Semester Final will cover:• Quadratics• Functions• Exponents• Logarithms

In class today:• Work on review problems• Ask your neighbor if you are confused• I will be around to answer questions

Skip 4d on the review packet - or anything else that you know we haven't covered yet.

Standard form: ax2 + bx + c = 0 -16t2 - 16t + 96 = 0

Factored form: a(x - p)(x - q) = 0 4(x - 12)(x + 5) = 0

Vertex Form: a(x - h)2 + k = 0 4(x - 12)2 + 3 = 0

Disguised Form: 3e2x - 4ex + 7 = 0

To Solve:> Factor> Graph> Use Square roots or Complete the square> Use a formula

A quadratic equation can have two solutions!

FactoringUnFOIL from x2 - 6x + 5 = 0 to (x - 5)(x - 1) = 0

Know and use Special Patterns (a + b)2 = a2 + 2ab+ b2

(a - b)2 = a2 - 2ab + b2

a2 - b2 = (a + b)(a - b)

Factoring to solve quadratic equations

1. Start by rearranging to standard form2. Remember that x = 0 might be a solution!3. Factor out any GCF

> Careful with dividing by x since x = 0 may be a solution4. Consider removing fractions by multiplying through by a common denominator5. Look for special patterns

> Square of a sum or difference?> Difference of two squares?

6. Look at the discriminant - is it worth trying?7. If a ≠ 1, consider guess & check or British method

Use Square RootsLook for a perfect square - don't expand if there is one!

(x - 3)2 - 4 = 0 3(t + 5)2 + 4 = 13

Complete the Square (so you can use square roots!)Add (b/2)2 to both sides to create a perfect square

x2 - 6x - 2 = 0x2 - 6x = 2

x2 - 6x + 9 = 2 + 9(x - 3)2 = 11x - 3 = ±√11x = 3 ± √11

Use Quadratic Formula

If b2 - 4ac > 0, the radical is real. There are two real solutions.If b2 - 4ac < 0, the radical is imaginary. There are two complex solutions.

If b2 - 4ac = 0, the radical is zero. There is one real solution at .

If b2 - 4ac is a perfect square, you can factor the quadratic into integers!If not, you can't!

Also known as Δ. Use it when a problem involves the number of roots.

Know the important features of a quadratic functionKnow the relationship between these features and the different forms of a quadratic equation:

y  =  a(x  -­‐  p)(x  -­‐  q)Factored  Form

y  =  a(x  -­‐  h)2  +  kVertex  Form

y  =  ax2  +  bx  +  cStandard  form

Up or down?How "sharp"?

Up or down?How "sharp"?

y-coordx-coord

Up or down?How "sharp"?

Positive Definite and Negative Definite

A quadratic is "Positive definite" if all of it's values are positive (not zero!).Happens when: a > 0 (opens up) and discriminant is negative (no roots)

A quadratic is "Negative definite" if all of it's values are negative (not zero!).Happens when: a < 0 (opens down) and discriminant is negative (no roots)

Really just about understanding the relationships above. Pick the form that's easiest.

Set the functions equal, solve the resulting equation

aka "Crosses"2 Solutions

aka "Tangent to"1 Solution No Solutions

Example: y = 2x + k is tangent to y = 2x2 - 3x + 4. Find k.

• Read the problem carefully. Pay attention to detail and, in particular, what the question is asking for.

• Sketch a diagram, make a table or plot a graph.• Define your variables carefully and clearly - write them down (including units of

measure)• Translate the problem into an equation that makes logical sense.• Do not try to write x = <blah> . When you do, you are essentially trying to solve the

problem in your head.• Check that the quantities in your equation are of the same units. You can only add and

subtract like units.• Use your algebra to solve the equation.• After you solve the equation, be sure to answer the question. It is often not the same as

the solution.

• The vertex is the extreme point. Understand the meaning of it's x and y coordinates.• The x coordinate is at -b/2a if the equation is in standard form.• The sign of the leading coefficient will tell you whether the function has a minimum or a

maximum.• A function sometimes needs to be manipulated before looking for the vertex.• A graphing calculator can be very helpful if it's allowed. See [2ND][CALC][MINIMUM] or

[MAXIMUM]

Changing forms:

Standard Form Vertex FormFactored FormFOIL

Factor

Expand

Complete the Square

Rewrite in vertex form: f(x) = 2x2 + 16x - 9

f(x) = 2(x2 + 8x ) - 9f(x) = 2(x2 + 8x + 16) - 9f(x) = 2(x + 4)2 - 41

+16 -16- 32

A function is a relation that has exactly one output for every input.

Vertical Line Test: If a vertical line passes through the graph of a relation exactly once for all x in its domain, then the relation is a function in that domain.

y = 100 + x f(x) = 100 + x

Domain: The set of all possible values of the input (x) to a relation.Range: The set of all possible values of the output (y) of a relation.

f(x) = 3x + 7

fx f(x)

Times 3 + 7x (g ○ f)(x) = (3x + 7)2 + 3

gg(f(x)) = (g ○ f)(x)

Square&

add 3

The composite of two functions is created by using the output of one function as the input to the other function.

Properties of Composite Functions

(f ○ g)(x) is not the same as (g ○ f)(x) in general

The range of the first function in a composition is the domain of the second.

Rational Functions

are functions that can be written as a ratio of two polynomials.

That is where p and q are polynomials.

