1. Review Topics (Chapter 0 & 1)
Exponents & Radical Expressions
Factoring
Quadratic Equations
Rational Expressions
Rational Equations & Clearing Fractions
Radical Equations
Solving Inequalities
Linear Graphing and Functions
Function Evaluation
Slope and Average Rate of Change
Difference Quotient

2. Review of Exponents 82 =8 • 8 = 64 24 = 2 • 2 • 2 • 2 = 16
82 =8 • 8 = = 2 • 2 • 2 • 2 = 16
x2 = x • x x4 = x • x • x • x
Base = x Base = x
Exponent = 2 Exponent = 4
Exponents of 1 Zero Exponents
Anything to the 1 power is itself Anything to the zero power = 1
51 = x1 = x (xy)1 = xy 50 = x0 = (xy)0 = 1
Negative Exponents
5-2 = 1/(52) = 1/ x-2 = 1/(x2) xy-3 = x/(y3) (xy)-3 = 1/(xy)3 = 1/(x3y3)
a-n = 1/an /a-n = an a-n/a-m = am/an

3. Raising Quotients to Powers
a n b an bn
a -n b
a-n
b-n bn an
b n a = = = =
Examples: = =
2x (2x) x3
y y y3 = =
2x (2x) y y3
y y y-3(2x) (2x) x3 = = = =

4. Product Rule am • an = a(m+n) x3 • x5 = xxx • xxxxx = x8
x4 y3 x-3 y6 = xxxx•yyy•yyyyyy = xy9
xxx
3x2 y4 x-5 • 7x = 3xxyyyy • 7x = 21x-2 y4 = 21y4
xxxxx x2

5. Quotient Rule am = a(m-n) an 43 = 4 • 4 • 4 = 41 = 4 43 = 64 = 8 = 4
43 = 4 • 4 • 4 = 41 = = = = 4 •
x5 = xxxxx = x x5 = x(5-2) = x3
x xx x2
15x2y3 = 15 xx yyy = 3y x2y3 = 3 • x -2 • y2 = 3y2
5x4y xxxx y x x4y x2
3a-2 b5 = 3 bbbbb bbb = b a-2 b5 = a(-2-4)b(5-(-3)) = a-6 b8 = b8
9a4b aaaa aa a a4b a6

6. Powers to Powers (am)n = amn (a2)3 a2 • a2 • a2 = aa aa aa = a6
(24)-2 = = = = 1/256
(24) • •
(x3)-2 = x – = x = x4
(x -5) x – x 6
(24)-2 = 2-8 = 1 = 1

7. Products to Powers (ab)n = anbn (6y)2 = 62y2 = 36y2
(2a2b-3)2 = 22a4b-6 = 4a = a
4(ab3) a3b a3b9b b15
What about this problem?
5.2 x = /3.8 x 109  x 109
3.8 x 105
Do you know how to do exponents on the calculator?

8. Square Roots & Cube Roots
A number b is a square root
of a number a if b2 = a
25 = 5 since 52 = 25
Notice that 25 breaks down into 5 • 5
So, 25 =  5 • 5
See a ‘group of 2’ -> bring it outside the
radical (square root sign).
Example: 200 = 2 • 100
= 2 • 10 • 10
= 10 2
A number b is a cube root
of a number a if b3 = a
8 = 2 since 23 = 8
Notice that 8 breaks down into
2 • 2 • So, 8 =  2 • 2 • 2
See a ‘group of 3’ –> bring it outside
the radical (the cube root sign)
Example: 200 = 2 • 100
= 2 • 10 • 10
= 2 • 5 • 2 • 5 • 2
= 2 • 2 • 2 • 5 • 5
= 2 25 3 3 3 3 3 3 3 3
Note: -25 is not a real number since no
number multiplied by itself will be negative
Note: -8 IS a real number (-2) since
-2 • -2 • -2 = -8 3

9. Nth Root ‘Sign’ Examples
Even radicals of positive numbers
Have 2 roots. The principal root
Is positive.  16
= 4 or -4
not a real number 
-16
Even radicals of negative numbers
Are not real numbers.  4
-16
not a real number
Odd radicals of negative numbers
Have 1 negative root.  5
-32
= -2  5 32
= 2
Odd radicals of positive numbers
Have 1 positive root.

