph, you should be able to determine the following:

Vertex:

Maximum:

Axis of Symmetry:

Domain:

Range:

x – intercepts:

y – intercept

Your turn: Using your graphing calculator, graph y = 2x2 – 6x + 20. From that, determine the following:

a) x – intercepts

b) y – intercepts

c) vertex

d) equation of axis of symmetry

e) domain

f) range

x – intercepts are very pivotal. They are also known as the zeros of the equation. Algebraically you can determine the zeros simply by solving for x.

y = -2x2 – 6x + 20

An object is fired upward with a speed of 60m/s. It’s height, h, in meters, after t, seconds is modelled by the equation:

h = -4.9t2 + 60t + 1.85

a) What does the vertex mean?

b) What do the intercepts mean?

c) What does the domain mean?

d) What does the range mean?

HMWK Pg. 257 #1 – 12

4.2/4.3 Solving Graphically & Transformations Math 20-1

Graph each of the following functions, and fill in the chart.

Function ZerosQuadratic

EquationDiscriminant

y = x2 + x – 6

y = x2 – 9x

y = 5x2 + 7x – 6

y = 4x2 – 28x + 49

**the discriminant helps determine how many roots there are. You find this out by using the

equation √b2−4 ac . If the result is +ve there will be 2 distinct roots, if the result is -ve there are

no real roots, and if the result is zero, there is 1 distinct root.

Knowing that the parent function is y = x2, and what it looks like, we will now look at how the graph changes as we alter the values of a, p and q. Using your graphing calculator fill in the following chart.

y = x2 y = -5x2 y = 0.2x2 y = (x – 3)2 y = (x + 3)2 y = (x + 2)2 + 4 y = (x + 2)2 - 3

Vertex

Axis of Symmetry

Max/Min

Domain

Range

Y-Int

X-Int

Putting it all together. Based on what you have found out from the chart above, answer the following questions:

a) What does the “a” value do to the graph of x2? What happens when a>0? What happens when 0<a>1?

b) What does the “p” value do to the graph of x2?

c) What does the “q” value do to the graph of x2?

HMWK Pg. 265 #3 – 5Pg. 269 #1 – 4

4.4 Analyzing in the form y = a(x – p)2 + q Math 20-1

Remember the following: f(x) = a(x – p)2 + q is the vertex form of the function. The standard form of the function is simply putting it in terms of y. This allows you to determine coordinates on the graph.

Standard form:

Determining the equation from its graph:

Step 1: Determine the vertex

Step 2: Does it open up or down?

Step 3: Pick a point on the graph and substitute in for x and y and solve for a.

Step 4: Write in its standard form

You try:

Graphing from the equation in standard form:

In order to sketch a graph from the equation you need to identify the following characteristics:

y = -2(x + 2)2 – 3 Direction of opening Vertex

Equation of axis of symmetry Intercepts (both x and y)

Domain and Range

Once you have these, you are able to sketch the graph.

You try y = 0.5(x – 3)2 + 2

Determining the equation from the characteristics: If you are given information about the function, you should be able to piece together the equation from the words that are used. You want to be very careful to pick up on words like “maximum” or “minimum”, “zeros” and any other word that give insight as to how the function might look.

The equation for the axis of symmetry of a quadratic function is x = -1. The graph passes through the points A(0, 3) and B(-3, 9). Determine the equation of the function.

A 12m long suspension bridge is made with rope and logs. The rope forms a parabola with its lowest point 2m above the centre of the bridge. The ropes are attached 6.5m above the bridge. Determine an equation to model the parabola.

Hmwk: Pg. 284 #1 - 14Chapter 4 Quadratic Functions Mid-Unit Review Math 20 - 1

Key Ideas Description or Example

Quadratic Functions

polynomial of degree two

Standard Form: f(x) = ax2 + bx + c, a 0

Vertex Form: f(x) = a(x p)2 + q , a 0

Factored Form: f(x) = (x + c)( x + d)

For a quadratic function, the graph is in the shape of a parabola

Parent Graph is y = x2

Characteristics of a Quadratic Function from the vertex form of the equation

f(x) = a(x - p)2 + q

The coordinates of the vertex are at (p, q). Note that the negative symbol from (x - p) does not transfer to the value of p.

f(x) = 2(x - 3)2 + 4 vertex at (3, 4)

f(x) = 2(x + 3)2 + 4 vertex at (-3, 4)

Horizontal Translations:

When p > 0 the graph moves (translated) to the right.