Rational Functions

GraphicallyGraph the reflection over y = x

AlgebraicallySwitch the x and y, then solve for y.

In mathematical notation:f(f -1(x)) = x and also f -1(f(x)) = x or

The range of f defines the domain of f-1The range of f-1 defines the domain of f

The domain of f-1 is the range of fThe domain of f is the range of f-1

Properties of Exponents

Let a and b be real numbers and m and n be integers. Then:

Rational Exponents

The familiar special patternsThey work for exponents!

4x + 2x - 20 = (2x)2 + 2x - 20 = (2x + 5)( 2x - 4)

If you can rewrite an exponential equation to have powers of the same base on either side, you can equate the exponents to solve the equation.

Equating Exponents

(1,0)(-1, 1/b)

(1, b)

Plot three points for the "mother" function.Stretch then shift (horizontally & vertically)

Verticalstretch

Verticalshift

Horizontalshift

Exponential Growth

The amount after t years of something growing at a rate of r % per year is given by:

A(t) = A0 (1 + r)t where A0 is the initial amount present. r is called the growth rate and is written as a decimal. The value 1 + r is called the growth factor. This idea can be extended for any time period as long as t and r are given in the same units of time (years, months, seconds, etc.)

Irrational, shows up in nature a lot.

Definition of Base 10 Logarithm

If 10x = a then x = log10 a

...or... log10a is the exponent to which you raise 10 to get x.

b

Definition of Logarithm

If bx = y then y = logb x

...or... logb x is the exponent to which you raise b to get x.

The domain of any log function must be restricted to positive arguments!

Solving Equations with Logartihms

You can take the logarithm of both sides of an equation and use the rules of logarithms to rewrite

an equation in logarithmic form.

Just like any other base, but eeeeeeasier, naturally

Solving equations with exponents when the bases are different!

1) Simplify both sides as much as possible first.2) Take the log of both sides, using an appropriate base depending on the situation.3) Manipulate the equation with log rules until you can isolate the variable.4) Simplify the result, using a calculator with proper rounding or as an exact answer.

Change of Base Formula for Logarithms

Use whatever base is convenient for c.

Properties of Graphs of Logarithms

The graph of y = logb(x) has the following properties for any base b:

The curve is a reflection of a corresponding exponential across the line y = x (they are inverses)• The domain is x > 0 (we can only find logs of positive numbers)• The graph has a vertical asymptote at x = 0 (the y-axis)• The domain of logb(g(x)) is given by the values of x that make g(x) > 0

More members of the family...

Summary of Graphs of Logarithmic CurvesFor f(x) = a logb(x - c) + da stretches the curve vertically

> |a| > 0 stretches the curve> 0 < |a| < 1 compresses the curve> a < 0 reflects over the x-axis (flips vertically)

• b increases or decreases the rate of change (steepness) of the curve> b > 1 curve is increasing> 0 < b < 1 curve is decreasing> b < 0 is for a future math class

• c shifts the curve horizontally> c > 0 shifts the curve to the right> c < 0 shifts the curve to the left

• d shifts the curve vertically> d > 0 shifts the curve up> d < 0 shifts the curve down

Same as the exponential case, but use logs when you are looking for the time involved (the exponent).Common applications: • Decay and half life.• Time for an investment to double.

Your final will be on Tuesday, December 17 in the second slot (10:20 am - 11:50 am)It will be in Ms. Lincoln's room - A203 (with SL2)

Guidelines:• Arrive and be ready to take the exam five minutes before the scheduled time. Students who

arrive late for the exam will have that much less time to finish, Five minutes late means 85 minutes for the exam, not 90 minutes. The exam will be closed after 15 minutes. Any student who arrives 15 minutes or later into the exam will not be allowed to take the exam, and will have to arrange a make-up exam- no later than Friday, December 20.

• No cell phones. Use of a cell phone during the exam will result in an F. No electronics of any kind.

• Use the bathroom before the test.• All students will remain in the exam area for the entire time. Students should bring a book to

read or a journal, etc.- nothing that makes any noise.• Cheating. Just Don't Do It. Consequences: F for the final exam, automatic decrease of 1-2

letter grades in the course.

Bring:• Your calculator - if you don't have one, check one out from me in advance• Scratch paper - graph paper preferably• Pencils with erasers• A copy of the SL formula sheet with nothing else written on it.• A confident, rested, focused attitude!

Hints:• Read through the exam first. You may very well not finish. Don't fret, collect all the points

you can on a problem, then move on. Come back to the more difficult problems if you have time.

• Just because you don't get one part of a problem does not mean you can't do subsequent parts. Look for information that is given in one part - use it in subsequent parts.

• Please show me your thinking so I can give you all the points you deserve.• Give exact answers or round to 3 significant figures (2 decimals for money only)• On problems that allow a calculator, expect to use it! You may not be able to solve the

problem without it - certainly not quickly.

You can do well!

Study Advice:• The exam will be about 50% on logs and exponents, the rest on previous ideas from the

semester.• Do questionbank problems from the review set.• Look at the markscheme to be sure you are correct and to understand the scoring.• Look through previous questionbanks and exams.• Create a formula sheet that you would like to have. Then look at the actual formula sheet

(here). Do some problems using the formulas that are not on the actual sheet to solidify your understanding of them.

• Relax and get a good night's sleep and have a good breakfast. That will be worth a lot more than another couple hours at midnight.