1 - quadratic functions and factoring · web viewsolving quadratic equations using a variety of...

CHAPTER 1 – QUADRATIC FUNCTIONS AND FACTORING

Big IDEAS:1) Graphing and writing quadratic function in several forms2) Solving quadratic equations using a variety of methods3) Performing operations with square roots and complex numbers

Section: 1 – 1 Graph Quadratic FUNctions in Standard Form

Essential Question

How are the values of a, b, and c in the equation related to the graph of a quadratic function?

Warm Up:

Key Vocab:

Quadratic Function

A function that can be written in the form where

Parabola “U” shape of quadratic function

Vertex

Highest or lowest point on a parabola. Always has coordinates

Student Notes Algebra II Chapter 1 – Quadratic Functions and Factoring KEY Page #1

f(x)=x^2

x=0

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

x

y

VertexAxis of Symmetry

Parabola

Axis of Symmetry

Vertical line that divides a parabola in half and passes through its vertex.


Key Concept:

Properties of the Graph

If (the a value is positive), then the graph opens up.

If (the a value is negative), then the graph opens down.

If (the a value is positive), then the vertex is a minimum.

If (the a value is negative), then the vertex is a maximum.

If , then the graph is narrower than

.

If , then the graph is wider than

.

The axis of symmetry is AND the vertex has an x-coordinate , so

are the coordinates for the vertex.

The y-intercept is c, so the coordinates is always on the graph of the parabola.

Show:

Ex 1: Graph using an x/y chart. Compare this graph with the graph of .


Both graphs open up because both a values are positive.

Both graphs have the same axis of symmetry because both of the b values are zero.

is wider than because the a

f(x)=(x^2)/2

Series 1

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

x

y x y1 y2

-4 8 4-2 2 10 0 02 2 14 8 4

Ex 2: Graph using an x/y chart. Compare this graph with the graph of

.

Ex 3: Graph using the vertex and axis of symmetry.

f(x)=-x^2+6x-8

x=3

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

x

y

Find vertex:

Vertex: Find the axis of symmetry:

Find the y-intercept:

y-intercept:

Use the axis of symmetry to duplicate points:


Both graphs open up because both a values are positive.


is wider than because the a

opens down because the a value is

negative and opens up because the a value is positive


is narrower than because the a value is greater than one.

f(x)=(-2x^2)+4

Series 1

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

x

y x y1 y2

-2 -4 4-1 2 10 4 01 2 12 -4 4

Ex 4: Tell whether the function has a maximum or minimum. Find the maximum/minimum value.

The function has a maximum value because the a value is negative

Closure: How is the vertex of a parabola related to its axis of symmetry?

The x-coordinate of the vertex yields the equation for the axis of symmetry

Why is it useful to know the axis of symmetry when graphing a parabola?

The axis of symmetry can be used to mirror and thereby duplicate points on the graph, so you only have to find half as many points.

Can a quadratic function have a maximum AND a minimum value? Why or why not?

No. The maximum/minimum of a quadratic function occurs at the vertex. A parabola has only one vertex.


Section: 1 – 2 Graph Quadratic FUNctions in Vertex or Intercept Form

Essential Question

When graphing a quadratic function, what are the advantages to having it written in vertex form? …in intercept form?

Warm Up:

Key Vocab:

Vertex Form

Quadratic equation of the form

Vertex has coordinates Axis of symmetry is

Intercept Form

Quadratic equation of the form

x-intercepts are located at and

The vertex is located midway between the x-intercepts and can

be found using

Key Concept:

T h e F O I L M e t h o dUsed to multiply two binomials (two term expressions) together

First Outer Inner Last


f(x )=x^2-3

x=0

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

x

y

f(x)=(x-2)^2-3

x=2

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

x

y

Show:

Ex 1: Graph

f(x)=(1/2)(x-3)^2-5

x=3

Series 1

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

x

y

Ex 2: The Tacoma Narrows Bridge in Washington has two towers that each rise 307 feet above the roadway and are connected by suspension cables as shown. Each cable

can be modeled by the function , where x and y are measured in feet.

What are the minimum and maximum distances between the suspension cables and the roadway?

The minimum distance occurs at the vertex , so the minimum distance above the roadway is 27 feet

The maximum distance occurs at the maximum rise of the towers which is 307 feet.


Find vertex:

Find axis of symmetry:

Find x-intercept:

Ex 3: Graph

f(x)=-x*(x-4)

x=2

Series 1

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

x

y

Ex 4: If an object is propelled straight upward from Earth at an initial velocity of 80 feet per second, its height after t seconds is given by the function , where t is the time in seconds after the object is propelled and h is the object’s height in feet.

a. How many seconds after it is propelled will the object hit the ground?

