last night’s hw 6. no 8. yes 32a. 2 32b. 5 32c. √x+2 33a. -1/9 33b. undefined 33c....

Last Night’s HWLast Night’s HW

6. no6. no 8. yes8. yes 32a. 232a. 2 32b. 532b. 5 32c. 32c. √x+2√x+2 33a. -1/933a. -1/9 33b. undefined33b. undefined 33c. 1/(y^2 + 6y)33c. 1/(y^2 + 6y)

66. D: (- ∞, 0) U (0, 2) U (2, ∞)68. D: (-6, ∞)70. D: (- ∞, -3) U (3, ∞)

Essential QuestionEssential Question

How do you determine the relative max, How do you determine the relative max, min, intervals of increase, decrease, even min, intervals of increase, decrease, even and odd functions?and odd functions?

1.2 Graphs of Functions1.2 Graphs of Functions

Example 1: Given the graph shown, answer the Example 1: Given the graph shown, answer the following questions?following questions?

a)a) What is f(0) ?What is f(0) ?b)b) What is f(9)?What is f(9)?c)c) What is the domain?What is the domain?d)d) What is the range?What is the range?e)e) What are the x-intercepts?What are the x-intercepts?f)f) How often does the line y = -1 intercept the How often does the line y = -1 intercept the

graph?graph?

a. f(0)=-3

(-4, 2)

(-2, 0)

(0, -3)

(1, -2)

(3, 0)

(5, 4)

(7, 0)

(9, -2)

(10, -1)

b. f(9)=is undefinedc. D=[-4, 9) U (9, 10)

d. R=[-3, 4]e. (-2, 0), (3, 0), (7, 0)

f. 4 times

Increasing and Decreasing Functions

Decreasing Constant Increasing

Example 2: Determine the open intervals on which each function is increasing, Example 2: Determine the open intervals on which each function is increasing, decreasing, or constant.decreasing, or constant.

a. b.

c.

(-1, 2)

(1, -2)

f(x) = x3 - 3x

(0, 1) (2, 1)

23

201

01

)(

tt

t

tt

tf

Increasing: (-∞,∞)

Increasing:(-∞, -1) U (1,∞)

Decreasing:(-1, 1)

Increasing: (-∞, 0)Constant: (0, 2)Decreasing: (2, ∞)

Relative Max and Min ValuesRelative Max and Min ValuesRelativeMax

Relative Min

Example 3: Use a graphing utility to approximate the relative Example 3: Use a graphing utility to approximate the relative maximum/minimum of the functionsmaximum/minimum of the functions

Sol: Min (0.67, -3.33)

a) f(x) = 3xf(x) = 3x2 2 - 4x - 2- 4x - 2 b) f(x) = -x3+x

Sol: Min (-0.58, -0.38)Max (0.58, 0.38)

Khan Academy Khan Academy

Recognizing Odd and Even FunctionsRecognizing Odd and Even Functions

Connection between even and odd Connection between even and odd numbers and functionsnumbers and functions

Even and Odd Functions: GraphicallyEven and Odd Functions: Graphically

Even FunctionSymmetric to y-axis

Odd FunctionSymmetric to origin

Not a Function:Symmetric to x-axis

Even and Odd Functions: Algebraically Even and Odd Functions: Algebraically

xxfa )(. xxxgb 3)(. 1)(. 2 xxhc

Even Function: A function f is even if, for each x in the domain of f, f(-x) = f(x).

Odd Function: A function f is odd if, for each x in the domain of f, f(-x) = -f(x)

Example 7: Determine whether each function is even, odd or neither.

1)(. 3 xxfd

xxxf )(

)(

)()()(

3

3

3

xx

xx

xxxg

1

1)()(2

2

x

xxh

)1(

1

1)()(

3

3

3

x

x

xxf

EVEN

ODDEVEN

NEITHER

HomeworkHomework

Pg96Pg96 #1-6all, #13-18 all, 21,22, 48, 50#1-6all, #13-18 all, 21,22, 48, 50