Why do exponential function graphs not have an [katex] \textbf{x} [/katex]-intercept?

There is no value of [katex] x [/katex] when [katex] y [/katex] is equivalent to [katex] 0 [/katex] in an exponential function.

How is the equation [katex] \bf{5}^\textbf{x}=\textbf{y} [/katex] different from [katex] \bf{5}^\textbf{x}+\bf{5}=\textbf{y}? [/katex]

The addition of the constant [katex] 5 [/katex] causes a horizontal shift in the graph of the function.

3. Identify the correct graph for the equation: $ y=10^x. $

4. Identify the correct graph for the equation: $ y=\left(\cfrac{1}{5}\right)^{x}. $

Exponential Function Graph - Math Steps, Examples & Questions

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Exponential function graph

Here you will learn about exponential function graphs, including recognizing and sketching exponential function graphs, plotting and interpreting exponential function graphs.

Students will first learn about exponential function graphs as part of functions in functions in high school.

What is an exponential function graph?

An exponential function graph is a representation of an exponential function of the form $y=k^x$ where $x$ and $y$ are variables and $k$ is a constant (a numerical value).

$x$ is the exponent and $k$ is the base.

The graph of an exponential function can represent either exponential growth or exponential decay.

When $k$ is greater than $1,$ it is a growth curve. It can be used to represent population growth or compound interest. As $x$ increases, so does $y.$

For example,

y=2^x

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When $k$ is less than $1,$ it is a decay curve. It can be used to represent (for example) radioactive decay.

For example,

y=(\cfrac{1}{2})^x

The $y$ -intercept $($ at $x=0)$ is $1$ since anything raised to the power $0$ is $1.$

See also: Anything to the power of $0$

The $x$ -axis is an asymptote to the curve. The curve gets very close to the horizontal asymptote but does not touch it. This is because $y$ ≠ $0.$

$y=k^x$ is a basic exponential function. There are more complex exponential functions of the form $y=ab^x.$

The graphs look similar to the ones above, they have an exponent $x,$ they have base $b$ and the $y$ -intercept is $a \, ($ this is because when $x=0, y=a\times{b^0}=a\times{1}=a).$ For example, $y=4\times 3^x$

What is an exponential function graph?

Common Core State Standards

How does this relate to high school math?

Functions – Interpreting Functions (HS.F-IF.C.7e)
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

How to recognize an exponential graph

In order to recognize an exponential graph:

Identify any linear/ quadratic/ cubic or inverse (reciprocal) functions.
Identify the exponential function.

Recognizing an exponential function graph examples

Example 1: recognize exponential function graphs

Which is the correct equation for the graph?

y=x^2+3 \hspace{1.5cm} y=3x+1 \hspace{1.5cm} y=\cfrac{3}{x} \hspace{1.5cm} y=3^x

Identify any linear/ quadratic/ cubic or inverse (reciprocal) functions.

$y=x^{2}+3$ is a quadratic function; its graph would be a parabola.

$y=3x+1$ is a linear function; its graph would be a straight line.

$y=\cfrac{3}{x}$ is an inverse function; its graph would be a hyperbola.

2Identify the exponential function.

$y=3^{x}$ is an exponential function. It has $x$ as the exponent and the base is $3.$ The base is greater than $1,$ so the graph is a growth curve.

Example 2: recognize exponential graphs

Which is the correct equation for the graph?

y=1-3x \hspace{1.5cm} y=x^2-3 \hspace{1.5cm} y=(\cfrac{1}{3})^x \hspace{1.5cm} y=x^3

Identify any linear/ quadratic/ cubic or inverse (reciprocal) functions.

$y=1-3x$ is a linear function; its graph would be a straight line.

$y=x^{2}-3$ is a quadratic function; its graph would be a parabola.

$y=x^{3}$ is a cubic function.

Identify the exponential function.

$y\left(\cfrac{1}{3}\right)^{x}$ is an exponential function. It has $x$ as the exponent and the base is $\cfrac{1}{3}.$ The base is less than $1,$ so the graph is a decay curve.

How to plot an exponential graph

In order to plot an exponential function graph:

Complete the table of values.
Plot the coordinates.
Draw a smooth curve through the points.

Plotting an exponential function graph examples

Example 3: plot exponential graphs

Draw the curve of $y=2^x$ for $- \, 1\leq{x}\leq3.$

Complete the table of values.

