MCR3U · Unit 2 test review

Exponential Functions: Test Review

Everything for the Unit 2 test on one page: the key facts from all 8 lessons, the mistakes that cost the most marks, and a practice test that points you back to the lessons you need.

Key facts

Exponent Laws

Product law: $a^{m} \cdot a^{n} = a^{m + n}$ (same base — add exponents).
Quotient law: $\frac{a ^{m}}{a ^{n}} = a^{m - n}$ (same base — subtract exponents).
Power of a power: $(a^{m})^{n} = a^{mn}$ (multiply exponents).
Zero exponent: $a^{0} = 1$ for any $a \neq = 0$ . Negative exponent: $a^{- n} = \frac{1}{a ^{n}}$ .
Rational exponent: $a^{m / n} = (n a)^{m}$ — denominator is the root, numerator is the power.
Power of a product or quotient: $(ab)^{n} = a^{n} b^{n}$ and $(\frac{a}{b})^{n} = \frac{a ^{n}}{b ^{n}}$ — but $(a + b)^{n} \neq = a^{n} + b^{n}$ .

Rational Exponents and Radicals

$a^{1/ n} = n a$ , the principal (non-negative) root when $n$ is even.
$a^{m / n} = (n a)^{m} = n a^{m}$ : the denominator is the root, the numerator is the power.
$a^{- m / n} = \frac{1}{a ^{m / n}}$ and $(\frac{a}{b})^{- m / n} = (\frac{b}{a})^{m / n}$ : flip first, and the value stays positive.
An even root of a negative number is not real; an odd root of a negative is negative: $(- 27)^{1/3} = - 3$ .
By hand, take the root before the power: $2 7^{4/3} = 3^{4} = 81$ .
The exponent laws still apply; add, subtract or multiply the fractional exponents (for $x > 0$ ).

Graphs of Exponential Functions

$y = b^{x}$ , $b > 0$ , $b \neq = 1$ : passes through $(0, 1)$ , horizontal asymptote $y = 0$ .
$b > 1$ : exponential growth (increasing). $0 < b < 1$ : exponential decay (decreasing).
Domain of $y = b^{x}$ : all real numbers. Range: $y > 0$ .
An exponential growth function ( $b > 1$ ) eventually exceeds any polynomial function, no matter the polynomial's degree.
$b = 1$ is excluded since $1^{x} = 1$ is constant, not exponential.
In a table with equal $x$ -steps: constant first differences mean linear, constant ratios mean exponential (the ratio is $b$ when $x$ steps by 1).

Linear, Quadratic or Exponential?

With equal $x$ -steps: constant first differences mean linear, constant second differences mean quadratic, constant ratios mean exponential.
An exponential's differences are exponential too: $a b^{x + 1} - a b^{x} = (b - 1) a b^{x},$ so every column of differences has the same ratio $b$ and is never constant.
For $y = a x^{2} + b x + c$ with $x$ -steps of 1, the second difference is $2 a$ . With a step of $h$ : linear differences are $mh$ , quadratic second differences $2 a h^{2}$ , exponential ratios $b^{h}$ .
Uneven $x$ -steps: compare $\frac{Δ y}{Δ x}$ for a line, and take the $h$ -th root of a ratio across a step of $h$ for an exponential.
Real data are exponential if the ratios are roughly constant with no trend; ratios drifting steadily toward 1 point to a polynomial.
An exponential with $b > 1$ eventually exceeds every linear and quadratic function, because its step-to-step factor stays $b$ while theirs shrinks toward 1.

Transformations of Exponential Functions

$y = a \cdot b^{k (x - d)} + c$ : same parameter roles as any transformed function.
The horizontal asymptote is always $y = c$ — read it directly off the equation.
Find the $y$ -intercept by substituting $x = 0$ and simplifying — there is no general shortcut.
$a < 0$ reflects in the $x$ -axis; $k < 0$ reflects in the $y$ -axis.
$2^{- x} = (1/2)^{x}$ — reflecting a growth curve in the $y$ -axis produces a decay-looking curve with a reciprocal base.
Mapping rule: $(x, y) \to (\frac{x}{k} + d, a y + c)$ . Range: $y > c$ if $a > 0$ , $y < c$ if $a < 0$ .