When p < 0 is translated or shifts to the left.

Vertical Translations:

When q > 0 the parabola shifts up.

When q < 0 the parabola shifts down.

The parameter “a” indicates the direction of opening as well as how narrow or wide the graph is in relation to the parent graph. When a > 0 , the graph opens up and the vertex is a minimum

min max

When a < 0, the graph opens downward and the vertex is a maximum.

When -1 < a < 1, the parabola is wider than the parent graph.

When a > 1 or a < -1, the parabola is narrower than the parent graph.

The Axis of Symmetry is an imaginary vertical line through the vertex that divides the function graph into two symmetrical parts. The equation for the axis of symmetry is represented by x – p = 0 or x = p

The domain of a quadratic function is all real numbers

. The only exception is for real life models. The range of a quadratic function depends on the value of

the parameters “a” and “q”.

When “a” is positive, the range is .

When “a” is negative, the range is .

x- and y- intercepts Calculate the x-intercept pt (x, 0) by replacing y with 0 in either form of the function equation. ax2 + bx + c = 0 or a(x p)2 + q = 0.

Calculate the y-intercept (0, y) by replacing x with 0 and solving for the value of y.

Summary of Characteristics.

Write a quadratic function in the form y = a(x – p)2 + q for a given graph or a set of characteristics of a graph.

Write the equation of the Quadratic Function in the form

4.5 Converting from Standard Form to General Form Math 20-1

There are some situations where it is more convenient to have the equation in standard form. Standard form allows us to sketch the graph easily, where general form allows us to determine the y-intercept quickly.

Write each in General form: (all you need to do is expand and simplify

a. 3(x + 1)2 – 4 b. –(x – 2)2 + 7

This process is called “completing the square”. By creating a “perfect square trinomial” you are able to simplify part of the function to show the vertex.

You try:

a. y = x2 – 6x + 2 b. y = x2 + 8x – 7

What about this one? y = x2 + 5x – 3

What changes if there is an “a” value other than 1? GO SLOW

y = 2x2 + 16x + 24 y = -4x2 + 9x – 2 y = ½ x2 + 5x + 1

You try: y = 3x2 – 12x + 7 y = -2x2 + 5x – 3

Hmwk: Pg. 295 #1 – 13

4.6 Analyzing Quadratic Function in General Form Math 20-1

y = ax2 + bx + c

What does each part of the equation tell us?

“a” * tells us if the curve opens up (+) or down (-)

* tells us if there is a MAXIMUM or MINIMUM with the vertex

* tells us how wide or narrow the curve will be

“b” * alters the parabola’s axis of symmetry

“c” * is the y-intercept of the parabola

How to break the general form of the equation down to determine some characteristics.

y = 0.5x2 -1.5x – 5

1. Factor out the 0.52. Factor the trinomial – tells you the zeros3. Use the zeros to determine the axis of symmetry. (remember the

parabola is symmetrical and the zeros are equidistant from the axis of symmetry.

4. Substitute the axis of symmetry value in, to find the full vertex.

You try: -2.5x2 – 7.5x + 10

Graphing

y = 4x2 + 4x – 15

a. Is it factorable? – use the discriminant (b2 – 4ac) to decide. If the discriminant is a perfect square, then the trinomial is factorable.

b. Factor it

c. Use the x – intercepts (zeros) to determine the axis of symmetry

d. Substitute the value of the axis of symmetry back into the equation to calculate the y value.

e. You now have the vertex and the x-intercepts to help sketch the graph. (use the step method)

What if it is not factorable? y= -3x2 + 9x + 1

a. Use the discriminant to determine if it is factorable.

b. Because it is not factorable, you now need to complete the square.

c. You now have the coordinates for the vertex.

d. Use the step method to create the points for the graph.

HMWK Pg. 306 #1 – 17

4.7 Solving Word Problems Math 20-1

When it comes to solving problems it is very important to realize the goal is to make a quadratic relationship. That means, at some point you need to have x2 somewhere. Keeping this in mind, it is equally important to read each question and be aware of what information they have given you. It will somehow relate back to what we have done throughout the chapter.

1. Two numbers have the sum of 20. Does the sum of their squares have a maximum value or a minimum value? What is this value, and what are the two numbers?

a. Make your “let” statements.b. Create your overall equation.c. Expandd. Determine if it is a maximum or a minimum (look at the “a” value)e. Complete the square to determine the values.f. Make your answer statements.