Plot the coordinates.

Draw an $x$ -axis and a $y$ -axis. It is best practice to use a pencil and plot the coordinates using small crosses.

The coordinates would be $(- \, 1, 0.5), (0, 1)$ and so on.

Draw a smooth curve through the points.

Use a pencil and turn the paper around if it makes it easier for you.

Note: Technically anything can be used as the initial values, but to see the $y$ -intercept and the overall behavior of the curve, values around $x=0$ are the best to use. These values are also typically the easiest to calculate when finding the equation of an exponential function $($ shown in Example $5$ and $6.)$

Example 4: plot exponential graphs

Draw the curve of $y=\left(\cfrac{1}{2}\right)^x$ for $- \, 2\leq{x}\leq2.$

Complete the table of values.

Plot the coordinates.

Draw an $x$ -axis and a $y$ -axis. It is best practice to use a pencil and plot the coordinates using small crosses.

The coordinates would be $(- \, 2, 4), (- \, 1, 2)$ and so on.

Draw a smooth curve through the points.

Use a pencil and turn the paper around if it makes it easier for you.

How to find the equation of an exponential function graph

In order to find the equation of an exponential function graph:

Substitute the pairs of values into the given equation.
Solve the two simultaneous equations.
Write down the equation of the exponential function.

Finding the equation of an exponential function graph example

Example 5: finding the equation of an exponential graph.

The sketch shows a curve with equation $y=ab^{x}$ where $a$ and $b$ are constants and $b>0.$ Find the equation of the curve.

Substitute the pairs of values into the given equation.

The given equation is $y=ab^{x}.$ Substituting $(0, 2)$ into it would become

$2=ab^0$

Substituting $(4, 162)$ into the equation would become

$162=ab^4$

Solve the two simultaneous equations.

2=ab^0

Since anything to the power $0$ is $1,$

$2=a\times{b}^{0}=a\times{1}$ so $a=1.$

Now look at the second equation $162=ab^4.$ Since $a=2,$ this becomes

$162=2\times b^4$

This can be solved to find the base $b.$

\begin{aligned}162&=2\times{b^4} \\\\ 81&=b^4 \\\\ \sqrt[4]{81}&=b \\\\ b&=3 \end{aligned}

Write down the equation of the exponential function.

The original equation was $y=ab^x.$ Since $a=2$ and $b=3,$ the equation is:

$y=2\times 3^x$

Example 6: finding the equation of an exponential graph.

The sketch shows a curve with equation $y=ab^x$ where $a$ and $b$ are constants and $b>0.$ Find the equation of the curve.

Substitute the pairs of values into the given equation.

The given equation is $y=ab^x.$ Substituting $(1, 10)$ into it would become

$10=ab^1$

Substituting $(3, 40)$ into it would become

$40=ab^3$

Solve the two simultaneous equations.

10=ab^1

Since anything to the power $1$ is itself, this becomes

$10=ab$

Now look at the second equation.

$40=ab^3$

Solve these two equations by dividing one by the other to eliminate $a.$

$\cfrac{40}{10}=\cfrac{ab^3}{ab}$

This becomes,

$4=b^2$

This can be solved to find the base $b$ .

\begin{aligned}4&=b^2 \\\\ \sqrt{4}&=b \\\\ b&=2 \end{aligned}

Substitute $b=2$ into one of the equations to find $a.$

\begin{aligned}10&=ab \\\\ 10&=a\times{2} \\\\ a&=5 \end{aligned}

Write down the equation of the exponential function.

The original equation was $y=ab^x.$ Since $a=5$ and $b=2,$ the equation is:

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

Teaching tips for exponential function graph

Start with easier subsets of real numbers as the values for $a$ and $b$ before moving on to more complex rational functions, so that students can focus on the overall shape of the function instead of the calculations themselves.

Choose worksheets that have a variety of question types and show the exponential function graph in a variety of ways.

Let struggling students use a graphing calculator to explore patterns in exponential function graphs and to check their work until they are confident solving on their own.

Easy mistakes to make

Using straight lines when graphing exponential functions
Do not join up the plotted points with straight lines like this

The points should be joined by a single, smooth curve.

All points are not on the curve
If there is a point that does not join up with the other points when you draw a smooth curve – go back and check the coordinate. In the example below there is a point that does not lie on the smooth curve so needs checking.

