Write an Exponential from Points

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Key Ideas Calculator Easy Medium Hard

f(x) = k a^x + c

c is the horizontal asymptote.

Divide the two point equations to find a.

What to Remember

Read c from the asymptote

In f(x) = k a^x + c, the horizontal asymptote is y = c.

Subtract c from each y-value before you look for k and a.

Two points give a and k

k a^x₁ = y₁ − c and k a^x₂ = y₂ − c.

Divide those equations. The k cancels, and a power of a remains.

Use the positive value of a.

When the power is not exact

If a and k simplify to whole numbers, fractions, or square roots, write that exact function. If they do not, round a and k to three decimal places.

Level 1

Easy

One point has x = 0, so k is immediate

Example

Write an exponential function f(x) = k a^x + c whose graph passes through (0, 6) and (2, 18) and has asymptote y = 2.

f(x) = k a^x + c

The asymptote is y = c.

c = 2

Substitute the point with x = 0.

f(0) = k a⁰ + 2

k + 2 = 6

k = 4

Substitute the second point.

f(2) = 4 a² + 2 = 18

4 a² = 16

a² = 4

a = 2

f(x) = 4(2^x) + 2

Answer: f(x) = 4(2^x) + 2

Write an exponential function f(x) = k a^x + c whose graph passes through (0, 5) and (1, 11) and has asymptote y = 3.

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f(x) = k a^x + c

The asymptote is y = c.

c = 3

Substitute the point with x = 0.

f(0) = k a⁰ + 3

k + 3 = 5

k = 2

Substitute the second point.

f(1) = 2 a¹ + 3 = 11

2 a¹ = 8

a¹ = 4

a = 4

f(x) = 2(4^x) + 3

Answer: f(x) = 2(4^x) + 3

Write an exponential function f(x) = k a^x + c whose graph passes through (0, 4) and (2, 19) and has asymptote y = −1.

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f(x) = k a^x + c

The asymptote is y = c.

c = −1

Substitute the point with x = 0.

f(0) = k · 1 − 1

k − 1 = 4

k = 5

Substitute the second point.

f(2) = 5 a² − 1 = 19

5 a² = 20

a² = 4

a = 2

f(x) = 5(2^x) − 1

Answer: f(x) = 5(2^x) − 1

Write an exponential function f(x) = k a^x + c whose graph passes through (0, 9) and (1, 21) and has asymptote y = 5.

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f(x) = k a^x + c

The asymptote is y = c.

c = 5

Substitute the point with x = 0.

f(0) = k a⁰ + 5

k + 5 = 9

k = 4

Substitute the second point.

f(1) = 4 a¹ + 5 = 21

4 a¹ = 16

a¹ = 4

a = 4

f(x) = 4(4^x) + 5

Answer: f(x) = 4(4^x) + 5

Write an exponential function f(x) = k a^x + c whose graph passes through (0, 8) and (1, 20) and has asymptote y = 4.

Show Answer

f(x) = k a^x + c

The asymptote is y = c.

c = 4

Substitute the point with x = 0.

f(0) = k a⁰ + 4

k + 4 = 8

k = 4

Substitute the second point.

f(1) = 4 a¹ + 4 = 20

4 a¹ = 16

a¹ = 4

a = 4

f(x) = 4(4^x) + 4

Answer: f(x) = 4(4^x) + 4

Write an exponential function f(x) = k a^x + c whose graph passes through (0, 6) and (3, 62) and has asymptote y = −2.

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f(x) = k a^x + c

The asymptote is y = c.

c = −2

Substitute the point with x = 0.

f(0) = k · 1 − 2

k − 2 = 6

k = 8

Substitute the second point.

f(3) = 8 a³ − 2 = 62

8 a³ = 64

a³ = 8

a = 2

f(x) = 8(2^x) − 2

Answer: f(x) = 8(2^x) − 2

Level 2

Medium

Neither x-value is 0

Example

Write an exponential function f(x) = k a^x + c whose graph passes through (1, 7) and (3, 19) and has asymptote y = 3.

f(x) = k a^x + c

The asymptote is y = c.

c = 3

Substitute each point.

k a¹ = 4

k a³ = 16

Divide the second equation by the first.

a² = 164

a² = 4

a = 2

Substitute a into the first equation.

2 · 2 = 4

k = 2

f(x) = 2(2^x) + 3

Answer: f(x) = 2(2^x) + 3

Write an exponential function f(x) = k a^x + c whose graph passes through (1, 8) and (3, 26) and has asymptote y = 2.

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f(x) = k a^x + c

The asymptote is y = c.

c = 2

Substitute each point.

k a¹ = 6

k a³ = 24

Divide the second equation by the first.

a² = 246

a² = 4

a = 2

Substitute a into the first equation.

3 · 2 = 6

k = 3

f(x) = 3(2^x) + 2

Answer: f(x) = 3(2^x) + 2

Write an exponential function f(x) = k a^x + c whose graph passes through (2, 11) and (4, 35) and has asymptote y = 3.

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f(x) = k a^x + c

The asymptote is y = c.

c = 3

Substitute each point.

k a² = 8

k a⁴ = 32

Divide the second equation by the first.

a² = 328

a² = 4

a = 2

Substitute a into the first equation.

