VertexMath · MCR3U · Free worksheet

Exponential Functions Review

Try each problem first. You don’t need to finish everything.

Core path ~55 min Full sheet ~1 h 50 min

Recommended: K1–K5, then one A and one T problem

K = Knowledge & Understanding · A = Application · T = Thinking · Boss = Challenge

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Quick reference

Exponent laws: $a^m\cdot a^n=a^{m+n}$ , $\dfrac{a^m}{a^n}=a^{m-n}$ , $(a^m)^n=a^{mn}$ , $(ab)^n=a^nb^n$ . The base stays the same: never multiply the bases.
$a^0=1$ , $a^{-n}=\dfrac{1}{a^n}$ , $a^{m/n}=\left(\sqrt[n]{a}\right)^m$ . A negative exponent means a reciprocal, not a negative number.
$y=b^x$ ( $b>0$ , $b\ne1$ ): domain $(-\infty,\infty)$ , range $(0,\infty)$ , $y$ -intercept $1$ , asymptote $y=0$ . Growth if $b>1$ , decay if $0<b<1$ .
$y=a\cdot b^{k(x-d)}+c$ : factor $k$ out of the exponent before reading the shift $d$ . The asymptote moves to $y=c$ .
Models: $A=A_0(1+r)^t$ , $A=A_0(1-r)^t$ , $A=A_0\left(\frac{1}{2}\right)^{t/h}$ . Linear data has constant differences; exponential data has constant ratios.

Show your work on separate paper. Show your work in the dot-grid space.

Core Skills (K)

Integer Exponents · Rational Exponents · Simplifying · Properties of $b^x$ · Transformations

K1 Integer Exponents

Write each answer as a fraction or with positive exponents.

a)
Evaluate $2^{-3}+3^0$ .
b)
Evaluate $\left(\dfrac{2}{5}\right)^{-2}$ .
c)
Simplify $\dfrac{(-2x^3)^4}{8x^{-2}}$ .

Stuck? Peek at a hint

$a^{-n}$ is the reciprocal $\frac{1}{a^n}$ . In (c), raise every factor inside the brackets to the 4th power, including the $-2$ .

K2 Rational Exponents

Evaluate without a calculator.

a)
$27^{2/3}$
b)
$16^{-3/4}$
c)
$\left(\dfrac{4}{9}\right)^{-3/2}$

Stuck? Peek at a hint

The denominator of the exponent is a root and the numerator is a power: take the root first, while the numbers are small.

K3 Simplifying

Simplify. Write each answer with positive exponents.

a)
$x^{1/2}\cdot x^{1/3}$
b)
$\dfrac{a^{3/4}b^{-2}}{a^{-1/4}b}$
c)
$\left(8x^6y^{-3}\right)^{-2/3}$

Stuck? Peek at a hint

Multiplying powers of the same base adds the exponents, so find a common denominator. In (c), the outer exponent applies to the $8$ too.

K4 Properties of $y=b^x$

a)
Determine which of these are exponential functions: $y=x^4$ , $y=4^x$ , $y=1^x$ , $y=(-4)^x$ , $y=0.4^x$ .
b)
For $f(x)=2^{-x}$ , state the domain and range in interval notation, the $y$ -intercept and the equation of the horizontal asymptote.
c)
Determine whether $f(x)=2^{-x}$ is increasing or decreasing, and evaluate $f(-3)$ .

Stuck? Peek at a hint

Rewrite $2^{-x}$ as a power with a single base between $0$ and $1$ .

K5 Transformations

Let $g(x)=2\cdot3^{2x-4}+1$ .

a)
Write $g$ in the form $y=a\cdot b^{k(x-d)}+c$ and state $a$ , $k$ , $d$ and $c$ .
b)
State the equation of the horizontal asymptote and the range of $g$ .
c)
The point $(1,3)$ is on $y=3^x$ . Determine its image on the graph of $g$ .

Stuck? Peek at a hint

Factor the coefficient of $x$ out of the exponent before you read any shift.

Real World (A)

Half-Life · Doubling · Linear vs Exponential

A1 Medical Tracer

A hospital in Hamilton prepares a dose of a medical tracer with an activity of 800 MBq. The tracer has a half-life of 6 hours.

a)
Write an equation for the activity $A$ , in MBq, after $t$ hours. Then find the activity one day after preparation.
b)
A scan needs an activity of at least 25 MBq. For how many hours after preparation can the dose be used?
c)
According to the model, when is the tracer completely gone? Does this match what happens to a real dose?

Stuck? Peek at a hint

Count half-lives: one day is $\frac{24}{6}$ of them. In (b), write $\frac{25}{800}$ as a power of $\frac{1}{2}$ .

A2 Petri Dish

A lab culture starts with 500 bacteria at 9:00 a.m. and doubles every 20 minutes.

a)
Write an equation for the number of bacteria $N$ after $t$ minutes.
b)
How many bacteria are there at 11:00 a.m.?
c)
At what time does the culture reach $256\,000$ bacteria?
d)
Evaluate $N(1440)$ , the model's count after one day, in scientific notation. Each bacterium has a mass of about $1\times10^{-12}$ g. Can the dish hold this many bacteria?

