Make a table of values for $y = 3^x$ from $x = -2$ to $x = 2$, and state the asymptote, domain and range.

Points $\left(-2, \frac{1}{9}\right)$, $\left(-1, \frac{1}{3}\right)$, $(0, 1)$, $(1, 3)$, $(2, 9)$; asymptote $y = 0$; domain all real numbers; range $y > 0$.

Describe how $y = 2^{x + 3} - 4$ transforms $y = 2^x$, and state its asymptote and range.

Left $3$ and down $4$; asymptote $y = -4$; range $y > -4$.

For $y = -2(3)^{x - 1} + 5$, state the asymptote, domain and range, and find the $y$-intercept.

Asymptote $y = 5$; domain all real numbers; range $y < 5$; $y$-intercept $\frac{13}{3}$.

Graph $y = \frac{1}{2}(4)^{2x - 4} + 1$ by mapping the key points $\left(-1, \frac{1}{4}\right)$, $(0, 1)$ and $(1, 4)$ of $y = 4^x$.

Points $\left(\frac{3}{2}, \frac{9}{8}\right)$, $\left(2, \frac{3}{2}\right)$, $\left(\frac{5}{2}, 3\right)$; asymptote $y = 1$; range $y > 1$.

The graph of $y = 2^x$ is reflected in the $x$-axis, compressed horizontally by a factor of $\frac{1}{3}$, then shifted $2$ units right and $8$ units up. Write its equation, state the asymptote and range, and find both intercepts.

$y = -2^{3(x - 2)} + 8$; asymptote $y = 8$; range $y < 8$; $x$-intercept $3$; $y$-intercept $\frac{511}{64} \approx 7.98$.

Grade 11 · MCR3U · Functions

MCR3U: Graphing Exponential Functions

An exponential function $y = b^x$ , with $b > 0$ and $b \ne 1$ , multiplies by the same factor every time $x$ goes up by $1$ . Graphing one comes down to a few key points and the asymptote, and every transformation in the unit moves those.

How It Works

The base function. $y = 2^x$ passes through $\left(-1, \frac{1}{2}\right)$ , $(0, 1)$ , $(1, 2)$ and $(2, 4)$ . It rises from left to right, and to the left it gets closer and closer to the $x$ -axis without touching it. Its horizontal asymptote is $y = 0$ , its domain is all real numbers and its range is $y > 0$ .

Growth or decay. If $b > 1$ the graph increases; if $0 < b < 1$ it decreases. $y = \left(\frac{1}{2}\right)^x$ is $y = 2^{-x}$ , the reflection of $y = 2^x$ in the $y$ -axis.

The transformed form is $y = a(b)^{k(x - d)} + c$ , read like any other transformation. $a$ stretches vertically, and reflects in the $x$ -axis if it is negative. $k$ compresses horizontally by a factor of $\frac{1}{k}$ , and reflects in the $y$ -axis if it is negative. $d$ shifts right and $c$ shifts up. Factor $k$ out of the exponent first: $2^{3x - 6} = 2^{3(x - 2)}$ .

Mapping rule. $(x, y) \to \left(\frac{x}{k} + d, \; ay + c\right)$ . Push the key points of the base function through it and join them smoothly.

The asymptote. Only $c$ moves it. Stretches, reflections and horizontal shifts leave $y = 0$ where it is, so the new asymptote is $y = c$ . Draw it first, as a dashed line.

Domain and range. The domain stays all real numbers. The range is $y > c$ when $a > 0$ , and $y < c$ when $a < 0$ because the graph is flipped below the asymptote.

Intercepts. Substitute $x = 0$ for the $y$ -intercept: it checks the sketch, and tests often ask for it.

5 Worked Examples

Easy first, test level last. Try each one on paper, then open the solution.

