Write an Exponential Function

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Writing an Exponential Function From Two Points

In this lesson, you will learn

  • the general form of an exponential function, y = abx
  • how to use two points on the graph to find a and b
  • how to work with a negative exponent and a fraction for b

You can write an exponential function from two points on the function's graph. For example, write an exponential function y = abx for a graph that includes (1, 1) and (2, 4).

Write an exponential function: two points (x1, y1) and (x2, y2) lead to the exponential function y = ab to the power x

The General Form and the Method

y = abx

The goal is to use the two given points to find a and b. Then, we can replace a and b in the equation y = abx with the values we found.

Rule

1. Substitute the first point into y = abx and solve for a.

2. Substitute the second point into y = abx.

3. Replace a with what you found in step 1, and solve for b.

4. Use b to find a, then write y = abx with the values of a and b.

Remember

In an exponential function, b is always positive and not equal to 1. So if you get bΒ² = 9, then b = 3, not βˆ’3.

What is y in y = 3(2)x when x = 2?

Show answer

y = 12. 3 Γ— 2Β² = 3 Γ— 4 = 12.

Example #1

Example #1: Write y = abx for a graph that includes (1, 1) and (2, 4)

(1, 1) and (2, 4)

  1. Use the general form of the exponential function y = abx and substitute for x and y using (1, 1).1 = ab11 = ab
  2. Divide both sides by b to solve for a.a = 1b
  3. Use the general form of the exponential function y = abx again and substitute for x and y using (2, 4).4 = ab2
  4. Substitute 1b for a in 4 = ab2.4 = 1b Γ— b24 = b2b4 = b
  5. Find a.a = 1b = 14

The exponential function is y = (1/4)(4)x

Check the answer: what is y in y = (1/4)(4)x when x = 3?

Show answer

y = 16. (1/4) Γ— 4Β³ = (1/4) Γ— 64 = 16, so (3, 16) is also on the graph.

A More Complicated Example Showing How to Write an Exponential Function

Example #2: Find y = abx for a graph that includes (1, 2) and (βˆ’1, 8)

(1, 2) and (βˆ’1, 8)

  1. Use the general form of the exponential function y = abx and substitute for x and y using (1, 2).2 = ab12 = ab
  2. Divide both sides by b to solve for a.a = 2b
  3. Use the general form of the exponential function y = abx again and substitute for x and y using (βˆ’1, 8).8 = abβˆ’1
  4. Substitute 2b for a in 8 = abβˆ’1.8 = 2b Γ— bβˆ’1
  5. Multiply both sides of the equation by b2.b2 Γ— 8 = 2b Γ— bβˆ’1 Γ— b2b2 Γ— 8 = 2b Γ— bb2 Γ— 8 = 2
  6. Divide both sides of the equation by 8.(b2 Γ— 8) / 8 = 2 / 8b2 = 28 = 14
  7. Take the square root.b = square root of 14 = 12b must be positive, so we use 1/2, not βˆ’1/2.
  8. Now, let us find a using the equation a = 2 / b.a = 2 Γ· 12 = 21 Γ— 21 = 41 = 4

The exponential function is y = 4(1/2)x

Common mistake

Forgetting what a negative exponent means. bβˆ’1 is 1b, not βˆ’b. For example, 4βˆ’1 = 14.

A graph of y = abx includes (0, 5) and (1, 15). What is the function?

Show answer

y = 5(3)x. At x = 0, b0 = 1, so 5 = a. Then 15 = 5b, so b = 3.

Summary

  • An exponential function has the form y = abx, with b positive and not equal to 1.
  • Substitute each point into y = abx to get two equations.
  • Solve the first equation for a, put it into the second, and solve for b. Then find a.
  • A negative exponent means a fraction: bβˆ’1 = 1/b.
  • If one point has x = 0, then a is its y-value, because b0 = 1.

Test Yourself: 10-Question Quiz

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Question 1 of 10Score: 0

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