10. Exponent Rules
(XY)m = xmym X Y m = Xm Ym

11. Examples to Work through

12. Some Rules for Simplifying Radical Expressions

13. Practice Problems

14. Operations on Radical Expressions
Addition and Subtraction (Combining LIKE Terms)
Multiplication and Division
Rationalizing the Denominator

15. Radical Operations with Numbers

16. Multiplying Radicals (FOIL works with Radicals Too!)

17. Rationalizing the Denominator
Remove all radicals from the denominator

18. Adding & Subtracting Polynomials
Combine Like Terms
(2x2 –3x +7) + (3x2 + 4x – 2) = 5x2 + x + 5
(5x2 –6x + 1) – (-5x2 + 3x – 5) = (5x2 –6x + 1) + (5x2 - 3x + 5)
= 10x2 – 9x + 6
Types of Polynomials
f(x) = 3 Degree 0 Constant Function
f(x) = 5x –3 Degree 1 Linear
f(x) = x2 –2x –1 Degree 2 Quadratic
f(x) = 3x3 + 2x2 – 6 Degree 3 Cubic

19. Multiplication of Polynomials
Step 1: Using the distributive property, multiply every term in the
1st polynomial by every term in the 2nd polynomial
Step 2: Combine Like Terms
Step 3: Place in Decreasing Order of Exponent
4x2 (2x3 + 10x2 – 2x – 5) = 8x5 + 40x4 –8x3 –20x2
(x + 5) (2x3 + 10x2 – 2x – 5) = x x3 – 2x2 – 5x
+ 10x x2 – 10x – 25
= 2x x3 + 48x2 –15x -25

20. Binomial Multiplication with FOIL
(2x + 3) (x - 7)
F. O. I. L.
(First) (Outside) (Inside) (Last)
(2x)(x) (2x)(-7) (3)(x) (3)(-7)
2x x x
2x x x
2x x

21. Division by a Monomial 3x2 + x 5x3 – 15x2 x 15x
4x2 + 8x – x2y + 10xy2
4x xy
15A2 – 8A A5 – 8A2 + 12
4A A

22. Review: Factoring Polynomials
To factor a number such as 10, find out
‘what times what’ = 10
10 = 5(2)
To factor a polynomial, follow a similar process.
Factor: 3x4 – 9x3 +12x2
3x2 (x2 – 3x + 4)
Another Example:
Factor 2x(x + 1) + 3 (x + 1)
(x + 1)(2x + 3)

23. Solving Polynomial Equations By Factoring
Zero Product Property : If AB = 0 then A = 0 or B = 0
Solve the Equation: 2x2 + x = 0
Step 1: Factor x (2x + 1) = 0
Step 2: Zero Product x = 0 or 2x + 1 = 0
Step 3: Solve for X x = 0 or x = - ½
Question: Why are there 2 values for x???

24. Factoring Trinomials
To factor a trinomial means to find 2 binomials whose product
gives you the trinomial back again.
Consider the expression:
x2 – 7x + 10
The factored form is:
(x – 5) (x – 2)
Using FOIL, you can multiply the 2 binomials and
see that the product gives you the original trinomial expression.
How to find the factors of a trinomial:
Step 1: Write down 2 parentheses pairs.
Step 2: Do the FIRSTS
Step3 : Do the SIGNS
Step4: Generate factor pairs for LASTS
Step5: Use trial and error and check with FOIL

25. Practice Factor: y2 + 7y –30 4. –15a2 –70a + 120
10x2 +3x – m4 + 6m3 –27m2
8k2 + 34k x2 + 10x + 25

26. Special Types of Factoring
Square Minus a Square
A2 – B2 = (A + B) (A – B)
Cube minus Cube and Cube plus a Cube
(A3 – B3) = (A – B) (A2 + AB + B2)
(A3 + B3) = (A + B) (A2 - AB + B2)
Perfect Squares
A2 + 2AB + B2 = (A + B)2
A2 – 2AB + B2 = (A – B)2

27. Quadratic Equations General Form of Quadratic Equation
ax2 + bx + c = 0
a, b, c are real numbers & a 0
A quadratic Equation: x2 – 7x + 10 = 0 a = _____ b = _____ c = ______
Methods & Tools for Solving Quadratic Equations
Factor
Apply zero product principle (If AB = 0 then A = 0 or B = 0)
Square root method
Completing the Square
Quadratic Formula 1 -7 10
Example1: Example 2:
x2 – 7x + 10 = 0 4x2 – 2x = 0
(x – 5) (x – 2) = 0 2x (2x –1) = 0
x – 5 = or x – 2 = 0 2x=0 or 2x-1=0
2x=1
x = or x = 2 x = 0 or x=1/2

