Write an exponential function from a description or two points

About this lesson

Integrated Math I › Linear and exponential functions

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Write an exponential function from a description or two points

Warm-up

Three quick questions from earlier lessons.

Discover

A table shows a function: f(0)=4f(0) = 4, f(1)=12f(1) = 12, f(2)=36f(2) = 36, f(3)=108f(3) = 108.

What do you multiply by to go from one value to the next? Can you write one formula that gives all four values?

Key idea

An exponential function can be written

f(x)=a⋅bxf(x) = a \cdot b^x, where a=f(0)a = f(0) and bb is the factor for each increase of 1 in xx

If the quantity is multiplied by bb every nn units instead, divide the input by nn: f(x)=a⋅bxnf(x) = a \cdot b^{\frac{x}{n}}.

Worked example

Write ff for the table above.

a=4a = 4
The starting value is f(0).
b=3b = 3
Divide consecutive values: 12 ÷ 4 = 3.
f(x)=4⋅3xf(x) = 4 \cdot 3^x
Put them in the form a · b^x.

Your turn, guided

Your turn: f(0)=5f(0) = 5 and f(1)=10f(1) = 10. Fill in the missing line.

a=5a = 5, b=2b = 2
10 ÷ 5 = 2.
f(x)=5⋅2xf(x) = 5 \cdot 2^x
Use the form a · b^x.

Practice ★

  1. Plot the points of the graph of f(x)=4⋅2xf(x) = 4 \cdot 2^x at x=0x = 0 and x=1x = 1.
  2. Plot the points of the graph of f(x)=3⋅4xf(x) = 3 \cdot 4^x at x=0x = 0 and x=1x = 1.
  1. An exponential function ff has f(0)=9f(0) = 9, and each time xx increases by 1, the value of ff is multiplied by 5. Write f(x)f(x).
  2. An exponential function ff has f(0)=9f(0) = 9, and each time xx increases by 1, the value of ff is multiplied by 4. Write f(x)f(x).
  3. An exponential function ff has f(0)=6f(0) = 6, and each time xx increases by 1, the value of ff is multiplied by 2. Write f(x)f(x).
  4. An exponential function ff has f(0)=6f(0) = 6, and each time xx increases by 1, the value of ff is multiplied by 4. Write f(x)f(x).
  5. An exponential function ff has f(0)=4f(0) = 4, and each time xx increases by 1, the value of ff is multiplied by 2. Write f(x)f(x).

Watch out!

Practice ★★

  1. A population is 7 thousand at t=0t = 0 and is multiplied by 3 every 4 years. Write the population P(t)P(t), in thousands, after tt years.
  2. A population is 8 thousand at t=0t = 0 and is multiplied by 3 every 3 years. Write the population P(t)P(t), in thousands, after tt years.
  3. A population is 4 thousand at t=0t = 0 and is multiplied by 2 every 3 years. Write the population P(t)P(t), in thousands, after tt years.
  4. A population is 8 thousand at t=0t = 0 and is multiplied by 2 every 2 years. Write the population P(t)P(t), in thousands, after tt years.
  5. A population is 9 thousand at t=0t = 0 and is multiplied by 3 every 5 years. Write the population P(t)P(t), in thousands, after tt years.

Exit ticket

Three questions to check you have got it.

  1. A population is 7 thousand at t=0t = 0 and is multiplied by 3 every 4 years. Write the population P(t)P(t), in thousands, after tt years.
  2. A population is 8 thousand at t=0t = 0 and is multiplied by 3 every 3 years. Write the population P(t)P(t), in thousands, after tt years.
  3. A population is 4 thousand at t=0t = 0 and is multiplied by 2 every 3 years. Write the population P(t)P(t), in thousands, after tt years.

Summary card

Writing an exponential function

  • f(x)=a⋅bxf(x) = a \cdot b^x: a=f(0)a = f(0), b=f(1)f(0)b = \dfrac{f(1)}{f(0)}.
  • Multiplied by bb every nn units: f(x)=a⋅bxnf(x) = a \cdot b^{\frac{x}{n}}.

Watch out: the starting value is f(0)f(0), and it goes in front, not in the base.

End of the lesson

Next lesson: Percent change and compound interest Back to the chapter

Answers: Write an exponential function from a description or two points

A

  1. (0,4), (1,8)(0, 4),\ (1, 8)
  2. (0,3), (1,12)(0, 3),\ (1, 12)

B

  1. 9⋅5x9 \cdot 5^{x}
  2. 9⋅4x9 \cdot 4^{x}
  3. 6⋅2x6 \cdot 2^{x}
  4. 6⋅4x6 \cdot 4^{x}
  5. 4⋅2x4 \cdot 2^{x}

C

  1. 7⋅3t47 \cdot 3^{\frac{t}{4}}
  2. 8⋅3t38 \cdot 3^{\frac{t}{3}}
  3. 4⋅2t34 \cdot 2^{\frac{t}{3}}
  4. 8⋅2t28 \cdot 2^{\frac{t}{2}}
  5. 9⋅3t59 \cdot 3^{\frac{t}{5}}

D

  1. 7⋅3t47 \cdot 3^{\frac{t}{4}}
  2. 8⋅3t38 \cdot 3^{\frac{t}{3}}
  3. 4⋅2t34 \cdot 2^{\frac{t}{3}}

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