2. Student parking costs $20 for a pass. At this price, 150 students will buy a pass. For every $5 increase, 20 fewer students will purchase passes.

a. What is the price of the parking pass that will maximize the revenue?

b. What is the maximum revenue?

Hmwk Pg. 322 #1 – 13

Chapter 4 Quadratic Functions Unit Review Math 20 – 1

Converting from vertex form to standard form.Follow order of operationsPEMDAS

Completing the square Converts the standard form into vertex form so the characteristics can be determined.

Complete the Square ProcessThe middle term coefficient must be divided by two and then squared.

Complete the Square when the leading coefficient is not 1.

The parameter “a” must be factored out of the terms involving x before completing the square.

Solving Problems

2y a x p q 2y ax bx c

Solving Problems Revenue

Vocabulary Definition

vertex form

Axis of symmetry

A line through a shape so that each side is a mirror image. Equation is x – p = 0

parabola A plane curve formed by the intersection of a right circular cone and a plane parallel to an element of the cone or by the locus of points equidistant from a fixed line and a fixed point not on the line.

domain The domain of a function is the set of input or argument values for which the function is defined.

range In mathematics a range of a function is the set of all of the output values made by the function. May be considered the height of the curve.

vertex The lowest point of the graph or the highest point of the graph. The axis of symmetry passes through the vertex.

maximum value

The parabola opens downward, vertex (p, q), the value of q gives the maximum value.y = q

minimum value

The parabola opens upward, vertex (p, q), the value of q gives the maximum value.y = q

5.1 Solving Quadratic Inequalities Math 20-1

Your Turn: Solve -3x2 – 8x + 3 ≥ 0

Now you have to verify

Your Turn Solve x2 + 5x – 24 > 0 (did you get x< - 8 or x > 3, xER?)

Sheila is going to rent a plot of public gardening land, and she can only afford to rent up to 180 ft2. She wants the length of the garden to be 3 ft longer than the width. What possibilities does Sheila have for the width of the garden? (did you get 0 – 12ft?)

5.2 Solving Linear Inequalities Math 20-1

Your Turn: Graph the linear inequality 2x + y > 2

1. Graph the associated equation 2x + y = 2

*use a dashed line, as it is >

2. Test a point that is not on the boundary line

3. Shade the half-plane that contains the solution set.4. Choose a point to verify

What if you are given the graph? How do you determine the equation?

Your Turn:

What about this situation?

5.3 Graphing Quadratic Inequalities Math 20-1

5.5 Solving Quadratic Inequalities Math 20-1 With Algebra

Your Turn: Solve using Substitution

y = -2x2 + 10x – 2y = -15

Your turn: Solve these systems y = x2 – 3x – 4 and y = 2x2 – 2x + 32x – y = 4 y = x2 + 5x – 7

Key Ideas Description or Example

Solving Linear Inequalities in Two Variables

The solution is a shaded half-plane region with a dashed boundary line if the inequality is < or >. The boundary line is solid if the inequality is < or >.

y is “less than or equal to”solid boundary line, shade below y is ”greater than”

dashed boundary line, shade above

Method One: Isolate the y-variable in the linear inequality and graph with technology.

Method Two: Graph using x- and y-intercepts and use Test point to determine shaded region.Method Three: Isolate the y-variable and graph using slope and y-intercept, then use test point to determine which side of the boundary line to shade. If the test point makes the inequality “true”, the point lies in the solution region, this side of the boundary should be shaded. If the test point makes the inequality “false” shade the region on the opposite side of the boundary.

Solving Quadratic Inequalities in One Variable

The solution set contains the intervals of x-values where the y-values of the graph are above or below the x-axis (depending on the inequality).

Graphical Method: Graph the related function; determine zeros and intervals of x-values where y-values are above or below x-axis.

Alternate Method: Determine the roots of the related function and use Case analysis or sign analysis with test points.

Determine zeros of the related function

Use test points in each domain region to

determine if the function is positive or negative

Indicate solution region.

The solution interval for

The solution interval for is written as

The solution interval for is written as

Quadratic Inequalities in Two Variables

The solution is a shaded region with a solid or broken boundary parabolic curve.

y “is greater than” dashed boundary, shade above

y is “greater than or equal to”solid boundary, shade above

Vocabulary Definition

Boundary Line Solid line if inequality symbol is .Dashed line if inequality symbol is < or >.