28. Square Root Method
If u2 = d then u = d or u = - d. If u2 = d then u = + d
Solving a Quadratic Equation with the Square Root Method
Example 1: Example 2:
4x2 = (x – 2)2 = 6 4
x – 2 = +6
x2 =
x = +  x = 6
So, x = 5 or - 5 So, x = 2 + 6 or 2 - 6

29. Completing the Square 2 2 (Example 1)
If x2 + bx is a binomial then by adding b 2 which is the square of half 2
the coefficient of x, a perfect square trinomial results:
x2 + bx + b 2 = x + b 2
Solving a quadratic equation with ‘completing the square’ method.
Example: Step1: Isolate the Binomial
x2 - 6x + 2 = 0
Step 2: Find ½ the coefficient of x (-3 )
x2 - 6x = and square it (9) & add to both sides.
x2 - 6x =
(x – 3)2 = 7
x – 3 = + 7
x = (3 + 7 ) or (3 - 7 )
Note: If the coefficient of x2 is not 1 you
must divide by the coefficient of x2 before
completing the square. ex: 3x2 – 2x –4 = 0
(Must divide by 3 before taking ½ coefficient of x)
Step 3: Apply square root method

30. (Completing the Square – Example 2)
Step 1: Check the coefficient of the x2 term. If 1 goto step 2
If not 1, divide both sides by the coefficient of the x2 term.
Step 2: Calculate the value of : (b/2)2
[In this example: (2/2)2 = (1)2 = 1]
Step 3: Isolate the binomial by grouping the x2 and x term together, then add (b/2)2 to both sides of he equation.
Step 4: Factor & apply square root method
2x2 +4x – 1 = 0
2x2 +4x – 1 = (x + 1) (x + 1) = 3/2
(x + 1)2 = 3/2
x2 +2x – 1/2 = 0
(x2 +2x ) = ½ √(x + 1)2 = √3/2
(x2 +2x ) = 1/ x + 1 = +/- √6/2
x = √6/2 – 1 or - √6/2 - 1

31. Quadratic Formula
General Form of Quadratic Equation: ax2 + bx + c = 0
Quadratic Formula: x = -b + b2 – 4ac discriminant: b2 – 4ac
2a if 0, one real solution
if >0, two unequal real solutions
if <0, imaginary solutions
Solving a quadratic equation with the ‘Quadratic Formula’
2x2 – 6x + 1= 0 a = ______ b = ______ c = _______
x = - (-6) + (-6)2 – 4(2)(1)
2(2)
= 6 + 36 –8 4
= 6 +  =  = 2 (3 + 7 ) = (3 + 7 ) 2 -6 1

32. Solving Higher Degree Equations
x3 = 4x
x3 - 4x = 0
x (x2 – 4) = 0
x (x – 2)(x + 2) = 0
x = x – 2 = 0 x + 2 = 0
x = x = -2
2x3 + 2x2 - 12x = 0
2x (x2 + x – 6) = 0
2x (x + 3) (x – 2) = 0
2x = 0 or x + 3 = 0 or x – 2 = 0
x = or x = or x = 2

33. Solving By Grouping x3 – 5x2 – x + 5 = 0 (x3 – 5x2) + (-x + 5) = 0
x – 5 = or x - 1 = or x + 1 = 0
x = or x = or x = -1

34. Rational Expressions
Rational Expression – an expression in which a polynomial is divided by
another nonzero polynomial.
Examples of rational expressions x
x x – x – 5
Domain = {x | x  0} Domain = {x | x  5/2} Domain = {x | x  5}

35. Multiplication and Division of Rational Expressions
A • C = A 9x = 3
B • C B 3x x
5y – = 5 (y – 2) = = 1
10y (y – 2)
2z2 – 3z – = (2z + 3) (z – 3) = 2z + 3
z2 + 2z – (z + 5) (z – 3) z + 5
A2 – B = (A + B)(A – B) = (A – B)
A + B (A + B)

36. Negation/Multiplying by –1- = OR

37. Examples x3 – x x + 1 x – 1 x x2 – 25 x2 –10x + 25• 
x2 – x2 + 8x
x2 + 5x x2 –10x + 25 •
(x3 – x) (x + 1)
x(x – 1) = =
(x + 5) (x – 5) • 2x(x + 4)
(x + 4)(x + 1) • (x – 5) (x – 5) = =
x (x2 – 1)(x + 1)
x(x – 1)
2x (x + 5)
(x + 1)(x – 5) =
x (x + 1) (x – 1)(x + 1)
x(x – 1) = =
(x + 1)(x + 1) = (x + 1)2

