Algebra 1 Exponential Functions and Growth Worksheet Solutions (PDF)

Algebra 1 math worksheet solutions: Exponential Functions and Growth – Maths PDF

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This algebra 1 exponential functions answers for Algebra 1 covers exponential functions and growth (Common Core F-LE.A.1, F-LE.A.2, F-LE.A.3, F-IF.C.7e, F-IF.C.8b), with a free PDF to print.

Use this key to check your work on the exponential functions worksheet. Every solution shows the steps, and the final answer appears in bold. If your answer differs, find the first line where your work and the key disagree, then redo the problem from there without looking.

Algebra 1 · Common Core State Standards: F-LE.A.1, F-LE.A.2, F-LE.A.3, F-IF.C.7e, F-IF.C.8b

Algebra 1 exponential functions answers: exercises 1 to 10

Exercise 1 – Identify a and b

In $y = a \cdot b^{x}$, $a = 3$ and $b = 2$. Since $b > 1$, it is growth.

Starting value 3; factor 2; growth

Exercise 2 – Decay factor

$b = 0.8$, and $1 – 0.8 = 0.2$.

Factor 0.8; 20% decrease

Exercise 3 – Complete a table

Each value is 3 times the one before.

$4 \cdot 3^{0} = 4$, $4 \cdot 3^{1} = 12$, $4 \cdot 3^{2} = 36$, $4 \cdot 3^{3} = 108$.

4, 12, 36, 108

Exercise 4 – Exponential table

Ratios: $\frac{6}{2} = 3$, $\frac{18}{6} = 3$, $\frac{54}{18} = 3$.

Exponential with ratio 3

Exercise 5 – Linear table

Differences: $9 – 5 = 4$, $13 – 9 = 4$, $17 – 13 = 4$. The ratios are not constant ($\frac{9}{5}$, $\frac{13}{9}$).

Linear, adding 4 each step

Exercise 6 – Write from a table

The starting value is $a = 5$ and each step doubles: $b = 2$.

$y = 5 \cdot 2^{x}$

Exercise 7 – Graph a growth function

Points and curve.

Compute $2^{-2} = \frac{1}{4}$, $2^{-1} = \frac{1}{2}$, $2^{0} = 1$, $2^{1} = 2$, $2^{2} = 4$, $2^{3} = 8$.

Plot the points and draw a smooth curve that approaches the x-axis on the left.

Curve through (0, 1), (1, 2), (2, 4), (3, 8)

Exercise 8 – Graph a decay function

Decay curve.

Values: $4, 2, 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}$.

The curve falls from left to right and gets close to the x-axis.

Decreasing curve through (0, 1)

Exercise 9 – Rate to factor

(a) $1 + 0.12 = 1.12$. (b) $1 – 0.08 = 0.92$.

1.12 and 0.92

Exercise 10 – Evaluate with a calculator

$1.05^{10} \approx 1.62889$.

$100(1.62889) \approx 162.89$.

162.89

Exercises 11 and up

Exercise 11 – Intercept and asymptote

At $x = 0$, $y = 1$. Any real number can be an exponent, and the output is always positive. As $x$ decreases, $2^{x}$ gets closer to 0.

(0, 1); all real numbers; $y > 0$; approaches the x-axis

Exercise 12 – Solve an exponential equation

$32 = 2^{5}$, so $x = 5$.

$81 = 3^{4}$, so $x = 4$.

$x = 5$ and $x = 4$

Exercise 13 – Linear versus exponential

Values of both functions.

$f(10) = 10(10) + 5 = 105$ and $g(10) = 2^{10} = 1024$.

f(10) = 105; g(10) = 1024; g is larger

Exercise 14 – Domain and range

Any real number can be an exponent, so the domain is all real numbers.

The output is always positive, so the range is $y > 0$.

Domain: all real numbers; range: $y > 0$

Exercise 15 – Rewrite the base

$2^{3t} = (2^{3})^{t} = 8^{t}$.

$y = 8^{t}$; factor 8

Exercise 16 – Town population

Growth factor $1 + 0.04 = 1.04$: $P = 20000(1.04)^{t}$.

$P = 20000(1.04)^{5} \approx 20000(1.21665) \approx 24333$.

$P = 20000(1.04)^{t}$; about 24,333 people

Exercise 17 – Car value

Decay factor $1 – 0.15 = 0.85$: $V = 24000(0.85)^{t}$.

$V = 24000(0.85)^{3} = 24000(0.614125) = 14739$.

$V = 24000(0.85)^{t}$; $14,739

Exercise 18 – Compound interest

$A = 1500(1.06)^{8}$.

$1.06^{8} \approx 1.593848$, so $A \approx 1500(1.593848) = 2390.77$.

$2,390.77

Exercise 19 – Medicine half-life

Substitute $t = 18$: $A = 80\left(\frac{1}{2}\right)^{3}$.

$= 80 \cdot \frac{1}{8} = 10$.

10 mg

Exercise 20 – Viral video

Day 2: $50 \cdot 3^{2} = 450$ and $50 + 200(2) = 450$ (equal).

Day 3: $50 \cdot 3^{3} = 1350$ and $50 + 200(3) = 650$.

From day 3 on, Video A is ahead and the gap grows.

Equal on day 2; A is larger from day 3

Exercise 21 – Doubling time

Solve $2000 \cdot 2^{t/12} = 8000$: $2^{t/12} = 4 = 2^{2}$.

$\frac{t}{12} = 2$, so $t = 24$.

24 years

Key facts behind this algebra 1 exponential functions answers

Key facts

Exponential form: f(x) = a·bˣ with a the starting value and b the factor.
Growth: b = 1 + r. Decay: b = 1 – r.
Linear functions add a constant; exponential functions multiply by a constant.
Compound interest: A = P(1 + r)ᵗ.
An exponential function eventually exceeds any linear function.

How to use this algebra 1 exponential functions answers

Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.

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