Solution Region All points in the Cartesian plane that makes the inequality true. The Shaded region contains all of the solutions to the inequality.

Case Analysis or Sign Analysis A number line is divided into intervals depending on the roots of the related equation. Inequality expressions describe each interval. A test point is used to determine if the interval is true or false.

Common Errors Description

simplifies to The direction of the inequality must be reversed when divided by a negative number.

Broken or Solid Boundary Line

< or > is shown with a broken line or parabolic curve.is shown with a solid line or parabolic curve.

Test point is on the Boundary Line

The test point (0, 0) may be used for most inequalities. Do not use the point (0, 0) when the boundary line or parabolic curve goes through the origin.

Section 6.1 Angles in Standard Position Math 20 – 1

In Geometry, an angle is formed by two rays with a common end point. In TRIGONOMETRY, angles may be interpreted as a rotation of a ray.

Sketch an angle in standard position. State the quadrant in which the terminal arm lies

50° 300° 200°

Reference Angle: The angle measurement from the horizontal (x) axis. Used as a method of measurement.

This means, when we are given a point, we should be able to determine all of the Primary Trigonometric ratios.

The point P(4, 7) is on the terminal arm of an angle Ө in standard position. Determine the primary trigonometric ratios of Ө. Also determine the measure of Ө to the nearest degree.

Step 1: Calculate the exact value of r

Step 2: Create your ratios

Step 3: Determine the measure of Ө

Your turn: Determine the primary ratios and the measure of Φ if it is the point P(3, 4) in standard position.

An aircraft made an emergency landing 200 km from an airport. Its heading from the airport was E50°N. The land-based rescue team has to travel East and then North to get to the aircraft. To the nearest kilometre, how far should the team travel in each direction?

DRAW A DIAGRAM

Homework Pg. 431 #3 – 15

QII

QIII

QI

QIV

Section 6.2 Angles in Standard Position Part 2 Math 20 – 1

The coordinate grid system is actually made up of 4 quadrants. The primary trig ratios that we worked with the other day, are set up to help us find angles that are in Quadrant I only, and as such, needs to be adjusted to help determine values in Quadrants II thru IV.

You will need to calculate the reference angle and then use that to determine the angle in standard position.

The point P(2, -5) lies on the terminal arm of an angle Ө in standard position. Determine the primary trigonometric ratios of Ө. And determine the measure of Ө to the nearest degree.

You try with the point B(-2, -4)

What if we are given a ratio? How could be determine the measure of the angle?

cosθ=¿−34

¿

Use what we know to break down our information

What if we are given tanθ=¿−43

¿ what would the measure of the angle be?

Now, using the ratio from our example above, how would you determine the other Trigonometric Ratios?

cosθ=¿−34

¿

Homework Pg. 448 #3 – 21

Section 6.2 Cont Special Angles Math 20 – 1

In Trigonometry there are 3 angles that are considered to be “Special Angles”. They are special, because of the relationship they have with rotational trigonometry, which will be discussed further in Math 30 – 1. Your only expectation is to learn the EXACT trigonometric ratios for each of these special angles and be able to manipulate this information to determine any angle rooted in the 3 defined values.

The special angles are the 30°, 45° and the 60°. The ratios are based on the following values:

How does this work?

1. Determine the angle and ratio for which we are interested.2. Determine which is the reference angle3. Match the exact ratio with the reference angle4. Use the CAST rule to determine the sign

Determine the exact ratio of

Sin 30° Tan 120° Cos 135° Tan 330°

Sin 210° Cos 240° Tan 300° Sin 150°

Section 6.4 The Sine Law Math 20 – 1

In grade 9 and 10 you worked with the primary trig ratios of Sin Cos and Tan to solve RIGHT triangles. However, not all triangles are right triangles. There are 2 options for this, depending on the information that you are given in the question. The first is the SINE LAW (the more complicated one )

Use the SINE law when you are given a triangle and you know a side and ITS angle.

How far apart are Stan and Paul?

If the instruction is to solve, you must determine ALL side measurements (100th ) and ALL angle measurements (10th )

The Ambiguous Case: When 2 sides and the NON-included angle are given, there is the potential for 3 different outcomes (0, 1, or 2 triangles may be drawn) This is referred to as the AMBIGUOUS case. It only happens with the SINE Law.