Applying the exponent to both variables in an exponential equation
It is common to confuse $y=ab^{x}$ with $y=(ab)^{x}.$ However, each will result in different $y$ values when substituting into a calculator or plotting a graph. $y=ab^{x}$ means $y=a\times(b^x)$ and treating it like $y=(ab)^{x}$ will not result in the same function.

Confusing exponent notation
This mistake is particularly common for fractional exponents and negative exponents.

For example, substituting $\cfrac{1}{2}$ for $x$ in the function $y=2\left(9^x\right),$ is equal to $y=2\cdot\sqrt{9}=2(3)=6.$

The $\cfrac{1}{2}$ fraction as an exponent operates like a square root and the square root of $9$ is $3.$

Substituting $- \, 2$ for $x$ in the function $y=2\left(9^x\right),$ is equal to $y=2\cdot\cfrac{1}{9^2}=\cfrac{2}{81}.$

The $- \, 2$ as an exponent operates the reciprocal to be multiplied by itself $2$ times.

Practice exponential graph questions

y=6^x

y=6x+1

y={x^2}+1

y=\cfrac{6}{x}

The curve is a growth curve; its equation will be an exponential function.

$y=6^{x}$ is an exponential function.

$y=6x+1$ is a linear function; its graph would be a straight line.

$y=x^{2}+1$ is a quadratic function; its graph would be a parabola.

$y=\cfrac{6}{x}$ is an inverse function; its graph would be a hyperbola.

y=1-10 x

y=1-x^2

y=0.1^x

y=x^3+1

The curve is a decay curve; its equation will be an exponential function.

$y=0.1^{x}$ is an exponential function.

$y=1-10x$ is a linear function; its graph would be a straight line.

$y=x^{2}+1$ is a quadratic function; its graph would be a parabola.

$y=x^{3}+1$ is a cubic function.

Exponential Function Graph 18 US A

Exponential Function Graph 18 US C

Exponential Function Graph 18 US B

Exponential Function Graph 18 US D

A correct table of values would be:

Exponential Function Graph 19 US

These $x$ -values and $y$ -values give coordinates such as $(0, 1), (1, 10)$ and $(2, 100).$

The coordinates are plotted on the grid. A smooth curve should be drawn through the points. The curve is increasing, because it is a growth function.

Exponential Function Graph 20 US A

Exponential Function Graph 20 US C

Exponential Function Graph 20 US B

Exponential Function Graph 20 US D

A correct table of values would be:

Exponential Function Graph 21 US

These $x$ -values and $y$ -values give coordinates such as $(- \, 2, 25), (- \, 1, 5)$ and so on.

The coordinates are plotted on the grid. A smooth curve should be drawn through the points. The curve is decreasing, because it is a decay function.

y=4\times{2^x}

y=4\times{3^x}

y=2\times{4^x}

y=3\times{4^x}

Substituting the coordinates into the given equation gives

4=ab^0=a

Now we need to find the base $b$

\begin{aligned}32&=ab^{3} \\\\ 32&=4\times{b^3} \\\\ 8&=b^{3} \\\\ b&=\sqrt[3]{8} \\\\ b&=2 \end{aligned}

So the equation is:

y=4\times{2^x}

y=7\times{2^x}

y=2\times{4^x}

y=7\times{3^x}

y=3\times{4^x}

Substituting the coordinates into the given equation gives

21=ab^1=ab

189=ab^3

Dividing the two equations to eliminate $a$

\begin{aligned}\cfrac{189}{21}&=\cfrac{ab^3}{ab} \\\\ 9&=b^2 \\\\ b&=\sqrt{9} \\\\ b&=3 \end{aligned}

Substitute $b=3$ into one of the equations to find $a.$

\begin{aligned}21&=ab \\\\ 21&=a\times{7} \\\\ a&=3 \end{aligned}

So the equation is:

y=7\times{3^x}

Exponential function graphs FAQs

Why do exponential function graphs not have an $\textbf{x}$ -intercept?

There is no value of $x$ when $y$ is equivalent to $0$ in an exponential function.

Is an asymptote always a horizontal line?

No, in different types of equations the asymptote may be vertical or any type of linear equation.

How is the equation $\bf{5}^\textbf{x}=\textbf{y}$ different from $\bf{5}^\textbf{x}+\bf{5}=\textbf{y}?$

The addition of the constant $5$ causes a horizontal shift in the graph of the function.

Exponential Function Graph 24 US

The next lessons are

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