38. Check Your Understanding
Simplify:
x2 –6x –7
x2 -1
Simplify:
x x2 + x - 6 
(x + 1) (x –7)
(x + 1) (x – 1)
x2 + x - 6
x – •
(x – 7)
(x – 1)
(x + 3) (x – 2)
x – •
(x + 3) 3

39. Addition of Rational Expressions
Adding rational expressions is like adding fractions
With LIKE denominators: =
x x = x - 1
x x x + 2
x (2 + x) (2 + x)
3x2 + 4x x2 + 4x (3x2 + 4x – 4) (3x -2)(x + 2) = = =
(3x – 2)

40. Adding with UN-Like Denominators
x2 – x + 3
(x + 3)(x – 3) (x + 3)
(x – 3)
(x + 3)(x – 3) (x + 3)(x – 3)
1 + 2(x – 3) x – x - 5
(x + 3) (x – 3) (x + 3) (x – 3) (x + 3) (x – 3)
+ 1 8
(3) (2) 7 = =

41. Subtraction of Rational Expressions
To subtract rational expressions:
Step 1: Get a Common Denominator
Step 2: Combine Fractions DISTRIBUTING the ‘negative sign’
BE CAREFUL!!
2x x + 1
x2 – x2 - 1
2x – (x + 1)
x2 -1
2x – x - 1
x2 -1 = = = 1
(x + 1) =
x – 1
(x + 1)(x –1)

42. Check Your Understanding
Simplify:
b b-1
2b b-2 -
b b-1
2(b – 2) b-2 -
b b+1
2(b – 2) b-2 + b
2(b – 2)
2(-b+1) + = -1 2
b –2b+2
2(b – 2)
-b + 2
2(b – 2) = =
-1(b – 2)
2(b – 2)

43. Complex Fractions Examples: 1 5 4 7 x x2 – 16 1 x - 4 1 x 2 x2 + 3 x 1
A complex fraction is a rational expression that contains
fractions in its numerator, denominator, or both.
Examples: 1 5 4 7 x
x2 – 16 1
x - 4 1 x 2 x2 + 3 x 1 x2 -
x + 2
3x - 1 x
x + 4
7/20

44. Rational Equations (2x – 1) (x - 2) (x + 1) 3x = 3 x + 1 = 3 6 = x
x = or x = 2
Careful! – What do
You notice about the answer?

45. Rational Equations Cont…
To solve a rational equation:
Step 1: Factor all polynomials
Step 2: Find the common denominator
Step 3: Multiply all terms by the common denominator
Step 4: Solve
(12x)
x x – 1 = 1
2x x =
6 (x + 1) -3(x – 1) = 4x
6x –3x = 4x
3x + 9 = 4x
-3x x
9 = x

46. Other Rational Equation Examples=
x – x x2 - 4
= 3
x x
(x + 2)(x – 2) =
x – x (x + 2) (x – 2)
4x = 3x2
3x x = 0
(3x + 2) (x – 2) = 0
3x + 2 = 0 or x – 2 = 0
3x = or x = 2
x = -2/ or x = 2
3(x + 2) (x – 2) =
3x x – =
8x – 4 = 12
8x = 16
x = 2

47. Check Your Understanding
Simplify: x
x2 – x2 – 1
x – x
x(x – 1) x2 – x(x + 1)
Solve x
2x – x + 1 x
x – x x2 + x - 2 4 1
x - 1 + -
= 1 = 5 -
2(x – 3)
x(x – 2) + = + -
-1/4 3
x(x – 1)(x + 1)
Try this one: =
F p q
Solve for p:

48. Radical Equations Continued…
Example1:
x + 26 – 11x = 4
26 – 11x = 4 - x
(26 – 11x)2 = (4 – x)2
26 – 11x = (4-x) (4-x)
x = 16 –4x –4x +x2
26 –11x = 16 –8x + x2
x x
0 = x2 + 3x -10
0 = (x - 2) (x + 5)
x – 2 = 0 or x + 5 = 0
x = x = -5
Example 2:
X2 = 64
Example 3:

49. Inequality Set & Interval Notation
Set Builder Notation
{1,5,6} { }  {6}
{x | x > -4} {x | x < 2} {x | -2 < x < 7}
x such that x such that x is less x such that x is greater
x is greater than –4 than or equal to 2 than –2 and less than
or equal to 7
Interval (-4, ) (-, 2] (-2, 7]
Notation
Graph -4 2 -2 7
Question: How would you write the set of all real numbers?
(-, ) or R