Finding an Exponential Equation

From the $y$ -intercept and one point: $a$ is the $y$ -intercept, then solve $a \cdot b^{x_{1}} = y_{1}$ for $b$ (positive root only).
From two points: $\frac{y _{2}}{y _{1}} = b^{x_{2} - x_{1}}$ cancels $a$ . Find $b$ first, then $a = \frac{y _{1}}{b ^{x_{1}}}$ .
Table with step $h$ and ratio $r$ : $b = r^{1/ h}$ , or write the model exactly as $y = a \cdot r^{x / h}$ .
Known asymptote $y = c$ : subtract $c$ from every $y$ -value, find $a$ and $b$ from $y - c = a \cdot b^{x}$ . The $y$ -intercept is then $a + c$ .
$a$ is the starting value and $b$ the factor per unit; $b - 1$ is the percent change per unit ( $b = 0.956$ means about 4.4% decrease).
Exponential regression fits $y = a \cdot b^{x}$ to all the data; check the residual pattern and keep predictions close to the data's range.

Solving Exponential Equations

Matching-base rule: if $b^{m} = b^{n}$ with $b > 0$ and $b \neq = 1$ , then $m = n$ , because $y = b^{x}$ never repeats an output.
Rewrite with the smallest common base, using negative exponents for reciprocals and rational exponents for roots: $\frac{1}{8} = 2^{- 3}$ , $39 = 3^{2/3}$ .
Replacing a base with a power multiplies the whole exponent: $(2^{3})^{x + 1} = 2^{3 x + 3}$ .
Isolate the power before matching. A coefficient is not part of the base: $3 (2)^{x} \neq = 6^{x}$ .
For a sum or difference of powers of one base, factor out the lowest power: $2^{x + 2} - 2^{x} = 2^{x} (2^{2} - 1)$ .
$b^{x} > 0$ always, so $b^{x}$ equal to a negative number or 0 has no solution; reject such factors.

Exponential Growth and Decay Applications

General model: $A (t) = A_{0} (1 + r)^{t}$ — growth if $r > 0$ , decay if $r < 0$ (or use $(1 - r)^{t}$ for a stated percentage decrease).
Half-life model: $A (t) = A_{0} (\frac{1}{2})^{t / h}$ , where $h$ is the half-life.
Doubling model: $A (t) = A_{0} (2)^{t / d}$ , where $d$ is the doubling time.
Matching bases: if $b^{m} = b^{n}$ then $m = n$ — rewrite both sides as powers of the same base to solve exactly.
Always keep $t$ and the rate's time period consistent (e.g. an annual rate needs $t$ measured in years).
Percent change per period = base − 1: a base of $0.92$ is 8% decay; a base of $1.075$ is 7.5% growth.

