Name _________________________________________ Date __________________ Period _________
8.7 Exponential Functions Guided Notes – Day 1
An exponential function is a function
of the form y  a  b x ,
where a  0, b  0, and b  1,
and the exponent must be a variable.
Writing Function Rules
1. Paper Folding Activity. Write a rule to describe the total number of new rectangles formed.
2. Rabbits. Oh no, 20 rabbits were released in the school! The rabbit population triples every half year. Write a
rule to describe the total population.
3. Mice. Suppose two mice live in a barn. If the number of mice quadruples every three months, how many mice
will be in the barn after two years?
4. Photo Re-Sizing. Many photo printing stations allow you to alter the size of the original document. The Kodak
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machine at CVS allows you to only reduce photos to 4 its original size. Write a rule to describe the width of the
new photograph after x reductions, if the original width measures 8 inches.
5. Mortality Rate. A farmer started with 1200 livestock on his farm, but a virus spread and half of his livestock
were killed off each week after the livestock were infected by the virus. Write a function rule to describe the
population of the livestock after x weeks.
Evaluating Function Rules
6.
Using the rule you created in #5, determine how much livestock will be left after 4 weeks.
7. Evaluate 𝑓(𝑥) = −6 ∙ 3𝑥 for 𝑥 = −2
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8. Evaluate 𝑔(𝑥) = −2 ∙ (3)𝑥 for 𝑥 = −1
Understanding Graphs of Exponential Functions
An exponential function is always in the form f(x) = abx. The graph of an exponential function is always the same general
shape. But the values of a and b determine the quadrants the graph is in, and the “direction” in which the graph is
facing. Analyze the information in the examples below.
y  2(3)x
y  2(3)x
a0
b1
a0
b1
y  abx
a0
0b1
a0
0b1
y
y
Respond to each statement by writing sometimes, always, or never.
9. When a is negative, the graph of y  abx is contained entirely in quadrants III and IV.
10. The y-intercept of y  abx is at a.
11. When b is a fraction between 0 and 1, the graph of y  abx is contained entirely in quadrants I and II.
12. The graph of y  abx is a straight line.
Graphing Exponential Functions (input/output at least 4 values)
13. Graph 𝑓(𝑥) = 4 ∙ 2𝑥
b>1, a>0 ….. exponential growth!
x
f(x)
14. Graph 𝑓(𝑥) = −2 ∙ 3𝑥
x
f(x)
x
f(x)
x
f(x)
1
15. Graph 𝑓(𝑥) = 2 ∙ (3)𝑥
0<b<1, a>0 ….. exponential decay!
1
2
16. Graph 𝑓(𝑥) = −3 ∙ ( )𝑥