IF a < b sin A : NO SOLUTION

IF a = b sin A: 1 SOLUTION

IF a > b sin A: 2 SOLUTIONS

In ΔABC <A = 40°, a = 10, b = 15. Find <B

a = 10

b sin A = 9.64

a > b Sin A --------- 2 solutions

Try this one:

Section 6.5 The Cosine Law Math 20 – 1

7.1 Equivalent Rational Expressions Math 20 - 1

How do we make EQUIVALENT rational expressions?Remember that a rational expression is basically a fraction. We use the same methods for making equivalent fractions to make equivalent rational expressions. We either MULTIPLY or DIVIDE, but we must remember that what you do to the top, you must do to the bottom.

By multiplying2 x2+6 x

4 x

By dividing4 x2+8 x

4 x

HMWK Pg. 527 #1 – 18***This is one of the most important sections of the course

7.2 Multiplying and Dividing Rational Expressions Math 20 - 1

Multiplying and dividing rational expressions is done following the same process as multiplying and dividing fractions. Here is the step by step process:

1. Factor all terms of the expression2. Flip the second fraction (if dividing)3. State all NPVs4. Eliminate any common factors5. Multiply Across6. Reduce if possible

( 2x2−12 x15 x )( 5 x

x−6 )( 12 x3

3 x2+6 x )( 4 x3+8 x2

5 )

( x−53 x2−9 x )÷( 5

6 x−18 )( 2 w4 w2−24 w )÷( 6 w2−6 w

9 w3+54 w2 )

Your Turn:

7.3/7.4 Adding and Subtracting Rational Expressions Math 20-1

Adding and subtracting rational expressions is done following the same process as adding and subtracting fractions.

Your turn:

But we all know, this chapter is not going to be this straight forward. As a rule, most of us struggle with adding and subtracting fractions, especially when the denominators are not the same. Here is the step by step instructions to help you out.

1. Factor all terms of the expression2. State your NPVs3. Eliminate ONLY IN THE SAME FRACTION4. Build your common denominator – use each factor only once5. Build your numerators6. Expand and simplify your numerators7. Reduce if possible

310 x2 +

615 x

36 x2 −

14 x

3 x2x+2

+ 45 x+5

2 x+5x2−16

− x−3x+4

Your turn:

HMWK Pg. 553 #1 – 17Pg. 565 #1 – 18

7.5 Solving Rational Expressions Math 20 – 1

x3−2 x−13

4=5 x

2

1x− 5

3 x=1

6

3x−5

− x+12x+10

= 3x2−25

Your Turn:

Hmwk Pg. 583 #1 – 17Tomorrow is a work period

7.6 Word Problems Math 20 – 1

The salt concentration in a tank is determined by the formula:**t in minutes

C= 10 t25+t

How long does it take for the concentration levels to reach 3.75g/L?

Paula and Mark can paint a room in 42 minutes if they work together. When Paula works alone, she can paint a room 13 minutes faster than Mark can when he works on his own. How fast can Mark paint a room?

Sam can wallpaper a bathroom in 3hrs. Rebecca can wallpaper the same sized bathroom in 5hrs. How long would it take if they did it together?

Find the value of two integers. One positive integer is 5 more than the other. When the reciprocal of the larger number is subtracted from the reciprocal of the smaller the result is 5

14 .

Sonny bought a case of concert t-shirts for $450. He kept 2 shirts for himself and sold the rest for $560. He made a profit of $10/shirt. How many t-shirts were in the case?Sonny organized his information in the following chart: Cost/shirt Sales/shirt Profit/shirt

450x

560x−2 10

A traveling salesman drives from home to a client’s store 150km away. On the return trip he drives 10 km/h slower and as a result added an hour and a half to the driving time. At what speed was the salesperson driving on the way to the store? Remember that d = rt

Hmwk Pg. 597 #1 – 15Tomorrow is a work period

Math 20-1 Chapter 7 Rational Expressions and Equations Concept Review

Key Ideas Description or Example

Simplifying Rational Expressions A rational expression is a fraction, where p and q are polynomials, q ≠ 0.

A non-permissible value is a value of the variable that causes an expression to be undefined. For a rational expression, this occurs when the denominator is zero.Indicate all non-permissible values for variables in a rational expression.

Rational expressions can be simplified by: factoring the numerator and the denominator determining non-permissible values for variables divide all common factors in both the numerator and denominator

Adding and Subtracting Rational Expressions.