Stuck? Peek at a hint

The exponent counts doublings, so it is $\frac{t}{20}$ with $t$ in minutes.

A3 Two Towns

The table shows the populations of two Ontario towns $t$ years after 2020.

$t$	$0$	$1$	$2$	$3$
$\text{Town A}$	$12\,000$	$12\,600$	$13\,200$	$13\,800$
$\text{Town B}$	$8\,000$	$8\,800$	$9\,680$	$10\,648$

a)
Determine which town grows linearly and which grows exponentially. Write an equation for each population.
b)
In which year does Town B first have more people than Town A?
c)
Use each model to estimate the population in 1995 ( $t=-25$ ). Which estimate is impossible, and what does that tell you about that model?

Stuck? Peek at a hint

For each town, compute the differences and the ratios of consecutive populations.

Think It Through (T)

Error Analysis · Building Functions

T1 Spot the Flaw

A student solves $x^{2/3}=16$ :
"Raise both sides to the power $\frac{3}{2}$ : $x=16^{3/2}=64$ . So $x=64$ is the only solution."

a)
Evaluate $(-64)^{2/3}$ . What does the result tell you about the student's claim?
b)
Identify the step where the student lost a solution, and give both solutions.
c)
Solve $x^{2/5}=4$ and $x^{3/4}=-8$ .

Stuck? Peek at a hint

Write $x^{2/3}$ as $\left(x^{1/3}\right)^2$ .

T2 Build It Backwards

a)
The graph shows a function of the form $y=a\cdot2^x+c$ , with its asymptote dashed. Each grid square is 1 unit. Determine $a$ and $c$ .
b)
Write an exponential function with range $(-\infty,4)$ and $y$ -intercept $1$ . Many answers are possible.
c)
Write a decreasing exponential function with asymptote $y=2$ whose graph passes through $(1,5)$ . Many answers are possible.

Stuck? Peek at a hint

The asymptote gives $c$ . Then use the $y$ -intercept, where $2^x=1$ .

Boss Level

Optional challenge

Hidden Quadratics

Solve over the real numbers.

a)
$4^x-3\cdot2^{x+1}+8=0$
b)
$9^x-3^{x+1}=54$
c)
Find every value of $k$ for which $4^x-k\cdot2^x+9=0$ has exactly one real solution.

Stuck? Peek at a hint

Write $4^x$ as $\left(2^x\right)^2$ and $2^{x+1}$ as $2\cdot2^x$ , then let $u=2^x$ .

Answer key

Final answers only. Try first, then check.

Core Skills

a) $\dfrac{9}{8}$
b) $\dfrac{25}{4}$
c) $2x^{14}$

a) $9$
b) $\dfrac{1}{8}$
c) $\dfrac{27}{8}$

a) $x^{5/6}$
b) $\dfrac{a}{b^3}$
c) $\dfrac{y^2}{4x^4}$

a) $y=4^x$ and $y=0.4^x$
b) $D=(-\infty,\infty)$ , $R=(0,\infty)$ ; $y$ -intercept $1$ ; asymptote $y=0$
c) Decreasing; $f(-3)=8$

a) $g(x)=2\cdot3^{2(x-2)}+1$ ; $a=2$ , $k=2$ , $d=2$ , $c=1$
b) $y=1$ ; $R=(1,\infty)$
c) $\left(\dfrac{5}{2},7\right)$

Real World

a) $A(t)=800\left(\frac{1}{2}\right)^{t/6}$ ; $50$ MBq
b) $30$ hours
c) Never: $A(t)>0$ for every $t$ , since $A=0$ is an asymptote. A real dose has a finite number of atoms, so it does run out; the model only works while many atoms remain.

a) $N(t)=500(2)^{t/20}$
b) $32\,000$ bacteria
c) 12:00 noon ( $t=180$ min)
d) $N(1440)=500\cdot2^{72}\approx2.4\times10^{24}$ bacteria, about $2.4\times10^{12}$ g (2.4 million tonnes). Impossible: food and space run out long before, so the growth stops being exponential.

a) Town A is linear: $P_A(t)=12\,000+600t$ . Town B is exponential: $P_B(t)=8000(1.1)^t$ .
b) 2028 ( $t=8$ : $P_B\approx17\,149$ , $P_A=16\,800$ )
c) $P_A(-25)=-3000$ is impossible: the linear model can't be extended that far back (it reaches $0$ at $t=-20$ ). $P_B(-25)\approx738$ is possible.

Think It Through

a) $16$ , so $x=-64$ is also a solution: $64$ is not the only one.
b) Raising to $\frac{3}{2}$ hides a square root: $\left(x^{1/3}\right)^2=16$ gives $x^{1/3}=\pm4$ , so $x=\pm64$ .
c) $x=\pm32$ ; $x^{3/4}=-8$ has no real solution, since $x^{3/4}=\left(\sqrt[4]{x}\right)^3\ge0$ .

a) $a=2$ , $c=-3$
b) e.g. $y=-3(2)^x+4$
c) e.g. $y=6\left(\frac{1}{2}\right)^x+2$

Boss Level

Boss

a) $x=1$ or $x=2$
b) $x=2$
c) $k=6$

Free cheat sheet

Exponential Functions Cheat Sheet

The whole unit on one page: every rule, a worked example for each idea, and the traps that cost marks.