The height of the ground is zero, so to find the time at height zero,

A is the start time, so is the end time

b. What is the object’s maximum height?

The maximum height occurs at the vertex which is midway between they-intercepts, so

The maximum height is 100 feet.

Ex 5: Write in standard form.

FOIL:


Find x-intercept:

Find vertex:

Find axis of symmetry:

Ex 6: Write in standard form.

FOIL:

Ex 7: A store sells about 150 Blu-Rays a week at a price of $20 each. The owner estimates that for each $1 decrease in price, about 25 more Blu-Rays will be sold each week. How can the owner maximize weekly revenue?

R e v e n u e = ( P r i c e ) ( Q u a n t i t y )

The maximum value occurs at the vertex, so

Closure: When you see an equation of a quadratic function in vertex form, how do you know

if the vertex is a maximum or a minimum point?

If the a value is positive, the function has a minimum. If the a value is negative, the function has a minimum.

When a quadratic function is in intercept form, how can you find the x-values of the intercepts AND the vertex?

The intercepts can be found by solving the equation (setting each quantity equal to zero)The vertex can be found by averaging the zeroes then plugging that value back into the

function.


Section: 1 – 3 Solve by Factoring

Essential Question

How can factoring be used to solve quadratic equations when a=1?

Warm Up:

Key Vocab:

Monomial A one term expression Examples:

Binomial A two term expression Examples:

Trinomial A three term expression Example:

Root Zero, x-intercept, solution

Key Concepts:

S p e c i a l P r o d u c t sType Formula Example

Difference of Squares

Perfect Square Trinomial

Z e r o P r o d u c t P r o p e r t yIf the product of two expressions is zero, then one or both of the expressions must be zero.

If A and B are expressions and , then

Example: If , then .


x

x

15 ft

10 ft

GARDEN

Show:

Ex 1: Factor the expression.

a. b.

PRIME

c.

d. e. f.

g. h. i.

Ex 2: What are the roots of the equation ?

A. -7, 6 C. -6, 7

B. -7, -6 D. 6,7

Ex 3: You have a rectangular vegetable garden in your backyard that measures 15 feet by 10 feet. You want to double the area of the garden by adding the same distance x to the length and width of the garden. Find the value of x and the new dimensions of the garden.


So, and the new dimensions

Ex 4: Find the zeroes of the function by rewriting the function in intercepts form.

a. b.

Closure: How can you recognize when a trinomial of the form is a perfect

square trinomial?

It must be of the form or , in other words, c must be a perfect square AND b must be twice c.

If an equation of the form with can be solved by factoring, what do you know about the signs of the roots?

One root must be positive, one root must be negative.

If an equation of the form with can be solved by factoring, what do you know about the signs of the roots?

Both roots must be positive or both roots must be negative.

If an equation of the form has only one root, what kind of trinomial is ?

It must be a perfect square trinomial.


So, and the new dimensions

Section: 1 – 4 Solve by Factoring

Essential Question

How do can factoring be used to solve quadratic equations when ?

Warm Up:

Key Vocab:

Greatest Common Factor

“GCF” Greatest value that divides evenly into each term of an expression

Show:


a. b. c.

d. e. f.


x

x

22 ft

15 ft

GARDEN


a. b. c.

Ex 3: Solve the equation.

a. b.

c. d.

Ex 4: You are designing a garden for the grounds of you high school. You want the garden to be made up of a rectangular flower bed surrounded by a border of uniform width to be covered with decorative stones. You have decided that the flower bed will be 22 feet by 15 feet, and your budget will allow for enough stone to cover 120 square feet. What should be the width of the border?

The border should be 1.5 feet.Student Notes Algebra II Chapter 1 – Quadratic Functions and Factoring KEY Page #14

Ex 5: An internet service provider sells high-speed internet service for $30 per month to 1500 customers. For each $1 increase in price, the number of customers will decrease by 25. How much should the company charge in order to maximize monthly revenue? What is the maximum monthly revenue?

R e v e n u e = ( P r i c e ) ( Q u a n t i t y )

The maximum value occurs at the vertex so,

price/month = 30 + 15 = $45

Maximum Revenue = $50,625

Closure After you have factored an expression, how can you check that it is factored

correctly?

FOIL to see if you get the original expression

What should ALWAYS be the first step to factoring?

Check for a GCF!