To add or subtract rational expressions, the expressions must have the same denominator.As with fractions, we add or subtract rational expressions with the same denominator by combining the terms in the numerator and then writing the result over the common denominator.For terms with Like Denominators, add or subtract the numerators only. The denominator does not change.

For unlike denominators, rewrite them in equivalent forms that have the same denominator

• Factor each denominator.• Find the least common denominator. The LCD is the product of all different

factors from each denominator, with each factor raised to the greatest power that occurs in any denominator.

Multiplying Rational Expressions

To Multiply Rational Expressions: (a common denominator is not required)● Factor the polynomials in each numerator and denominator ● Simplify the expression by dividing out common factors in both the

numerator and denominator. *Don’t forget to simplify before you multiply!● State the NPV’s

Dividing Rational Expressions

To Divide Rational Expressions:● Factor the polynomials in the numerators and denominators if possible● List all non-permissible values for the variables. ● Multiply the first term by the reciprocal of the second term● Divide out common facots

Solving Rational Equations

To Solve a rational equation:1. Determine the LCD of the denominators, list all NPV’s2. Multiply both sides of the equation by the LCD. Reduce common factors.3. Solve the resulting polynomial equation.4. Verify all solutions

Solving Problems

Solving Problems

Vocabulary Definition

Non-Permissible Values (NPV’s)

Restrictions on the variable.

NPV’s are any values for a variable that would make a denominator equal to zero. The denominator may need to be factored to determine the restrictions on the variable.

Rational Equation A rational equation is an equation containing at least one rational expression.

Common Errors Description

When simplifying rational expressions, an error is to divide only one term in the dividend by the divisor.

Incorrect Correct

NPV’S 1. Forget to identify the non-permissible values before simplifying and multiplying.2. Forget to identify the non-permissible values for the numerator of the divisor in a division statement.3. Forget to identify the non-permissible values for the equivalent forms of a rational expression. The non-permissible values must be determined in each case before the expression is simplified.

Section 8.1 Part 1 Absolute Value Functions Math 20 – 1

Absolute Value is defined as the distance from zero in the number line. Absolute value of -6 is 6, and the absolute value of 6 is 6. Both are 6 units from 0 in the number line.

You can express the distance between these 2 points in 2 ways.

|6 – (-6)| and |-6 – 6|

The graph of an absolute value is such that the range will never be negative.

There is a second way of writing this relationship, and it is called a PIECEWISE DEFINITION. What this is, is the definition of the absolute value in 2 pieces. The first

piece is for the values of x that are 0 and greater, and the second piece is for the values of x that are less than 0.

Graphing the absolute value of the function y = |2x – 4| and y = |x2 – 4|

Method 1: Use a table of Values

y = |2x – 4|

Domain: x-Int:

Range: y-Int:

The x-intercept of the linear function is the x-intercept of the corresponding absolute value function. This point may be called an INVARIANT POINT.

y = |x2 – 4|

Domain: x-Int:

Range:

Method 2: Using the graph of the Linear Function y = |2x – 4|

1. Graph the actual function y = 2x - 4Slope = 2x-int = (2, 0)y-int = (0, -4)

2. Reflect the negative x values on the x axis

3. Create the final graph of the absolute function

Method 2: Using the graph of the Linear Function y = |x2 – 3x - 4|

1. Graph the actual function y = x2 – 3x – 4

2. Reflect in the x-axis the part of the graph of y = x2 – 3x – 4, that is below the x-axis

3. The final graph of the function

Domain: x-Int:

Range:

Section 8.1 Part 2 Expressing as a Piecewise function Math 20 – 1

Now, let us use the functions that we spoke about last class:

Find the invariant point: (remember this is the x intercept of the positive portion of the expression)

Its important to realize it’s a positive slope

Set up your number line

Putting it all together: Graph the absolute value function y = |-2x + 3| and express it as a piece wise function.

Graphing: remember it’s a negative slope Piecewise: Find x intercept

Domain:

Range:

Piecewise Function |-2x + 3| =

**Remember the domains of the function y = f(x) is always the same as the function y = |f(x)|

**Remember, the ranges of the function y = f(x) is NOT the same as the function y = |f(x)|.

Expressing a quadratic as a piecewise function: f(x) = |x2 – 3x – 4|

Determine the Critical Points: *** Opens UP

Make your line:

Using the breakdown that you have created above, make your piecewise statements.

