Example 1

Algebra II:

Use your graphing calculator to find all the real zeros of the function.

f(x) = x3 – 15x2 +3x + 5

Example 2

Algebra II:

Use your graphing calculator to find all the real zeros of the function on the given interval.

g(x) = x6 – 5x4 + 3x3 + 2x2 – x, [-2, 2]

Example 3

Algebra I or II:

Use the graphing calculator to find the solution of the system y = x and y = -2x + 15.

- Graph both, change window, find intersection

- Set equal to each other, set = 0, graph and find zeros.

- Show them both ways.

Example 4

PreCal:

Use the graphing calculator or find all the zeros on the interval 0,2

for

sin 5 4f x x .

Example 5

All Classes: Teach students how to store numbers and use Y1 or f1(x).

For f(x) = x2 – 8 and g(x) = 2x3 – 3x2 + 8x + 1, find the value of g(x) at the positive zero of

f(x).

Example 6

Determine analytically if each function is even, odd or neither by evaluating f(-x).

6 22 3f x x x 3 5f x x

21f x x x

Example 7

Graphically:

Complete each graph as indicated. f(x) is even f(x) is odd

Example 8

Numerically:

Complete each table as indicated. f(x) is even g(x) is odd h(x) is even

x f(x)

-7 5

-4 6

-1 -2

x g(x)

-8 -6

-3 0

-1 2

x h(x)

12 -7

5 -9

2 4

0 0

f x f x

Example 9

Verbally:

Determine whether each function is even or odd.

Justify your answer.

Example 10

Algebra I or II:

22 3 5

2

a b c

a b

x g(x)

-7 13

-5 8

-3 -2

0 0

3 -2

5 8

7 13

x f(x)

-3 -8

-2 -7

-1 -3

0 0

1 3

2 7

3 8

Example 11

Calculus:

1 1

2 22 2

2

14 4 2

2

4

x x x x

x

Example 12

Add problems like this to Algebra II or PreCal:

1 1

2 2

2

x x

x

Example 13

Easy: 2 1f x x and g x x

(a) Find f g (b) Find g f

Example 14

More Difficult: 2

3

1f x

x

and 1g x x

(a) Find f g (b) Find g f

Example 15

Even More Difficult: 1

2

4x xf x

x

and

12 24g x x

Find f g

Example 16

Easy question: for f(x) = x2, evaluate the difference quotient f x h f x

h

.

Example 17

Easy question: for f x x , evaluate the difference quotient 9

9

f x f

x

.

Example 18

More difficult: for 2

1f x

x , evaluate the difference quotient

f x h f x

h

.

Example 19

More difficult: for 1

f xx

, evaluate the difference quotient f x h f x

h

.

Example 20

3ln 1 6 4x

Example 21

Example 21 – A

Find the function y = f(x) that satisfies the equation with the given initial conditions.

1ln 1y C

x 2 0f

Example 21 – B

Graph each function and label as increasing at an increasing rate, increasing at a decreasing

rate, decreasing at a decreasing rate, or decreasing at an increasing rate.

Example 22

Proof of Pythagorean Theorem

Example 23

Proof of Area of a Trapezoid

Example 24

Proof/Explanation for Surface Area of a Cylinder

Example 25

Geometry: Introduce the integral as a new mathematical symbol meaning to find

the area between a function and the x-axis. Area under the x-axis is negative.

Find 0

3

f x dx

. Find 0

5

f x dx

. Find 4

0

f x dx .

Example 26

Find the area under the curve f(x) from -5 to 0.

Example 26.5

Graph the line that produces a cone of radius 4 and height of 3 when rotated about the y-axis.

Example 27

The function h (x) is defined as h (x) = f (g (x)) – 6. Refer to the table below to

answer questions #1-5.

1. Find h (1).

2. Find h (2).

3. Find h (3).

Example 28

Inverse Functions:

(Using above table)

4. The inverse of g (x) is g -1(x). Write the equation of the line that intersects

g -1(x) at x = 2 and has a slope of 1

5.

5. The inverse of f (x) is f -1(x). Write the equation of the line that intersects

f -1(x) at x = 10 and has a slope of 1

4 .

Example 29

Let f x be the function defined as 5 1f x x . If 1g x f x , find the following

values.

(a) 4g (b) 0g (c) 1g

Example 30

Algebra II: 3f x x h k new TEK. Find inflection point (h, k) and state concavity.

Example 31

Take the function 1f x x and transform it so that y = 3 becomes the x-axis.

Take the function 1f x x and transform it so that x = 10 becomes the y-axis.

Example 32

Write the area, A, of a square as a function of its perimeter, P.

Example 33

Algebra II or PreCal: An open box is to be made from a square piece of material 24 centimeters on a side by cutting equal squares from the corners and turning up the sides. Express the volume of the box as a function of x.

Example 34

Algebra II or PreCal

Use the IVT and The Fundamental Theorem of Algebra to determine the minimum

number of zeros that exist for each continuous function. Justify your answer.

f(x) is a cubic function with real coefficients.

x f(x)

-5 -3

-4 -1

-1 2

1 2

2 1

3 -2

6 -4

Example 35

Find the average rate of change of the function f(x) = x2 – 10 over the interval 2 4x .

Example 36

Given that 2 2 3x t t t (in feet), find the average velocity over the time interval

1 5t , where t is measured in seconds.

Example 37

Example 38

Example 39

a) Find the minimum value of the function 2 4f x x .

b) Where does the function 24g x x reach a maximum?