Radical Sign

Radicand

332x

Section: 1 – 5 Solve Quadratic Equations by Finding Square Roots

Essential Question How can you use square roots to solve a quadratic equation?

Warm Up:

Key Vocab:

Square Root

If , then

If the square of a number r is a number s , then a number r is a square root of a number s

Examples: two is the square root of four

four is the square root of sixteen

Radical An expression of the form

Radicand Number inside the radical sign

Conjugate

A pair of binomials whose middle signs are opposite.

Examples: and

and


Key Concepts:

P r o p e r t i e s o f R a d i c a l s:Name Formula Translation

Product PropertyThe square root of a product is the product of the square roots

You can break apart multiplication

Division Property

The square root of a quotient is the quotient of the square roots

You can break apart division

R a t i o n a l i z i n g t h e D e n o m i n a t o rRationalizing the denominator is a process of removing a radical from the denominator of a fraction.

Example: Step 1: Step 2: Step 3:

Form of the denominator: Multiply numerator AND denominator by:

multiply by the same square root

multiply by the conjugate

multiply by the conjugate

Show:

Ex 1: Simplify the expression.

a. b.

c. d.


Ex 2: Simplify the expression

a. b.

c. d.

Ex 3: Solve .

Ex 4: What are the solutions of the equation ?

A.-

C.


B. D.


Ex 5: When an object is dropped, its height h in feet above the ground after t seconds can

be modeled by the function where is the object’s initial height in feet. If you drop an object off the roof of an apartment building that is 240 feet tall, about how long will it take the object to hit the ground?

The object will take approximately 3.9 seconds to hit the ground

Closure: How does multiplying conjugates relate to the difference of squares formula?

They are the same concept. The difference of squares formula factors using conjugates.


Section: 1 – 6 Perform Operations with Complex Numbers

Essential Question How do you perform operations on complex numbers?

Warm Up:

Key Vocab:

Imaginary Unit, i

Represents the solution to the quadratic equation

Examples:

Complex Number A number , where a and b are real numbers and i is the

imaginary unit.

Standard Form of a Complex NumberImaginary Number A complex number , where

Pure Imaginary number

A complex number , where and

Examples: bi, ,

Complex Conjugates A pair of complex numbers whose middle signs are opposite.

The product of two complex conjugates will always be a real number.

Examples: and Student Notes Algebra II Chapter 1 – Quadratic Functions and Factoring KEY Page #21

and


Key Concepts:

P r o p e r t i e s o f i

O p e r a t I o n s o n C o m p l e x N u m b e r sOperation Method Translation

Addition

Real number + real number, imaginary number + imaginary number

Add like terms

Subtraction

Real number - real number, imaginary number - imaginary number

Subtract like terms

Multiplication FOIL

Division

Rationalize the denominator.

Multiply numerator AND denominator by the complex conjugate

Show:

Ex 1: Solve


Ex 2: Write the expression as a complex number in standard form.

a. b. c.

Ex 3:

Ex 4: Write the expression as a complex number in standard form.

a. b.


Ex 5: Write the quotient in standard form

Key Vocab continued:

Complex Plane

A coordinate plane in which each

coordinate represents a

complex number .

The horizontal axis (x-axis) is the real axis.

The vertical axis (y-axis) is the imaginary axis.

Absolute Value of a Complex Number

The absolute value of a complex

number , denoted , is a nonnegative real number

defined as .

Represents the distance between z and the origin in the complex plane.

Ex 6: Plot each complex number in the same complex plane.


imaginary

Ex 7: Find the absolute value the expression.

a. b.

Closure: If you FOIL two complex numbers of the form , which of the

products: F, O, I, L contribute to the real part of the product and which contribute to the imaginary part.

F and L yield the real parts of the product. O and I yield the imaginary parts of the product.

When plotting numbers on the complex plane, what type of numbers occur on the imaginary axis?

Only pure imaginary numbers occur on the imaginary axis.


real

A.

B.

C.

D.

AB

C

D

Section: 1 – 7 Complete the Square

Essential Question

How is the process of completing the square used to solve quadratic equations?

Warm Up:

Key Vocab:

Completing the Square

The process of adding a term to a quadratic expression of the form

to make it a perfect square trinomial.

Key Concept:

C o m p l e t i n g t h e S q u a r e


Steps: Example:

1. Transform the equation so that the constant term c is alone on the right side.

Step 1:

2. If a is not equal to 1, then divide both sides by a.

Step 2:

3. Add the square of half the coefficient of

the first degree term to both sides.