How does this change if you have a function that **Opens DOWN?

y = |-x2 + 2x + 8|

Determine critical points:

x = 4 or x = -2

Make your line:

Now create your piecewise function:

|-2x2 + 2 x + 8|

Section 8.2 Absolute Value Equations Math 20 – 1

To solve absolute value equations, you need to remember the actual definition of absolute value:

This should act as a reminder that there are always 2 equations to consider when solving an absolute value function.

Solve |x| = 3 what this means is that the output cannot be negative

Equation 1: Domain x ≥ 0

Just use the equation |x| = x, so in this case x = 3

Equation 2 : Domain < 0

Just use the equation |x| = -x, so in this case x = -3

Therefore the solution is 3 or -3. On a number line, this solution looks like this:

Solve |3x + 2| = 4x + 5

Solving graphically: graph the absolute value of the function in your y1 and the solution in the y2. Then calculate the intersections. Your solution will be the x value.

What if the equation is more complex? |x – 5| = 2 **output cannot be negative

Use the piecewise version of the absolute value:

Equation 1: domain ≥ 5

x – 5 = 2 x = 7 and 7 satisfies the condition x ≥ 5

Equation 2: domain x < 5 -(x – 5) = 2

x – 5 = -2 and 3 satisfies the condition x < 5 x = 3

The roots of the equation are x = 3 and x = 5

***always VERIFY your solutions to see that they are valid!!!!

Graphically!!!!

What if the equation is even MORE complicated?

23 2 if 33 22(3 2) if 3

x xx

x x

Use the piecewise statements:

You now have to implement a restriction. Remember an absolute value statement MUST RESULT IN A POSITIVE. This means that 4x + 5 must be ≥ 0. In order for this to occur, x ≥ -5/4. Now set up your 2 equations. (graph it)

What about quadratic absolute values?

Now Graph it. What do you notice?

Word problems

Before a bottle of water can be sold, it must be filled with 500mL of water with an absolute error of 5mL. Determine the minimum and maximum acceptable volumes for the bottle of water that are to be sold.

Solve an Absolute Value Equation graphically.

Ron guessed there were 50 Jelly Beans in the jar. His guess was off by 15 jelly beans. How many jelly beans are in the jar? Write an absolute value equation that could be used to represent this situation.

Section 8.3 Reciprocal Functions Math 20 – 1

The reciprocal function is a special case of the rational function. For a reciprocal,

the numerator is always 1. A reciprocal function has the form , where f(x) is a polynomial and f(x) ≠ 0.

The reciprocal of a number is obtained by interchanging the numerator and the denominator.

What do reciprocals look like graphically?

You should notice that the x intercept of the linear function creates a point or line that the reciprocal graph never actually touches. This is called an ASYMPTOTE.

Things to consider.

You try these:

Sketch the graphs of f(x) = 2x – 6, and its reciprocal:

Sketch the graphs of f(x) = x2 – 9 and its reciprocal

Absolute Value and Reciprocal Functions Concept Review

Key Ideas Description or Example

Absolute value Represents the distance from zero on a number line, regardless of direction. Absolute value is written with a vertical bar around a

number or expression. It represents a positive value.

Example: |24| = 24 |-2| = 12

The absolute value of a positive number is positive, The absolute value of a negative number is positive, and the absolute value of

zero is zero.

Piecewise Definition of an absolute value function

Graphing an Absolute Value Function

To graph the absolute value of a linear function:

● Method 1: Create a table of values, then graph the function using the points. Note: There are two pieces and all points are on or above the x-axis.

● Method 2: Graph the related linear function y = 2x + 3. Reflect, in the x-axis, the part of the graph that is below the x-axis. (Negative y-values become positive.)

The domain is all real numbers.

The range is .

To graph the absolute value of a quadratic function:

● Method 1: Create a table of values, and then graph the function using the points. Note there are three pieces and all points are on or above the x-axis.

● Method 2: Graph the related quadratic function y = x2 – 4. Then reflect in the x-axis the part of the

graph that is below the x-axis.

The domain is all real numbers.

The range is .

Writing Absolute Value as a Piecewise Function

1. Determine the x-intercepts by setting the expression within the

absolute value equal to zero.

2. Use slope (linear function) or direction of opening (quadratic

function) to determine which parts of the graph are above or below the

x-axis.

