1. Write Exponential Functions and Constant Ratios

2. Constant Ratio / Common Ratio
Remember that in a geometric sequence the terms have a constant ratio (common ratio / constant multiplier).
Each term is multiplied by the same value to create the next term
Example: 4, 12, 36, 108, …
the constant ratio is 3 because each term is multiplied by 3 to make the next term
Example: 9, 3, 1, 1/3, …
the constant ratio is 1/3 because each term is multiplied by 1/3 to make the next term (like dividing by 3)
This makes an exponential function (only if the constant ratio is positive)
To identify a constant ratio, divide a term by the preceding (previous) term. If the ratio between all terms is the same (constant) it is an exponential function.

3. Write Exponential Functions
Exponential Functions use form f(x)=abx
a is the y-intercept (0,a)
b is the common ratio
First identify the constant ratio by comparing y output values from a table or graph (if it is hard to spot the constant ratio mentally, set up a division problem)
Second identify the y-intercept
Write in f(x)=abx form

4. Write Exponential Functions from Tables
To find the constant ratio: divide one term by the previous term. That ratio should match for all y values
To find the constant ratio: divide one term by the previous term. That ratio should match for all y values 5
= 𝟏 𝟐 -6
= 3 10 -2
2 𝟏 𝟐
-18
= 𝟏 𝟐
= 3 5 -6
Constant ratio: 3
y-intercept: −𝟐 𝟑 how do we find this?
Exponential function:
g(x) = −𝟐 𝟑 (3)x
Constant ratio: 𝟏 𝟐
y-intercept: 10
Exponential function:
g(x)=10( 𝟏 𝟐 )x

5. Write Exponential Functions from Tables
To find the constant ratio: divide one term by the previous term. That ratio should match for all y values
To find the constant ratio: divide one term by the previous term. That ratio should match for all y values 2
= 𝟐 𝟑 3 15
= 𝟓 𝟑
𝟒 𝟑 9
= 𝟐 𝟑 2
Constant ratio: 𝟓 𝟑
y-intercept: 9
Exponential function:
g(x)=9( 𝟓 𝟑 )x
Constant ratio: 𝟐 𝟑
y-intercept: 3
Exponential function:
g(x)=3( 𝟐 𝟑 )x

6. Write Exponential Functions from Graphs
To find the constant ratio: divide one term by the previous term. That ratio should match for all y values 9
= 1.5 6 -5
= 5 6 -1
= 3 2
Constant ratio: 3
y-intercept: 2
Exponential function:
g(x)=2(3)x
Constant ratio: 5
y-intercept: -1
Exponential function:
g(x)=-1(5)x
Constant ratio: 1.5
y-intercept: 6
Exponential function:
g(x)=6(1.5)x