Unit Summary

One printable page with every rule.

↓ Unit Summary PDF

Build your own

The same page with blanks: fill it in from memory, then check.

↓ Build-your-own PDF

Phone card

The essentials on one screen. Save it to your photos.

↓ Phone card (image)

Most testedA car worth $32 000 loses $15\%$ of its value each year. What is it worth after $4$ years? (card 8)

1Exponent laws

$a^ma^n=a^{m+n}\qquad\dfrac{a^m}{a^n}=a^{m-n}$

$(a^m)^n=a^{mn}\qquad(ab)^n=a^nb^n$

UseKeep the base; never multiply bases. A power applies to every factor in the bracket, coefficients included.

Trap $4^2\cdot4^3=4^5$ , not $16^5$ : add the exponents, keep the base.

Trap $(-2x^2)^4=16x^8$ : an even power of a negative is positive.

2Zero and negative exponents

$a^0=1\qquad a^{-n}=\dfrac{1}{a^n}$

UseA negative exponent means a reciprocal, not a negative number. It applies only to its own base.

Trap $3x^{-2}=3/x^2$ , not $1/(3x^2)$ : the $-2$ belongs to $x$ only.

3Rational exponents

$a^{1/n}=\sqrt[n]{a}\qquad a^{m/n}=\left(\sqrt[n]{a}\right)^m$

UseTake the root first, then the power: the numbers stay small. The same laws apply: $x^{1/2}x^{1/3}=x^{5/6}$ .

Example $8^{-2/3}=1/(\sqrt[3]{8})^2=1/4$ .

Trap $x^{2/3}=9$ gives $x=\pm9^{3/2}=\pm27$ : an even numerator allows two answers.

4Scientific notation

Use $a\times10^n$ with $1\le a<10$ . Multiply the $a$ 's, add the exponents, then put $a$ back between $1$ and $10$ .

Trap $(3.0\times10^{-4})(5.0\times10^2)=15\times10^{-2}=1.5\times10^{-1}$ : a smaller $a$ raises the exponent.

5The graph of $y=b^x$

Use $b>0$ , $b\ne1$ . Domain $\mathbb{R}$ , range $y>0$ , $y$ -intercept $1$ , asymptote $y=0$ . $b>1$ grows; $0<b<1$ decays.

Trap $y=-5^x$ means $-(5^x)$ : at $x=2$ it is $-25$ , not $25$ .

6Which model fits?

UseEqual $x$ -steps: constant differences mean linear; constant second differences, quadratic; a constant ratio $y_{n+1}/y_n$ , exponential. Check every difference.

Example $5, 15, 45, 135$ : ratio $3$ , so $y=5(3)^x$ if the table starts at $x=0$ .

TrapIf the table starts at $x=1$ , $a$ is not the first $y$ -value: $a$ is the value at $x=0$ .

7Transformations

$y=a\,b^{k(x-d)}+c$

UseFactor $k$ out of the exponent first. Map $(x,y)\to(x/k+d,\ ay+c)$ . The asymptote moves to $y=c$ .

Trap $5^{3x-6}=5^{3(x-2)}$ : right $2$ , not $6$ .

8Growth and decay

$A=A_0(1+r)^t\qquad A=A_0(1-r)^t$

$A=A_0(2)^{t/d}\qquad A=A_0\left(\dfrac12\right)^{t/h}$

Use $d$ is the doubling time and $h$ the half-life, in the same units as $t$ .

Example$32 000 losing $15\%$ a year: $32\,000(0.85)^4\approx$ $16 704.20 after $4$ years.

TrapLosing $12\%$ a year: the base is $1-0.12=0.88$ , not $0.12$ .

TrapDoubling every $8$ h: the exponent is $t/8$ , not $8t$ .

Linear, quadratic or exponential?

Constant first differences Linear

Constant second differences Quadratic

Constant ratio Exponential

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Core Skills (K)

K1 Integer Exponents

K2 Rational Exponents

K3 Simplifying

K4 Properties of y=bxy=b^xy=bx

K5 Transformations

Real World (A)

A1 Medical Tracer

A2 Petri Dish

A3 Two Towns

Think It Through (T)

T1 Spot the Flaw

T2 Build It Backwards

Boss Level

Hidden Quadratics

Answer key

Core Skills

Real World

Think It Through

Boss Level

Exponential Functions Cheat Sheet

Unit Summary

Build your own

Phone card

1Exponent laws

2Zero and negative exponents

3Rational exponents

4Scientific notation

5The graph of y=bxy=b^xy=bx

6Which model fits?

7Transformations

8Growth and decay

Linear, quadratic or exponential?

More free MCR3U worksheets

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What is VertexMath?

K4 Properties of $y=b^x$

5The graph of $y=b^x$