50. Inequality Example 6 Statement Reason 7x + 15 > 13x + 51 [Given]
x < 6 [Divide by –6, so must ‘flip’ the inequality sign
Set Notation: {x | x < 6}
Interval Notation: (-, 6]
Graph: 6

51. Graphs y axis x axis y - $$ in thousands x Yrs Quadrant II (-, +)
Origin (0, 0)
(6,0) 2
4 6 y
intercept x
intercept
(5,-2)
(-6,-3)
(0,-3)
Quadrant III (-, -)
Quadrant IV (+, -)
When distinct points are plotted as above
the graph is called a scatter plot – ‘points
that are scattered about’
Graphs represent trends in data.
For example:
x – number of years in business
y – thousands of dollars of profit
Equation : y = ½ x – 3
A point in the x/y coordinate plane is
described by an ordered pair of
coordinates (x, y)

52. Linear Equations The graph of a linear equation is a line.
A linear function is of the form y = mx + b, where m and b are constants.
y = 3x + 2
y = 3x + 5x
y = -2x –3
y = (2/3)x -1
y = 4
6x + 3y = 12 y x
x y=3x x y=2/3x –1
All of these equations are linear.
Three of them are graphed above.

53. X and Y intercepts -3 6 y Equation: y = ½ x – 3 (6,0) x y intercept x
(0,-3)
The y intercept happens where y is something & x = 0: (0, ____)
Let x = 0 and solve for y: y = ½ (0) – 3 = -3
The x intercept happens where x is something & y = 0: (____, 0)
Let y = 0 and solve for x: 0 = ½ x – 3 => 3 = ½ x => x = 6 -3 6

54. Slope Slope is the ratio of RISE (How High) y2 – y1 y (Change in y)
RUN (How Far) x2 – x x (Change in x) =
Slope = 5 – 2 = 3
1 - 0
Slope = 1 – (-1) = 2
3 –
y = mx + b
m = slope
b = y intercept y x
Things to know:
Find slope from graph
Find a point using slope
Find slope using 2 points
Understand slope between
2 points is always the same
on the same line
x y=3x x y=2/3x –1

55. The Possibilities for a Line’s Slope (m)
Positive Slope x y
m > 0
Line rises from left to right.
Negative Slope x y
m < 0
Line falls from left to right.
Zero Slope x y
m = 0
Line is horizontal.
m is undefined
Undefined Slope x y
Line is vertical.
Example:
y = ½ x + 2
Example:
y = -½ x + 1
Example:
y = 2
Example:
x = 3
Question: If 2 lines are parallel do you know anything about their slopes?
Things to know:
Identify the type of slope given a graph.
Given a slope, understand what the graph would look like and draw it.
Find the equation of a horizontal or vertical line given a graph.
Graph a horizontal or vertical line given an equation
Estimate the point of the y-intercept or x-intercept from a graph.

56. Linear Equation Forms (2 Vars)
Standard Form Ax + By = C A, B, C are real numbers.
A & B are not both 0.
Example: 6x y = 12
Things to know:
Graph using x/y chart
Know this makes a line graph.
Slope Intercept Form y = mx + b m is the slope
b is the y intercept
Example: y = - ½ x - 2
Things to know:
Find Slope & y-intercept
Graph using slope & y-intercept
Application meaning of of slope & intercepts
Point Slope Form y – y1 = m(x – x1)
Example: Write the linear equation through point P(-1, 4) with slope 3
y – y1 = m(x – x1)
y – 4 = 3(x - - 1)
y – 4 = 3(x + 1)
Things to know:
Change from point slope to/from other forms.
Find the x or y-intercept of any linear equation

57. Practice Problems
Find the slope of a line passing through (-1, 2) and (3, 8)
Graph the line passing through (1, 2) with slope of - ½
Is the point (2, -1) on the line specified by: y = -2(x-1) + 3 ?
Find the equation of a line with slope = 4 through the point (-1,5)
Find the equation of a line passing through the points (-2, 1) and (3, 7)
8. Graph (using an x/y chart – plotting points) and find intercepts of
any equation such as: y = 2x + 5 or y = x2 – 4

58. A Rational Function Graph
y = 1 x
x y /2
-1/
Undefined ½ 1
Intercepts:
No intercepts exist
If y = 0, there is no solution for x.
If x = 0, y is undefined
The line x = 0 is called a vertical asymptote.
The line y = 0 is called a horizontal asymptote.

