Linear, Quadratic, or Exponential
How can you tell from a table if a function is exponential?
There is a constant ratio between outputs
What does 1.05 represent in an exponential model?
5% growth factor
What does P represent?
Initial value
What is the inverse of an exponential function?
Logarithmic function
What is the relationship between logs and exponents?
Logs are inverses of exponentials
What does +3 outside the function do?
Vertical translation up 3
When do you use exponential regression?
When data has multiplicative growth/decay
Identify the function type:
(1, 3), (2, 6), (3, 9), (4, 12)
Linear (constant difference +3)
Convert 8% growth into a multiplier.
1.08
Write an equation for the model: initial 200, grows continuously at 3%
y=200e^{0.03t}
Find the inverse of the function.
f(x)=2^x
f^{-1}(x)=log_2x
Rewrite the equation in logarithmic form.
2^3=8
log_2(8)=3
Describe the transformation:
f(x)=e^x+5
Vertical translation up 5
Write the equation: starts at 10, doubles each step
y=10(2)^x
Identify the function type:
(1, 2), (2, 4), (3, 8), (4, 16)
Exponential (ratio ×2)
Convert 0.94 to a percent change.
6% decay
A car loses value at 10% continuously, starting at 30,000. Write an equation for the scenario.
y=30000e^{-0.10t}
Find inverse of the function:
f(x)=e^x
f^{-1}(x)=lnx
Rewrite the equation in exponential form.
log_5(125)=3
5^3=125
Describe the transformation:
y=e^{x-2}
Horizontal translation right 2
Write the model: Data grows by 6% each step, starts at 100.
y=100(1.06)^x
A function has a constant second difference. What type is it?
Quadratic
Write an exponential model for 12% growth starting at 50.
y=50(1.12)^x
Find r if the value triples in 4 years
3=e^{4r}=>r=ln3/4
Find inverse of the function.
f(x)=3^x+4
f^{-1}(x)=log_3(x-4)
Solve for x.
log_3x=4
x = 81
Describe the transformation:
y=-e^x
Reflection over x-axis
estimate the growth factor:
3 -> 9 -> 27 -> 81
x3
Which type of function is best fit for data with increasing growth rate?
Exponential
A population doubles every year. What is the growth factor per year?
2
A quantity follows the given model. After 2 years, y=800. Find r.
y=500e^{rt}
r=ln(8/5)/2
Find the inverse of the function.
f(x)=2^{x-1}-5
f^{-1}(x)=log_2(x+5)+1
Solve for x.
lnx=2.5
x=e^2.5 or 12.182
Describe the transformation:
y=3ln(x-4)+2
- Vertical dilation x3
- Horizontal translation right 4
- Vertical translation up 2
Write exponential model for data:
(0, 12), (1, 10.8), (2, 9.72)
y=12(0.9)^x