Where marks get lost

Thinking a negative exponent makes the value negative — it actually moves the base to the other side of a fraction (reciprocal), and the value stays positive if the base is positive. (Exponent Laws)
Applying the product law to different bases (e.g. treating $2^{3} \cdot 3^{2}$ as if the exponents could combine — they can't unless the bases match). (Exponent Laws)
Forgetting that $a^{0} = 1$ applies to the whole base, not to a coefficient in front of it (e.g. in $3 x^{0}$ , only $x^{0} = 1$ ; the 3 stays). (Exponent Laws)
Computing the power before the root in a rational exponent when the root would have given a much simpler number to work with. (Exponent Laws)
Using the laws across addition: writing $(x + 3)^{2} = x^{2} + 9$ or $2^{3} + 2^{4} = 2^{7}$ . The laws only combine products and quotients; $(x + 3)^{2} = x^{2} + 6 x + 9$ and $2^{3} + 2^{4} = 24$ . (Exponent Laws)
Swapping the root and the power: writing $3 x^{2}$ as $x^{3/2}$ . The index of the root always goes in the denominator, so $3 x^{2} = x^{2/3}$ . (Rational Exponents and Radicals)
Treating a negative exponent as a negative answer: $1 6^{- 1/2}$ is $\frac{1}{4}$ , not $- 4$ . The minus sign in the exponent means reciprocal. (Rational Exponents and Radicals)
Applying the exponent to the numerator only: $(\frac{4}{9})^{3/2} = \frac{8}{9}$ is wrong. Both parts get the exponent: $\frac{4 ^{3/2}}{9 ^{3/2}} = \frac{8}{27}$ . (Rational Exponents and Radicals)
Mixing up which negatives fail: $(- 27)^{1/3} = - 3$ is real (odd root), while $(- 16)^{1/2}$ is not real (even root). Writing $(- 16)^{1/2} = - 4$ is wrong because $(- 4)^{2} = 16$ , not $- 16$ . (Rational Exponents and Radicals)
Getting the fraction arithmetic wrong: $x^{1/2} \cdot x^{1/3}$ is $x^{5/6}$ (add, using sixths), not $x^{1/6}$ (that multiplies) and not $x^{2/5}$ (that adds tops and bottoms). (Rational Exponents and Radicals)
Believing an exponential graph touches or crosses the $x$ -axis — it only ever approaches $y = 0$ , never reaches it. (Graphs of Exponential Functions)
Assuming a graph that rises is automatically growth without checking whether the base could still be between 0 and 1 with a negative leading coefficient (a different situation, covered under transformations). (Graphs of Exponential Functions)
Confusing the roles of base and exponent — thinking $y = x^{2}$ (polynomial) and $y = 2^{x}$ (exponential) behave the same way for all $x$ . (Graphs of Exponential Functions)
Forgetting that the range of $y = b^{x}$ excludes zero and all negative numbers. (Graphs of Exponential Functions)
Testing a table for an exponential by looking at differences. The first differences of $1, 3, 9, 27$ are $2, 6, 18$ (not constant), which tells you only that it is not linear; you must divide consecutive values to see the constant ratio 3. (Graphs of Exponential Functions)
Running differences or ratios on a table whose $x$ -values are not equally spaced, then concluding the data fit no pattern. Check the $x$ -steps first and correct for uneven ones. (Linear, Quadratic or Exponential?)
Deciding "the first differences aren't constant, so it must be exponential". Non-constant first differences only rule out linear; a quadratic also fails that test, so check second differences and ratios too. (Linear, Quadratic or Exponential?)
Deciding from three points. $(0, 2), (1, 4), (2, 8)$ fit both $y = 2^{x + 1}$ and $y = x^{2} + x + 2$ ; only a fourth value (16 or 14) tells them apart. (Linear, Quadratic or Exponential?)
Rejecting real data as "not exponential" because the ratios are not exactly equal, or accepting it when the ratios drift steadily in one direction. Look for small scatter around one value, not perfect equality. (Linear, Quadratic or Exponential?)
