Algebra 1 Exponential Worksheet Answer Key

Contents

This algebra 1 exponential worksheet answer key solves all 24 problems of the exponential growth and decay worksheet step by step.

Each problem has a complete worked solution with the final answer in bold. Compare your method with the steps shown, especially how a percent rate becomes the factor, and check any rounding in the money problems.

Algebra 1Common Core: F-LE.A.1a · F-LE.A.1c · F-LE.A.2 · F-LE.A.3 · F-IF.C.8b · F-LE.B.5

Algebra 1 Exponential Worksheet Answer Key: problems 1–5

Problem 1: Evaluate an exponential function

$f(0)=3\cdot1=$$3$, $f(1)=3\cdot2=$$6$, $f(4)=3\cdot16=$$48$.

Problem 2: Constant factor

The differences are 5, 10, 20 (not equal). The ratios are $10: 5=2$, $20: 10=2$, $40: 20=2$. The function is exponential with factor 2.

Each y is multiplied by 2.

Problem 3: Linear or exponential tables

Table B: differences $7-4=3$, $10-7=3$, $13-10=3$. Linear, difference +3.
Table C: ratios $27: 81=\frac{1}{3}$, $9: 27=\frac{1}{3}$, $3: 9=\frac{1}{3}$. Exponential, factor $\frac{1}{3}$.

Problem 4: Identify a and b

$a=7$, $b=3$: growth (b > 1).
$a=250$, $b=0.9$: decay (0 < b < 1).
$a=0.5$, $b=4$: growth.

Problem 5: Percent rate from the factor

$1.08-1=0.08$: 8% growth.
$1-0.95=0.05$: 5% decay.
$1.5-1=0.5$: 50% growth.
$1-0.6=0.4$: 40% decay.

Algebra 1 Exponential Worksheet Answer Key: problems 6–10

Problem 6: Factor from the percent rate

$b=1+0.12=$$1.12$
$b=1-0.07=$$0.93$
$b=1+1=$$2$
$b=1-0.35=$$0.65$

Problem 7: Equation from a table

At $x=0$, $y=3$, so $a=3$. The ratio is $15: 3=5$, so $b=5$. The function is $y=3(5)^{x}$. Check: $3(5)^{3}=3\cdot125=375$.

Each y is multiplied by 5.

Problem 8: Decay from a table

$a=1000$. $b=600: 1000=0.6$. Check: $360: 600=0.6$ and $216: 360=0.6$. The function is $y=1000(0.6)^{x}$.

Problem 9: Equation from two points

$a=5$. Then $5b^{2}=45$, so $b^{2}=9$ and $b=3$ (b is positive). $y=5(3)^{x}$.
$a=96$. Then $96b^{3}=12$, so $b^{3}=\frac{1}{8}$ and $b=\frac{1}{2}$. $y=96(\frac{1}{2})^{x}$.

Problem 10: Equation from a graph

The graph crosses the $y$-axis at 2, so $a=2$. From (0, 2) to (1, 6): $6: 2=3$, so $b=3$. Check: $2(3)^{2}=18$. The equation is $y=2(3)^{x}$.

Starting value 2, factor 3.

Algebra 1 Exponential Worksheet Answer Key: problems 11–15

Problem 11: Evaluate growth

$1.05^{3}=1.157625$. $400\times 1.157625=463.05$. Answer: $463.05$.

Problem 12: Evaluate decay

$0.8^{4}=0.4096$. $900\times 0.4096=368.64$. Answer: $368.64$.

Problem 13: Exponential versus linear

$g$: 1, 2, 4, 8, 16, 32. $h$: 3, 8, 13, 18, 23, 28. The values of $g$ are below $h$ for $x=0$ to $4$ (for example 16 < 23). At $x=5$, $32>28$. Answer: $x=5$.

At x = 5 the exponential is first above the line.

Problem 14: Interpret the parameters

12,000 is the population in 2020 ($t=0$). The factor 0.97 means the population is 97% of the previous year, a 3% decrease each year. For 2023, $t=3$: $12000(0.97)^{3}=12000\times 0.912673=10952.076\approx$$10,952$ people.

Problem 15: Compound interest, yearly

$A=2000(1.03)^{4}$. $1.03^{4}=1.12550881$. $A=2000\times 1.12550881\approx$$\$2,251.02$.

Algebra 1 Exponential Worksheet Answer Key: problems 16–20

Problem 16: Compounding twice a year

Yearly: $1000(1.06)^{2}=\$1,123.60$. Twice a year: $r/n=0.03$ and $nt=4$, so $1000(1.03)^{4}=\$1,125.51$. Difference: $1125.51-1123.60=$$\$1.91$.

Problem 17: Equivalent forms of an exponential

$12t$ counts months and the factor per month is 1.01: 1% per month.
$1.01^{12}\approx1.1268$, so the yearly factor is about 1.1268, a yearly rate of about 12.7%.
$300(1.01)^{24}=300\times 1.2697\approx$$\$380.92$.

Problem 18: Half-life model

$t=20$: $80(0.5)^{2}=80 imes0.25=$$20 ext{ g}$. $t=30$: $80(0.5)^{3}=80 imes0.125=$$10 ext{ g}$. Every 10 years the amount is cut in half, so the half-life is 10 years.

Problem 19: Video views

$a=150$ and $b=2$, so $V(d)=150(2)^{d}$. $V(7)=150\times 128=$$19,200\text{ views}$.

Problem 20: Car depreciation

$b=1-0.15=0.85$, so $V(t)=24000(0.85)^{t}$. $0.85^{3}=0.614125$. $V(3)=24000\times 0.614125=$$\$14,739$.

Algebra 1 Exponential Worksheet Answer Key: problems 21–24

Problem 21: Two savings plans

A: 20, 35, 50, 65, 80, 95, 110, 125. B: 2, 4, 8, 16, 32, 64, 128, 256 for $w=0$ to $7$. At $w=5$, $95>64$. At $w=6$, $128>110$. Plan B first has more in week 6, and it stays ahead.

At week 6, Plan B passes Plan A.

Problem 22: Medicine in the body

$b=0.88$, so $M(h)=200(0.88)^{h}$. $M(3)=200 imes0.681472pprox$$136.29 ext{ mg}$. Table: $M(4)pprox119.94$, $M(5)pprox105.55$, $M(6)pprox92.88$. Since $105.55>100>92.88$, the first hour is hour 6.

Problem 23: Growing cell culture

$N(t)=500(1.2)^{t}$. $N(3)=1000>864$ and $N(4)=500\times 2.0736=1036.8>1000$, so the first hour is hour 4. $N(5)=500\times 2.48832=1244.16\approx$$1,244\text{ cells}$.

Cell count by hour.

Problem 24: Interest: simple versus compound

Priya: $P(t)=800(1.05)^{t}$. Diego: $D(t)=800+40t$. After 10 years, $P(10)=800\times 1.628895\approx\$1,303.12$ and $D(10)=800+400=\$1,200$. Priya has $103.12 more. They tie after 1 year ($840), then the exponential pulls ahead.

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