Solutions
Name________________________
Date __________
Algebra III: 5.4 Exponential Functions
Bell Work Write recursive equations for explicit equations that each represent a sequence.
𝑡(0) = 4
𝑡(0) = 5
1 𝑥
1
1. 𝑡(𝑛) = −2𝑥 + 5
2. 𝑡(𝑛) = 4 �2�
𝑡(𝑛 + 1) = 𝑡(𝑛) − 2
𝑡(𝑛 + 1) = 𝑡(𝑛) ∙
2
Graph the following functons in the graphing calculator, plot points and graph on the grids below:
A. 𝑦 = (2)𝑥
10
B. 𝑦 = (3)𝑥
y
10
8
8
6
6
4
4
2
y
2
x
−10
−8
−6
−4
−2
2
4
6
8
10
x
−10
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
−10
−10
Answer the following questions.
1.
Are the slopes of the graphs constant or changing? changing
2.
Are the graphs increasing or decreasing? increasing
3.
What are the intercepts for each graph?
The y-intercepts are both 1, there are
no x-intercepts on either graph.
4
6
8
10
Graph the following functons in the graphing calculator, plot points and graph on the grids below:
1 𝑥
1 𝑥
C. 𝑦 = �2�
10
D. 𝑦 = �3�
y
10
8
8
6
6
4
4
2
2
y
x
x
−10
−8
−6
−4
−2
2
4
6
8
10
−10
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
−10
−10
4
Answer the following questions.
4.
Are the slopes of the graphs constant or changing?
5.
Are the graphs increasing or decreasing?
6.
What are the intercepts for each graph?
Changing
Decreasing
The y-intercepts are both 1, there are
no x-intercepts on either graph.
Use Graphs A-D to answer the following
7.
8.
Decrease
If 𝑦 = (𝑏)𝑥 and b is between 0 and 1, the graph will ___________________________
Increase
If 𝑦 = (𝑏)𝑥 and b is greater than 1, the graph will ___________________________
6
8
10
(Below are only the answers. SHOW work for full credit.)
Solve the following problems. Show sufficient work.
7-7. A grocery store is offering a sale on bread and soup. Khalil buys four cans of soup and three loaves of
bread for $11.67. Ronda buys eight cans of soup and one loaf of bread for $12.89.
a. Write equations for both Khalil’s and Ronda’s purchases.
b. Solve the system to find the price of one can of soup and the price of one loaf of bread.
7-8. If two expressions are equivalent, they can form an equation that is considered to be always true. For
example, since 3(x − 5) is equivalent to 3x −15, then the equation 3(x − 5) = 3x − 15 is always true,
that is, true for any value of x.
If two expressions are equal only for certain values of the variable, they can form an equation that is
considered to be sometimes true. For example, x + 2 is equal to 3x − 8 only when x = 5, so the
equation x + 2 = 3x − 8 is said to be sometimes true.
If two expressions are not equal for any value of the variable, they can form an equation that is
considered to be never true. For example, x − 5 is not equal to x + 1 for any value of x, so the
equation x − 5 = x + 1 is said to be never true.
Is the equation (x + 3)2 = x2 + 9 always, sometimes or never true? Justify your reasoning completely.
7-9. Consider the sequence that begins 40, 20, 10, 5, …
40
a. Based on the information given, is this sequence
arithmetic or geometric? Why?
b. Plot the sequence on a graph up to n = 6.
30
20
c. Will the values of the sequence ever become
zero or negative? Explain.
10
2
4
6
7-10. If a ball is dropped from 160 cm and rebounds to 120 cm on the first bounce, how high will the ball
be (hint: this pattern is exponential):
a. On the 2nd bounce?
b. On the 5th bounce?
c. On the nth bounce?