Easy
Make a table of values for $y = 3^x$ from $x = -2$ to $x = 2$ , and state the asymptote, domain and range.
Show solution
1. $3^{-2} = \frac{1}{9}$ , $3^{-1} = \frac{1}{3}$ , $3^0 = 1$ , $3^1 = 3$ , $3^2 = 9$ .
2. The values approach $0$ to the left but never reach it.
Answer: Points $\left(-2, \frac{1}{9}\right)$ , $\left(-1, \frac{1}{3}\right)$ , $(0, 1)$ , $(1, 3)$ , $(2, 9)$ ; asymptote $y = 0$ ; domain all real numbers; range $y > 0$ .
Easy
Describe how $y = 2^{x + 3} - 4$ transforms $y = 2^x$ , and state its asymptote and range.
Show solution
1. $x + 3 = x - (-3)$ , so $d = -3$ : a shift left $3$ .
2. $c = -4$ : a shift down $4$ , which moves the asymptote to $y = -4$ .
Answer: Left $3$ and down $4$ ; asymptote $y = -4$ ; range $y > -4$ .
Medium
For $y = -2(3)^{x - 1} + 5$ , state the asymptote, domain and range, and find the $y$ -intercept.
Show solution
1. $c = 5$ , so the asymptote is $y = 5$ .
2. $a = -2 < 0$ flips the graph below the asymptote, so the range is $y < 5$ . The domain is all real numbers.
3. At $x = 0$ : $y = -2(3)^{-1} + 5 = -\frac{2}{3} + 5 = \frac{13}{3}$ .
Answer: Asymptote $y = 5$ ; domain all real numbers; range $y < 5$ ; $y$ -intercept $\frac{13}{3}$ .
Medium
Graph $y = \frac{1}{2}(4)^{2x - 4} + 1$ by mapping the key points $\left(-1, \frac{1}{4}\right)$ , $(0, 1)$ and $(1, 4)$ of $y = 4^x$ .
Show solution
1. Factor the exponent: $2x - 4 = 2(x - 2)$ , so $a = \frac{1}{2}$ , $k = 2$ , $d = 2$ , $c = 1$ .
2. Mapping rule: $(x, y) \to \left(\frac{x}{2} + 2, \; \frac{y}{2} + 1\right)$ .
3. $\left(-1, \frac{1}{4}\right) \to \left(\frac{3}{2}, \frac{9}{8}\right)$ , $(0, 1) \to \left(2, \frac{3}{2}\right)$ , $(1, 4) \to \left(\frac{5}{2}, 3\right)$ .
4. The asymptote is $y = 1$ and the graph rises above it.
Answer: Points $\left(\frac{3}{2}, \frac{9}{8}\right)$ , $\left(2, \frac{3}{2}\right)$ , $\left(\frac{5}{2}, 3\right)$ ; asymptote $y = 1$ ; range $y > 1$ .
Test level
The graph of $y = 2^x$ is reflected in the $x$ -axis, compressed horizontally by a factor of $\frac{1}{3}$ , then shifted $2$ units right and $8$ units up. Write its equation, state the asymptote and range, and find both intercepts.
Show solution
1. $a = -1$ , $k = 3$ , $d = 2$ , $c = 8$ : $y = -2^{3(x - 2)} + 8$ .
2. Asymptote $y = 8$ ; with $a < 0$ the range is $y < 8$ .
3. $x$ -intercept: $2^{3(x - 2)} = 8 = 2^3$ , so $3(x - 2) = 3$ and $x = 3$ .
4. $y$ -intercept: $y = -2^{-6} + 8 = 8 - \frac{1}{64} = \frac{511}{64}$ .
Answer: $y = -2^{3(x - 2)} + 8$ ; asymptote $y = 8$ ; range $y < 8$ ; $x$ -intercept $3$ ; $y$ -intercept $\frac{511}{64} \approx 7.98$ .

Learn It on Martian Lab

Martian Lab teaches this in the Transformations of Exponential Functions lesson of its MCR3U Exponential Functions unit. Watch it, then take the Exponential Functions practice tests to see if it stuck.

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Free Exponential Functions practice test, with solutions

More Exponential Functions topics: Exponent Laws and Rational Exponents · Exponential Growth and Decay

More MCR3U topics: Inverse Functions · Transformations of Functions · Sine Law and the Ambiguous Case · Domain, Range and Function Notation · Simplifying Rational Expressions · Operations with Rational Expressions · Max/Min and Vertex Form · Linear-Quadratic Systems · Quadratic Word Problems · Simplifying Radicals · Special Angles and the CAST Rule · Cosine Law and 3D Problems · Transformations of Sine and Cosine · Sinusoidal Word Problems · Arithmetic Sequences and Series · Geometric Sequences and Series · Pascal's Triangle and the Binomial Theorem · Simple and Compound Interest · Annuities: Future and Present Value

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