Assuming a function with a big coefficient, such as $1000 x$ or $100 x^{2}$ , stays ahead of $2^{x}$ forever because it is ahead for the first few values. (Linear, Quadratic or Exponential?)
Assuming the asymptote is always $y = 0$ , forgetting that $c$ relocates it to $y = c$ . (Transformations of Exponential Functions)
Trying to find the $y$ -intercept using a shortcut (like reading off a constant term) instead of actually substituting $x = 0$ . (Transformations of Exponential Functions)
Mixing up which parameter reflects in which axis ( $a$ for $x$ -axis, $k$ for $y$ -axis — same convention as every other function family). (Transformations of Exponential Functions)
Forgetting that the range shifts along with the asymptote — writing $y > 0$ even after a vertical translation has moved it. (Transformations of Exponential Functions)
Reading the horizontal shift before factoring out $k$ : in $y = 2^{3 x - 6}$ the shift is 2 right, because $3 x - 6 = 3 (x - 2)$ , not 6 right. (Transformations of Exponential Functions)
Taking $a$ to be the first $y$ -value in a table that does not start at $x = 0$ . If the first row is $x = 2$ , step back to $x = 0$ by dividing by $b^{2}$ . (Finding an Exponential Equation)
Using a table's ratio as $b$ when the $x$ -step is 2 or 5. A ratio of 1.5 per 5 units means $b = 1. 5^{1/5} \approx 1.0845$ , not 1.5. (Finding an Exponential Equation)
Forgetting to subtract the asymptote. For a drink cooling toward 21 °C, the ratios of the raw temperatures are not constant; the ratios of (temperature − 21) are. (Finding an Exponential Equation)
Keeping a negative root: $b^{2} = 9$ gives $b = 3$ only, since an exponential base must be positive. (Finding an Exponential Equation)
Rounding $b$ early, or trusting a regression model far outside the range of the data (a town of 8000 that "reaches 280 000 by 2100"). (Finding an Exponential Equation)
Dropping the brackets when replacing a base: $(2^{3})^{x + 1}$ written as $2^{3 x + 1}$ instead of $2^{3 x + 3}$ . The 3 multiplies the whole exponent. (Solving Exponential Equations)
Merging a coefficient into the base: turning $3 (2)^{x} = 48$ into $6^{x} = 48$ . Only the 2 is raised to the power, so divide by 3 first: $2^{x} = 16$ , $x = 4$ . (Solving Exponential Equations)
Trying to match exponents across a sum, or writing $2^{x + 2} - 2^{x} = 2^{2}$ . Factor out the common power instead: $2^{x} (4 - 1)$ . (Solving Exponential Equations)
"Solving" an impossible equation: answering $x = - 2$ for $3^{x} = - 9$ , or keeping $2^{x} = - 3$ as a solution. Since $b^{x} > 0$ , these have no solution and must be rejected. (Solving Exponential Equations)
Rewriting fractions and roots wrongly, such as $\frac{1}{8} = 2^{1/3}$ or $2 = 2^{2}$ . Reciprocals give negative exponents ( $\frac{1}{8} = 2^{- 3}$ ) and roots give fractional ones ( $2 = 2^{1/2}$ ). (Solving Exponential Equations)
Using $r$ directly as the base instead of $1 + r$ (or $1 - r$ ) — e.g. writing $(0.03)^{t}$ instead of $(1.03)^{t}$ for 3% growth. (Exponential Growth and Decay Applications)
Mismatching the units of $t$ and the rate (using $t$ in months with an annual rate, without converting). (Exponential Growth and Decay Applications)
In half-life/doubling problems, forgetting to divide $t$ by the half-life or doubling time inside the exponent. (Exponential Growth and Decay Applications)
When matching bases, forgetting to apply the power-of-a-power rule correctly after rewriting one side (e.g. leaving $(2^{2})^{2 x - 1}$ instead of simplifying to $2^{4 x - 2}$ ). (Exponential Growth and Decay Applications)
Matching bases before isolating the power: in $3 (2)^{x} = 48$ you must divide by 3 first ( $2^{x} = 16$ , so $x = 4$ ); combining it into $6^{x} = 48$ is wrong because the 3 is not raised to the power $x$ . (Exponential Growth and Decay Applications)