2 · 4 = 8

k = 2

f(x) = 2(2^x) + 3

Answer: f(x) = 2(2^x) + 3

Write an exponential function f(x) = k a^x + c whose graph passes through (2, 13) and (5, 97) and has asymptote y = 1.

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f(x) = k a^x + c

The asymptote is y = c.

c = 1

Substitute each point.

k a² = 12

k a⁵ = 96

Divide the second equation by the first.

a³ = 9612

a³ = 8

a = 2

Substitute a into the first equation.

3 · 4 = 12

k = 3

f(x) = 3(2^x) + 1

Answer: f(x) = 3(2^x) + 1

Write an exponential function f(x) = k a^x + c whose graph passes through (1, 14) and (2, 32) and has asymptote y = 5.

Show Answer

f(x) = k a^x + c

The asymptote is y = c.

c = 5

Substitute each point.

k a¹ = 9

k a² = 27

Divide the second equation by the first.

a¹ = 279

a¹ = 3

a = 3

Substitute a into the first equation.

3 · 3 = 9

k = 3

f(x) = 3(3^x) + 5

Answer: f(x) = 3(3^x) + 5

Write an exponential function f(x) = k a^x + c whose graph passes through (1, 7) and (3, 4) and has asymptote y = 3.

Show Answer

f(x) = k a^x + c

The asymptote is y = c.

c = 3

Substitute each point.

k a¹ = 4

k a³ = 1

Divide the second equation by the first.

a² = 14

a = 12

Substitute a into the first equation.

8 · 12 = 4

k = 8

f(x) = 8(12)^x + 3

Answer: f(x) = 8(12)^x + 3

Level 3

Hard

An exact root, or a rounded base

Example

Write an exponential function f(x) = k a^x + c whose graph passes through (1, 4) and (3, 10) and has asymptote y = 1.

f(x) = k a^x + c

The asymptote is y = c.

c = 1

Subtract c from each y-value.

k a¹ = 3

k a³ = 9

Divide the second equation by the first.

a² = 93

a² = 3

The positive base is a square root.

a = square root of 3

Substitute a into the first equation.

k (square root of 3)¹ = 3

k · square root of 3 = 3

k = 3square root of 3

Multiply the numerator and the denominator by square root of 3.

k = 3 · square root of 33

k = square root of 3

f(x) = square root of 3 (square root of 3)^x + 1

Answer: f(x) = square root of 3 (square root of 3)^x + 1

Write an exponential function f(x) = k a^x + c whose graph passes through (2, 7) and (6, 25) and has asymptote y = 1.

Show Answer

f(x) = k a^x + c

The asymptote is y = c.

c = 1

Subtract c from each y-value.

k a² = 6

k a⁶ = 24

Divide the second equation by the first.

a⁴ = 246

a⁴ = 4

The positive base is a square root.

a = square root of 2

Substitute a into the first equation.

k (square root of 2)² = 6

(square root of 2)² = 2

k · 2 = 6

k = 3

f(x) = 3(square root of 2)^x + 1

Answer: f(x) = 3(square root of 2)^x + 1

Write an exponential function f(x) = k a^x + c whose graph passes through (2, 12) and (4, 30) and has asymptote y = 6.

Show Answer

f(x) = k a^x + c

The asymptote is y = c.

c = 6

Substitute each point.

k a² = 6

k a⁴ = 24

Divide the second equation by the first.

a² = 246

a² = 4

a = 2

Substitute a into the first equation.

32 · 4 = 6

k = 32

f(x) = 32(2^x) + 6

Answer: f(x) = 32(2^x) + 6

Write an exponential function f(x) = k a^x + c whose graph passes through (1, 5) and (3, 12) and has asymptote y = 2. Round a and k to three decimal places when they are not exact.

Show Answer

f(x) = k a^x + c

The asymptote gives c.

c = 2

k a¹ = 3

k a³ = 10

Divide the second equation by the first.

a² = 103

Take both sides to the power 12.

a ≈ 1.826

k = 3 ÷ a¹

k ≈ 1.643

Round a and k to three decimal places.

f(x) = 1.643(1.826)^x + 2

Answer: f(x) = 1.643(1.826)^x + 2

Write an exponential function f(x) = k a^x + c whose graph passes through (2, 10) and (5, 40) and has asymptote y = 4. Round a and k to three decimal places when they are not exact.

Show Answer

f(x) = k a^x + c

The asymptote gives c.

c = 4

k a² = 6

k a⁵ = 36

Divide the second equation by the first.

a³ = 366

Take both sides to the power 13.

a ≈ 1.817

k = 6 ÷ a²

k ≈ 1.817

Round a and k to three decimal places.

f(x) = 1.817(1.817)^x + 4

Answer: f(x) = 1.817(1.817)^x + 4

Write an exponential function f(x) = k a^x + c whose graph passes through (1, 6) and (4, 20) and has asymptote y = 2. Round a and k to three decimal places when they are not exact.

Show Answer

f(x) = k a^x + c

The asymptote gives c.

c = 2

k a¹ = 4

k a⁴ = 18

Divide the second equation by the first.

a³ = 184

Take both sides to the power 13.

a ≈ 1.651

k = 4 ÷ a¹

k ≈ 2.423

Round a and k to three decimal places.

f(x) = 2.423(1.651)^x + 2

Answer: f(x) = 2.423(1.651)^x + 2

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