Writing an Exponential Function
Read the Table
Growth or Decay?
Write the Equation
Evaluate and Interpret
Graph It
100
Write an exponential function for this table.
y=4(2)^x
100
A plant doubles in height each week. It starts at 3 cm. This table shows its growth. The pattern multiplies by what number each week?
The pattern multiples by 2 each week (the growth factor is 2).
100
A phone loses half its battery every hour. This word describes the type of exponential function that models it.
What is "exponential decay"?
100
A savings account starts with $500 and triples every year. What is the equation for the amount of money y after x years.
y=500(3)^x
100
Diego saw a video. The views (v) after t days since he saw the video are modeled by the equation below.
For the equation v(t) = 480 · 3ᵗ, what is the value of v(0) and what does it represent.
480, the number of views when Diego first saw the video
100
All exponential GROWTH functions share this general shape on a graph.
What is a J-shaped curve (curving upward)?
200
Write an exponential function for this table.
y=3(4)^x
200
What is the equation for this table?
y=5(3)^x
200
Which two equations represent exponential GROWTH and which two represent exponential DECAY? f(x) = 500 · (3/2)ᵗ, g(x) = 200 · 3ᵗ, h(x) = 800 · (1/4)ᵗ, k(x) = 600 · (2/3)ᵗ.
GROWTH: f(x) = 500 · (3/2)ᵗ, g(x) = 200 · 3ᵗ
DECAY: h(x) = 800 · (1/4)ᵗ, k(x) = 600 · (2/3)ᵗ.
200
A new aquarium opens with 200 fish and the population doubles each year.
1. Write the equation for the number of fish y after x years
2. Find the number of fish after 10 years.
1.
y=200(2)^x
2. 204,800 fish
200
Diego saw a video. The views (v) after t days since he saw the video are modeled by the equation below.
For the equation v(t) = 480 · 3ᵗ, this is the value of v(2) and what it represents.
What is 4,320, the number of views 2 days after seeing the video?
200
The graph of y = 8 · 2ˣ crosses the y-axis at this point.
What is (0,8)?
300
Write an equation for this function.
y=4(3)^x
300
What is the initial value and growth factor?
a=4 and b=3
300
A population of 1,000 rabbits grows by a factor of 1.08 each year. After one year there are 1,080. How many rabbits are there after two years?
1,116 rabbits
300
A car worth $24,000 loses 1/4 of its value every year. What is the equation for the car's value y after x years?
y = 24,000 · (3/4)ˣ?
300
Diego saw a video. The views (v) after t days since he saw the video are modeled by the equation below.
For the equation v(t) = 480 · 3ᵗ, this is the value of v(−1) and what it represents.
What is 160, the estimated number of views one day BEFORE Diego saw the video?
300
This is the key difference between the graph of y = 2ˣ and the graph of y = 5 · 2ˣ.
What is the y-intercept — y = 2ˣ crosses at (0, 1) and y = 5 · 2ˣ crosses at (0, 5), but both have the same shape and growth factor?
400
Write an equation for this table.
y=0.2(3)^x
400
For the equation
y=6(2)^x
fill out the table for y.
1.5, 3, 6, 12
400
List these four equations from the SMALLEST to LARGEST growth factor:
h(t) = 300 · (1/10)ᵗ
d(t) = 50 · (7/3)ᵗ
f(t) = 900 · (2/3)ᵗ
c(t) = 100 · 2ᵗ
Smallest (top)
h(t) = 300 · (1/10)ᵗ
f(t) = 900 · (2/3)ᵗ
c(t) = 100 · 2ᵗ
d(t) = 50 · (7/3)ᵗ
Largest (bottom)
400
An algae population starts at 1000 sq. ft. and is 1/3 of the previous population every hour. Write the equation for y area after x hours.
y=1000(1/3)^x
400
The equation a = 500 · (3/4)ᵗ gives milligrams of aspirin in a body t hours after taking it. This is the amount remaining after 2 hours, rounded to the nearest milligram.
What is 281 mg?
400
The graph of y = 80 · (2/3)ˣ starts at this y-intercept and moves in this direction as x increases, because the base is less than 1.
What is a y-intercept of (0, 80), and the graph curves downward toward zero (exponential decay)?