3. Keep the parts that are positive (above x-axis) and indicate the

domain.

4. Reflect the negative parts in the x-axis, multiply the expression by -1

for this part and indicate the domain.

Be careful when assigning domain, it changes depending on which piece of the graph was below the

x-axis.

Note the linear expression would have a negative slope, examine how this changes the domain pieces.

Linear Expressions

Quadratic Expressions

Analyzing Absolute Value Functions Graphically

To analyze an absolute value graphically:

● first, graph the function.● then, identify the characteristics of the graph, such as x-

intercept, y-intercept, minimum values, domain and range

The domain of an absolute value function, y = |f(x)|, is the same as the domain of y = f(x).

The range of the absolute value function will be greater or equal to zero.

Solving an Absolute Value Equation Graphically

An absolute value equation includes the absolute value of an expression involving a variable. To solve an graphically:

● Graph the left side and the right side of the equation on the same set of axes.

● The point(s) of intersection are the solutions.

Solving an Absolute Value Equation Algebraically

Determine the zero of the function inside the abs.

Use sign analysis to determine which parts of the domain are

positive or negative.

To solve algebraically consider the two cases:

● Case 1- the expression inside the absolute value symbol is greater than or equal to zero.

● Case 2- the expression in the absolute value symbol is less than zero.

5. The roots in each case are the solutions.6. There may be extraneous roots that need to be identified

and rejected.7. Verify the solution by substituting into the original

equation. There may be no solution, one solution or two solutions if the

absolute value expression is a line.

There may be no solution, one, two or three solutions if the absolute value expression is quadratic.

Solving Linear Abs Equation Determine zero of the abs function. Determine domain pieces.

Case 1: positive y values Case 2: negative y values

Verify each solution in the original absolute value equation.

|x + 3| = - 2 does not have any solution, absolute value must be positive.

Solving Quadratic Abs Equations Determine the zeros of the abs function.

continued on next page

Case 1: positive y values Case 2: negative y values

Solutions are in the domain. Neither solution is in domain.

These solutions are extraneous.

Reciprocal Functions

A reciprocal function has the

form y =

1 ( )f x

where f(x) is a polynomial and

f(x) ≠ 0.

For any function f(x), the reciprocal function is . The reciprocal of y = x is y = 1/x.

To Graph a reciprocal Function: Plot invariant points. Invariant points are where the y values

are 1 or -1 x-intercepts become vertical asymptotes. The x-axis is a horizontal asymptote Take the reciprocal of the y values of the original function to

plot the reciprocal of the function.

Restrictions on the denominator of the reciprocal function.

The reciprocal function is not defined when the denominator is 0. These non-permissible values relate to the asymptotes of the

graph of the reciprocal function.

The non-permissible values for a reciprocal function (position of asymptotes) also come from the x-intercepts of the original graph.

Linear Reciprocal Graphs

Quadratic Reciprocal Graphs

Common Errors Description

No vertical bars around the value. A small error that sometimes seems to happen are students forgetting to put vertical bars around the value, which does not get

the absolute value of whatever the value is.

Not eliminating extraneous solutions when it comes to Absolute Value equations.

When verifying a solution, if the solution does not satisfy the equation, the solution is considered extraneous.

Not plotting the invariant points in the proper areas.

An error that may occur is plotting invariant points, on the wrong co-ordinate of the line.

Not plotting the Vertical or Horizontal asymptote in the proper

place.

If the vertical asymptote is not plotted in the proper area, the entire graph is wrong.

Not properly graphing Absolute Value functions.

Some students graph absolute value functions as if it was a linear function and forget that it is an absolute value graph; no part of the

graph will be in the negative portion of the co-ordinate plane.

Vocabulary Definition

Absolute Value Distance from zero on a number line.

Piecewise definition

Absolute Value Function

A function that involves the absolute value of a variable.

Piecewise Function A function composed of two or more separate functions or pieces, each with its own specific domain, that combine to define the overall function.

Invariant Point A point that remains unchanged when a transformation is applied to it.

Absolute Value Equation

An equation that includes the absolute value of an expression involving a variable.

Asymptote An imaginary line whose distance from a given curve approaches zero.

Vertical Asymptote For reciprocal functions, vertical asymptotes occur at the non-permissible values of the function, the x-intercepts of the original function graph.

Horizontal Asymptote

For our reciprocal functions, there will always be a horizontal asymptote at y = 0.