Step 3:

4. Factor the left side. Step 4:

5. Solve by taking the square root on both sides of the equation

Step 5:

Show:

Ex 1: Solve the equation by finding square roots.


Ex 2: Find the value of c that makes a perfect square trinomial, then write the expression as the square of a binomial.

Ex 3: Solve by completing the square.

Ex 4: Solve by completing the square.

Ex 5: The area of the triangle shown is 144 square units. What is the value of x?


Ex 6: Write in vertex form. Identify the vertex.

Vertex:

Ex 7: The height y (in feet) of a ball that was thrown up in the air from the roof of a

building after t seconds is given by the function . Find the maximum height of the ball.

The maximum height is 114 feet.

Closure: If is a perfect square trinomial and b is an odd integer, what do you

know about the value of c?The c value will be a fraction.


Section: 1 – 8 Use Quadratic Formula and the Discriminant

Essential Question How do you use the quadratic formula and the discriminant?

Warm Up:

Key Vocab:

Quadratic Formula

Let a, b, and c be real numbers such that , then

can be used to solve ANY quadratic equation in standard form

.

Discriminant Radicand of the quadratic formula:


Key Concept:

U s i n g t h e D i s c r i m i n a n tValue of the Discriminant: Number of Roots: Graphical Representation:

Discriminant is positive.Two Real Roots

Discriminant is zero.One Real Root

Discriminant is negative.

Two Imaginary Conjugate Roots

Show:

Ex 1: Use the quadratic formula to solve.

a. b. c.


Ex 2: Find the discriminant of the quadratic equation. Use the discriminant to determine the number of solutions for the equation.

a.

PositiveTwo Real Roots

b.

ZeroOne Real Root

c.

NegativeTwo Imaginary Conjugate Roots

Ex 3: For an object that is launched or thrown the equation , where

stands for an initial velocity and stands for the initial height. A basketball player passes the ball to a teammate. The ball leaves the player’s hand 5 feet above the ground and has an initial vertical velocity of 55 feet per second. The teammate catches the ball when it returns to a height of 5 feet. How long is the ball in the air?

The ball is in the air for approximately 3.4 seconds

Closure: Does the discriminant give the solution of a quadratic equation?

No, it tells how many and what type of roots.

How can you tell if a quadratic equation has one or two solutions?

If the discriminant is positive or negative, the equation has two solutions. If the discriminant is zero, the equation has one solution.

How are the solutions of a quadratic formula related if the discriminant is negative?


If the discriminant is negative, there will be two imaginary conjugate roots because there solution contains the square root of a negative number.


Section: 1 – 9 Graph and Solve Quadratic Inequalities

Essential Question How do you solve quadratic inequalities in one variable?

Warm Up:

Key Concept:

Graphing a Quadratic Inequality in Two Variables

1. Graph the parabola with the equation

Use a solid curve for

Use a dotted curve for < and >

2. Test a point inside the parabola to determine whether that point is in the solution of the inequality.

3. Shade

If the test point IS a solution, shade inside the parabola.

If the test point IS NOT a solution, shade outside the parabola.


Show:

Ex 1: Graph

Ex 2: A computer desk with a solid glass top can safely support a weight W (in pounds) provided , where x is the thickness of the desktop (in inches). Graph the inequality.

-0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

100

200

300

400

500

600

700

x

y


Ex 3: Graph the system of inequalities.

y>x^2-3

y<-2x^2+4x+2

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

x

y

Key Concept:

Graphing a Quadratic Inequality in One Variable

1. Solve the equivalent equation

2. Plot the solutions as critical values on the number line/coordinate plane.

Use closed circles for

Use open circles for < and >

3. Use the critical values to divide the number line/coordinate plane into partitions. Choose a test point from each partition.

4. Shade

If the test point IS a solution, shade interval.

If the test point IS NOT a solution, do not shade the interval.

Ex 4: Solve algebraically


Ex 5: Solve by graphing


f(x)=x^2/2+3x-2

f(x)=x^2/2+3x-2

f(x)=x^2/2+3x-2

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

x

y

Ex 6: Solve by using a table.

Closure:

What is the difference between solving a quadratic inequality in one variable and solving a quadratic inequality in two variables?

When solving a quadratic inequality in one variable, you are only looking for a range of x-values. You shade a number line. When solving a quadratic inequality in two variables, you are looking for a range of coordinates: x-values and y-values. You shade a coordinate plane.


x y-5 7-4 00 -82 03 7

1 - quadratic functions and factoring · web viewsolving quadratic equations using a variety of...

Documents