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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?
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