59. Functions and Graphs Thinking Exercise: Draw a ‘line’ in the x/y axes.
Year
1997
1998
independent variable (x)
1999
2000
$3111
$3247
$3356
$3510
dependent variable (y)
Cost
The cost depends on the year.
The table above establishes a relation between the year and the cost of tuition
at a public college. For each year there is a cost, forming a set of ordered pairs.
A relation is a set of ordered pairs (x, y). The relation above can be written
as 4 ordered pairs as follows:
S = {(1997, 3111), (1998, 3247), (1999, 3356), (2000, 3510)}
x y x y x y x y
Domain – the set of all x-values. D = {1977, 1998, 1999, 2000}
Range – the set of all y-values. R = {3111, 3247, 3356, 3510}
Year(x) Cost(y)
Thinking Exercise: Draw a ‘line’ in the x/y axes.
What is the Domain & Range?

60. Functions & Linear Data Modeling
Input x
Functions & Linear Data Modeling
y – Profit in thousands of $$
(Dependent Var)
x - Years in business
(Independent Var)
Function f
(6,0)
Output
y=f(x) y
intercept x
intercept
(0,-3)
Equation: y = ½ x – 3
Function: f(x) = ½ x – 3
x y = f(x)
f(0) = ½(0)-3=-3
f(2) = ½(2)-3=-2
f(6) = ½(6)-3=0
f(8) = ½(8)-3=1
A function has exactly one output value (y)
for each valid input (x).
Use the vertical line test to see if an equation is
a function.
If it touches 1 point at a time then FUNCTION
If it touches more than 1 point at a time then
NOT A FUNCTION.

61. How to Determine if an equation is a function
Graphically: Use the vertical line test
Symbolically/Algebraically: Solve for y to see if there is only 1 y-value.
Example 1: x2 + y = 4
y = 4 – x2
For every value of x there
Is exactly 1 value for y, so
This equation IS A FUNCTION.
Example 2: x2 + y2 = 4
y2 = 4 – x2
y = 4 – x2 or y = – x2
For every value of x there
are 2 possible values for y, so
This equation IS NOT A
FUNCTION.

62. Are these graphs functions?
Use the vertical line test to tell if the following are functions:
y = x2
Y-axis
Symmetry
x = y2
X-axis
Symmetry
y = x3
Origin
Symmetry

63. More on Evaluation of Functions
f(x) = x2 + 3x + 5
Evaluate: f(2)
f(2) = (2)2 + 3(2) + 5
f(2) =
f(2) = 15
Evaluate: f(x + 3)
f(x + 3) = (x + 3)2 + 3 (x + 3) + 5
f(x + 3) = (x + 3)(x + 3) + 3x
f(x + 3) = (x2 + 3x + 3x + 9) + 3x + 14
f(x + 3) = (x2 + 6x + 9) + 3x + 14
f(x + 3) = x2 + 9x + 23
Evaluate: f(-x)
f (-x) = ( -x)2 + 3( -x) + 5
f (-x) = x2 - 3x + 5

64. More on Domain of Functions
A function’s domain is the largest set of real numbers for which the value f(x)
is a real number. So, a function’s domain is the set of all real numbers
MINUS the following conditions:
specific conditions/restrictions placed on the function
Bounds relating to real-life data modeling
(Example: y = 7x, where y is dog years and x is dog’s age)
values that cause division by zero
values that result in an even root of a negative number
What is the domain the following functions:
f(x) = 6x 2. g(x) = x h(x) = 2x + 1
x2 – 9

65. Slope & Average Rate of Change
y - $$ in thousands x
Yrs
y = x2 - 4x + 4
y = ½ x – 3
(6,0)
(0,-3)
Non-linear equations do not have a constant rate of change. But you can
Find the average rate of change from
x1 to x2 along a secant to the graph.
f(x2) – f(x1)
x2 – x1
The slope of a line may be
interpreted as the rate of change.
The rate of change for a line is constant (the same for any 2 points)
y2 – y1
x2 – x1

66. Definition of a Difference Quotient
The average rate of change for f(x) is called the “difference quotient”
and is defined below. (This is an important concept in calculus – it
becomes the mathematical definition of the derivative you will learn
About this semester.
Example: Find the difference quotient for : f(x) = 2x2 -3
f(x + h) = 2(x + h ) = x2 + 4xh + 2h2 -3 – 2x2 + 3 h
= 2(x + h)(x + h) = xh + 2h2
= 2(x2 + 2xh + h2 ) h
= 2x2 + 4xh + 2h2 -3
= x + 2h
So, the difference quotient is:
2x2 + 4xh + 2h2 -3 – (2x2 -3)

