Download Lesson 7-6

Notes for Lesson 7-6: Exponential Functions
7-6.1 – Identifying Linear and Exponential Functions
Vocabulary:
Exponential Function – A function that repeatedly multiplies an initial amount by the same positive
number.
An exponential function is a function that is in the form of 𝑦 = 𝑎 ∗ 𝑏 𝑥 where 𝑏 > 0 𝑎𝑛𝑑 𝑏 ≠ 1. This will give
you a function where the outputs all have a common ratio, which is a multiplication factor. Remember, a
common difference (Arithmetic Sequence) where you are adding the same value to each consecutive output is a
linear function. This is how you can tell the difference between the two.
Examples:
X
Y
0
-1
1
-3
2
-9
3
-27
The x values differ by adding 1 but the y values differ by a multiple of 3, therefore this is an exponential
function and the common ratio is 3
X 1
Y -1
2
1
3
3
4
5
The x values differ by 1 and the y values are found by adding 2 each time, therefore this is a linear function and
the common difference is 2
X 1
Y 2
2
8
3
32
4
128
The x values differ by 1 and the y values are found by multiplying 4 each time, therefore it is an exponential
function with a common ratio of 4
7-6.2 – Evaluating an Exponential Function
We can evaluate an exponential function by substituting in a value for the variable exponent and solving the
equation. Remember that the exponent is done before the multiplication.
Example:
An initial population of 20 rabbits triples every half year. The function 𝑓(𝑥) = 20 ∗ 3𝑥 gives the
population after x half year periods. How many rabbits will there be after 3 years?
𝑓(3) = 20 ∗ 33
𝑓(3) = 20 ∗ 27
𝑓(3) = 540 𝑟𝑎𝑏𝑏𝑖𝑡𝑠
Suppose 30 flour beetles are left undisturbed in a warehouse bin. The beetle population doubles each
week. The function 𝑓(𝑥) = 30 ∗ 2𝑥 give the population after x weeks. How many beetles will there be after 56
days?
𝑓(8) = 30 ∗ 28 𝑓(8) = 30 ∗ 256 𝑓(3) = 7680 𝑏𝑒𝑒𝑡𝑙𝑒𝑠
7-6.3 & 4 – Graphing an Exponential Function.
To graph an exponential function make a table of values that include negatives, zero and positive values. You
will find the curve that represents the function.
Examples:
Graph 𝑦 = 3 ∗ 2𝑥
X
-2
-1
0
1
2
Y
¾
1½
3
6
12
Plot the points and connect them with a smooth curve
Graph 𝑦 = −0.5 ∗ 3𝑥
X
-2
-1
0
1
2
Y
-1/18
-1/6
-0.5
-1.5
-4.5
7-6.5 – Solving one variable equations
To solve an exponential function you need to isolate the power and then figure out what would the exponent
have to be to make the equation equal.
Examples:
3 ∗ 2𝑥 = 24
2𝑥 = 8 𝑠𝑜 𝑥 = 3
5 ∗ 2𝑥 − 152 = 8
5 ∗ 2𝑥 = 160 2𝑥 = 32 𝑠𝑜 𝑥 = 5
If there are multiple x’s you will need to use the graphing calculator. Using the graphing function, input
both sides of the equation and graph each separately on the same graph. Use the 2nd figure button and hit f4 to
get to the calc menu and hit 5 for intersection. The x and y values will appear at the bottom of the screen.
There many be 2 possible solutions.
Examples:
2𝑥 = 0.5𝑥 + 2
1.25𝑥 = −2𝑥
−3.86 𝑎𝑛𝑑 1.45
−0.45