Practice test

24 questions from across the unit. Each one shows the lesson it comes from.

Exponent Laws
1
Evaluate $1 6^{3/4}$ .
Exponent Laws
2
Evaluate $3 2^{2/5}$ .
Exponent Laws
3
What is the value of $- 1 6^{1/4}$ ?
Rational Exponents and Radicals
4
Evaluate $8 1^{3/4}$ without a calculator.
Rational Exponents and Radicals
5
Simplify $x^{2/3} \cdot x^{1/2}$ for $x > 0$ . Write the answer as a single power of $x$ .
Rational Exponents and Radicals
6
Match each radical to its exponent form.
1. A. $x^{- 1/3}$
2. B. $x^{1/2}$
3. C. $x^{3/2}$
4. D. $x^{3/4}$
$x$
$\frac{1}{3 x}$
$(4 x)^{3}$
$x^{3}$
Graphs of Exponential Functions
7
Which function represents exponential decay?
Graphs of Exponential Functions
8
The graphs of $y = 2^{x}$ and $y = x^{2}$ meet at $x = 2$ and at one larger whole number. What is that larger value of $x$ ?
Graphs of Exponential Functions
9
Match each function to its description.
$y = 3^{x}$
$y = (0.4)^{x}$
$y = 3 x + 1$
$y = x^{2}$
Linear, Quadratic or Exponential?
10
For $x = 0, 1, 2, 3, 4$ a table gives $y = 7, 10, 15, 22, 31$ . What type of function is it?
Linear, Quadratic or Exponential?
11
Each data set was measured at equal time intervals. Which is best modelled by an exponential function?
Linear, Quadratic or Exponential?
12
A table has $x = 0, 1, 3, 4, 7$ and $y = 1, 3, 7, 9, 15$ . The first differences are $2, 4, 2, 6$ . What should you conclude?
Transformations of Exponential Functions
13
State the horizontal asymptote of $y = 5 (2)^{x} - 7$ .
Transformations of Exponential Functions
14
Reflecting y = 2ˣ in the y-axis produces a graph identical to which function?
Transformations of Exponential Functions
15
Match each function to its range.
1. A. $y < - 1$
2. B. $y < 3$
3. C. $y > - 1$
4. D. $y > 3$
$y = 2^{x} + 3$
$y = - 2^{x} + 3$
$y = 5 (2)^{x} - 1$
$y = - (0.5)^{x} - 1$
Finding an Exponential Equation
16
A town's population model $P = 8200 (1.04)^{t}$ was fitted to data from 2010 ( $t = 0$ ) to 2020. Which prediction from it deserves the most trust?
Finding an Exponential Equation
17
An exponential function passes through $(0, 3)$ and $(1, 6)$ . Write its equation in the form $y = a (b)^{x}$ .
Finding an Exponential Equation
18
Order the steps to find $y = a \cdot b^{x}$ through $(2, 36)$ and $(5, 972)$ .
1. 1Check: $4 (3)^{5} = 972$
2. 2Substitute back: $36 = 9 a$ , so $a = 4$
3. 3Take the cube root: $b = 3$
4. 4Write $36 = a b^{2}$ and $972 = a b^{5}$
5. 5Divide to cancel $a$ : $b^{3} = 27$
Solving Exponential Equations
19
Solve $2 5^{x} = 12 5^{x - 1}$ .
Solving Exponential Equations
20
Solve $5^{x + 1} + 5^{x} = 750$ .
Solving Exponential Equations
21
Solve $2 7^{2 x} = 9^{x + 4}$ .
Exponential Growth and Decay Applications
22
An investment of 5000 dollars grows at 4% per year. Find its value after 3 years, to the nearest cent.
dollars
Exponential Growth and Decay Applications
23
Solve $9^{x} = 27$ by matching bases (write x as a fraction).
Exponential Growth and Decay Applications
24
A bacteria culture of 500 cells doubles every 20 minutes. After how many minutes will there be 16000 cells?
min

Endless drill

Fresh numbers every time, for the calculation questions in this unit.

Endless drillnew numbers every time

Generating a problem…

Exponential Functions: Test Review

Key facts

Where marks get lost

Practice test

Endless drill

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