67. Increasing, Decreasing, and Constant Functions
A function is increasing on an interval if for any x1, and x2 in the interval, where x1 < x2, then f (x1) < f (x2).
A function is decreasing on an interval if for any x1, and x2 in the interval, where x1 < x2, then f (x1) > f (x2).
A function is constant on an interval if for any x1, and x2 in the interval, where x1 < x2, then f (x1) = f (x2).
Constant
f (x1) = f (x2)
(x1, f (x1))
(x2, f (x2))
Increasing
f (x1) < f (x2)
Decreasing
f (x1) > f (x2)

68. More Examples a. b. Observations Decreasing on the interval (-oo, 0)-5 -4 -3 -2 -1 1 2 3 4 5 a. b.
Observations
Decreasing on the interval (-oo, 0)
Increasing on the interval (0, 2)
Decreasing on the interval (2, oo).
Observations
a. Two pieces (a piecewise function)
b. Constant on the interval (-oo, 0).
c. Increasing on the interval (0, oo).
Challenge Yourself: What might be
the definition of the piecewise function
for this graph? (You will learn about these
Later. Can you guess what it might be?)

69. Relative (local) Min & Max
f(x) = sin (x)
x y /
3/ 2
The point at which a function changes its increasing or decreasing
behavior is called a relative minimum or relative maximum. y 2
(90, f(90)) f(90), or 1, is a local max 1 x
180
360 90
270 -1
(270, f(270))
f(270), or -1, is a local min -2
A function value f(a) is a relative maximum of f if there exists an open interval about a such that f(a) > f(x) for all x in the open interval.
A function value f(b) is a relative minimum of f if there exists an open interval about b such that f(b) < f(x) for all x in the open interval.

70. Library of Functions/Common Graphs
y = c x
y = x x
y = x2 x
y = x3 x
y = x x
y = |x| x
y = 1/x
y = x1/3 x

71. Piecewise Functions
A function that is defined by two (or more) equations over
a specified domain is called a piecewise function.
f(x) = x if x < 0
5x if x>=0
f(-5) = (-5) = = 28
f(6) = 5(6) + 3 = 33
See Page 247 for more examples

72. Sum, Difference, Product, and Quotient of Functions
Let f and g be two functions. The sum of f + g, the difference f – g,
the product fg, and the quotient f /g are functions whose domains are the set of
all real numbers common to the domains of f and g, defined as follows:
Sum: (f + g)(x) = f (x)+g(x)
Difference: (f – g)(x) = f (x) – g(x)
Product: (f • g)(x) = f (x) • g(x)
Quotient: (f / g)(x) = f (x)/g(x), provided g(x) does not equal 0
Example: Let f(x) = 2x+1 and g(x) = x2-2.
f+g = 2x+1 + x2-2 = x2+2x-1
f-g = (2x+1) - (x2-2)= -x2+2x+3
fg = (2x+1)(x2-2) = 2x3+x2-4x-2
f/g = (2x+1)/(x2-2)

73. Adding & Subtracting Functions
If f(x) and g(x) are functions, then:
(f + g)(x) = f(x) + g(x)
(f – g)(x) = f(x) – g(x)
Examples: f(x) = 2x and g(x) = -3x – 7
Method Method1
(f + g)(4) = 2(4) (4) – (f – g)(6) = 2(6) + 1 – [-3(6) – 7]
= – = [-18 – 7]
= = [-25]
= =
Method Method2 =
(f + g)(4) = 2x x – (f - g)(6) = 2x [-3x – 7]
= x – = 2x x + 7
= – = 5x + 8
= = 5(6) + 8 = =
Adding/subtracting also extends to non-linear
functions you will see in a subsequent chapter.

74. Exponential Functions
– any function whose equation contains a variable in the exponent.
[measures rapid increase or decrease (Example: epidemic growth)]
f(x) = bx
f – exponential function b - constant base (b > 0, b  1) x = any real number -- domain is (-∞ , ∞)
f(x) = 2x g(x) = 10x h(x) = 3x+1
Shift up c units
f(x) = bx + c
Shift down c units
f(x) = bx – c
Shift left c units
f(x) = bx+c
Shift right c units
f(x) = bx-c
f (x) = 3x
g(x) = 3x+1
(0, 1)
(-1, 1) 1 2 3 4 5 6 -5 -4 -3 -2 -1
Graphing Exponential Functions:
32+1 = 33 = 27
32 = 9 2
31+1 = 32 = 9
31 = 3 1
30+1 = 31 = 3
30 = 1
3-1+1 = 30 = 1
3-1 = 1/3 -1
3-2+1 = 3-1 = 1/3
3-2 = 1/9 -2
g(x) = 3x+1
f (x) = 3x x

75. The Natural Base e
An irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. This irrational number is approximately equal to More accurately,
The number e is called the natural base. The function f (x) = ex is called the natural exponential function.
f (x) = ex
f (x) = 2x
f (x) = 3x
(0, 1)
(1, 2) 1 2 3 4
(1, e)
(1, 3) -1

76. Logarithmic Functions
A logarithm is an exponent such that for b > 0, b  1 and x > 0
y = logb x if and only if by = x
Logarithmic equations Corresponding exponential forms
2 = log5 x 1) 52 = x
3 = logb ) b3 = 64
log3 7 = y 3) 3y = 7
y = loge 9 4) ey = 9
log25 5 = 1/2 because 251/2 = 5.
25 to what power is 5?
log25 5
log3 9 = 2 because 32 = 9.
3 to what power is 9?
log3 9
log2 16 = 4 because 24 = 16.
2 to what power is 16?
log2 16
Logarithmic Expression Evaluated
Question Needed for Evaluation
Evaluate the
Logarithmic Expression

77. Logarithmic Properties
Logb b = 1 1 is the exponent to which b must be raised to obtain b. (b1 = b).
Logb 1 = 0 0 is the exponent to which b must be raised to obtain 1. (b0 = 1).
logb bx = x The logarithm with base b of b raised to a power equals that power.
b logb x = x b raised to the logarithm with base b of a number equals that number.
Graphs of f (x) = 2x and g(x) = log2 x [Logarithm is the inverse of the exponential function] 4 2 8 1
1/2
1/4
f (x) = 2x 3 -1 -2 x 2 4 3 1 -1 -2
g(x) = log2 x 8
1/2
1/4 x
Reverse coordinates.
Properties of f(x) = logb x
Domain = (0, ∞)
Range = (-∞, +∞)
X intercept = 1 ; No y-intercept
Vertical asymptote on y-axis
Decreasing on 0<b<1; increasing if b>1
Contains points: (1, 0), (b, 1), (1/b, -1)
Graph is smooth and continuous -2 -1 6 2 3 4 5
f (x) = 2x
f (x) = log2 x
y = x

78. Common Logs and Natural Logs
A logarithm with a base of 10 is a ‘common log’
log = ______ because 103 = 1000
If a log is written with no base it is assumed to be 10.
log 1000 = log = 3 3
A logarithm with a base of e is a ‘natural log’
loge 1 = ______ because e0 = 1
If a log is written as ‘ln’ instead of ‘log’ it is a natural log
ln 1 = loge 1 = 0

79. Properties & Rules of Logarithms
Basic Properties
Logb b = 1 1 is the exponent to which b must be raised to obtain b. (b1 = b).
Logb 1 = 0 0 is the exponent to which b must be raised to obtain 1. (b0 = 1).
Inverse Properties
logb bx = x The logarithm with base b of b raised to a power equals that power.
b logb x = x b raised to the logarithm with base b of a number equals that number.
For M>0 and N > 0
Product Rule
logb(MN) = logb M + logb N
Quotient Rule
logb M = logb M - logb N N
Power Rule p
logb M = p logb M

80. Logarithmic Property Practice
Quotient Rule
Product Rule
logb M = logb M - logb N N
logb(MN) = logb M + logb N
log3 (27 • 81) =
2) log (100x) =
3) Ln (7x) =
1) log x
2) Ln e5 11 = =
Power Rule
log5 74
Log (4x)5
Ln x =
Ln x =
logb M = p logb M p = =

81. Expanding Logarithms logb(MN) = logb M + logb N
logb M = logb M - logb N N
logb M = p logb M p
1) Logb (x2 y )
3) log5 x
25y3 4)
log2 5x2 3
2) log6 3 x
36y4

82. Condensing Logarithms
logb(MN) = logb M + logb N
logb M = logb M - logb N N
logb M = p logb M p
Note: Logarithm coefficients
Must be 1 to condense.
(Use power rule 1st)
log4 2 + log ) 2 ln x + ln (x + 1)
Log 25 + log 4 5) 2 log (x – 3) – log x
Log (7x + 6) – log x 6) ¼ logb x – 2 logb 5 – 10 logb y

83. The Change-of-Base Property
Example: Evaluate log3 7
Most calculators only use:
Common Log [LOG] (base 10)
Natural Log [LN] (base e)
It is necessary to use the change
Of base property to convert to
A base the calculator can use.