Graphs of exponential equations of the form \\ y=a\left(B^{x-h}\right)+k (where a,\,B, and h and k are any number and B>0) are called exponential graphs.

Like lines, exponential graphs will always have a y-intercept. This is the point on the graph which touches the y-axis. We can find this by setting x=0 and finding the value of y. For example, the y-intercept of y=2^x is (0,1).

Similarly, we can look for x-intercepts by setting y=0 and then solving for x. Because this is an exponential equation, there could be 0 or 1 solutions, and there will be the same number of x-intercept. For example, the graph of y=2^x has no x-intercept.

Exponential graphs have a horizontal asymptote which is the horizontal line which the graph approaches but does not touch. For example, the horizontal asymptote of y=2^x is y=0, (the x-axis).

Examples

Example 1

Consider the equation y = 4^{x}.

Complete the table of values:

x	-3	-2	-1	0	1	2	3
y

Worked Solution

Create a strategy

Substitute each of the x-values in the table into the equation and evaluate.

Apply the idea

For x=-3:

\displaystyle y	\displaystyle =	\displaystyle 4^x
	\displaystyle =	\displaystyle 4^{-3}	Substitute x=-3
	\displaystyle =	\displaystyle \dfrac{1}{4^3}	Use the negative index rule
	\displaystyle =	\displaystyle \dfrac{1}{64}	Evaluate

Similarly, by substituting the remaining x-values into y=4^x, we get:

x	-3	-2	-1	0	1	\, 2	\, 3
y	\, \, \dfrac{1}{64}	\, \, \dfrac{1}{16}	\, \, \dfrac{1}{4}	1	4	16	64

Using some of these points, graph the equation y = 4^{x} on the number plane.

Worked Solution

Create a strategy

Plot some of the points in the table of values and draw the curve passing through each plotted point.

Apply the idea

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Some of the ordered pairs of points to be plotted on the coordinate plane are (-1,\dfrac{1}{4}), (0,1), and (1,4).

This curve of the equation y = 4^{x} must pass through each of the plotted points.

Which of the options completes the statement?

As x increases, the y-values ...

increase

decrease

stay the same

Worked Solution

Create a strategy

Look at the trend of the graph as x increases.

Apply the idea

As we move along the x-axis from left to right, the graph trend upwards. So this means that as x increases, the corresponding y-values increase.

So option A is the correct answer.

Which of the options completes the statement?

As x decreases, the y-values ...

increase

decrease

stay the same

Worked Solution

Create a strategy

Look at the trend of the graph as x decreases.

Apply the idea

As we move along the x-axis from right to left, the graph trend downwards. So this means that as x decreases, the corresponding y-values decrease.

So option B is the correct answer.

Which of the following statements is true?

The curve crosses the x-axis at a very small x-value that is beyond the scale of the graph shown.

The curve never crosses the x-axis.

The curve crosses the x-axis at exactly one point on the graph shown.

Worked Solution

Create a strategy

Find the x-intercept of the equation.

Apply the idea

\displaystyle y	\displaystyle =	\displaystyle 4^x	Write the equation
\displaystyle 0	\displaystyle =	\displaystyle 4^x	Substitute y=0

There is no value of x that satisfies the equation 4^x=0, since 4^x \gt 0 for all x. So the graph has no x-intercept and will never cross the x-axis.

So option B is the correct answer.

At what value of y does the graph cross the y-axis?

Worked Solution

Create a strategy

Find the y-value when x=0.

Apply the idea

The graph crosses the y-axis when x=0.

In part (a), we filled out the table of values for y=4^x. The y-value we found for x=0 is 1.

So the graph crosses the y-axis at y=1.

Idea summary

The graph of an exponential equation of the form y=A\left(B^{x-h}\right)+k is an exponential graph.

Exponential graphs have a y-intercept and can have 0 or 1 x-intercepts, depending on the solutions to the exponential equation.

Exponential graphs have a horizontal asymptote which is the horizontal line that the graph approaches but does not intersect.

Transformations of exponential graphs

An exponential graph can be vertically translated by increasing or decreasing the y-values by a constant number. So to translate y=2^x up by k units gives us y=2^x + k.

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This graph shows y=2^x translated vertically up by 2 to get y=2^{x} + 2, and down by 2 to get y=2^{x} -2.

Similarly, an exponential graph can be horizontally translated by increasing or decreasing the x-values by a constant number. However, the x-value together with the translation must both be in the index. That is, to translate y=2^x to the left by h units we get y=2^{x+h}.

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This graph shows y=2^{x} translated horizontally to the left by 2 to get y=2^{x+2}, and to the right by 2 to get y=2^{x-2}.

An exponential graph can be vertically scaled by multiplying every y-value by a constant number. So to expand the exponential graph y=2^x by a scale factor of a we get y=a\left(2^{x}\right). We can compress an exponential graph by dividing by the scale factor instead.

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This graph shows y=2^{x} vertically expanded by a scale factor of 4 to get y=4\left(2^{x}\right) and compressed by a scale factor of 4 to get y=\dfrac{1}{4} \left(2^{x}\right). Notice that in this case, we get equivalent graphs from the horizontal translations y=2^{x+2} and y=2^{x-2} respectively. This is a consequence of the index laws.

We can vertically reflect an exponential graph about the x-axis by taking the negative of the y-values. So to reflect y=2^x about the x-axis gives us y=-2^x. Notice that this is distinct from \\ y=(-2)^x which will not give us an exponential graph.

We can similarly horizontally reflect an exponential graph about the y-axis by taking the negative of the x-values. So to reflect y=2^x about the y-axis gives us y=2^{-x}.

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This graph shows y=2^{x} horizontally reflected or graph about the y-axis to get y=2^{-x} and vertically reflected or graph about the x-axis to get y=-2^{x}.

Exploration

Use the following applet to explore transformations of the graph of an exponential function by dragging the sliders.

Loading interactive...

Changing B changes the steepness of the graph. Changing A changes the steepness of the graph and negative values of A flip the curve vertically. Changing h shifts the curve horizontally, and changing k shifts the curve vertically.

Examples

Example 2

Consider the graph of y = 3^{-x}:

Find the y-value of the y-intercept of the curve 3^{-x}.

Worked Solution

Create a strategy

Substitute x=0 into the equation.

Apply the idea

\displaystyle y	\displaystyle =	\displaystyle 3^{-x}	Write the equation
	\displaystyle =	\displaystyle 3^{0}	Subsitute x=0
	\displaystyle =	\displaystyle 1	Evaluate

Fill in the table of values for 3^{-x}.

x	-3	-2	-1	0	1	2	3
y

Worked Solution

Create a strategy

Substitute each of the x-values in the table into 3^{-x} and evaluate.

Apply the idea

For x=-3:

\displaystyle y	\displaystyle =	\displaystyle 3^{-x}
	\displaystyle =	\displaystyle 3^{-(-3)}	Substitute -3 to x
	\displaystyle =	\displaystyle 3^{3}	Use the power to a power rule
	\displaystyle =	\displaystyle 27	Evaluate

Similarly, by substituting the remaining x-values into y=3^{-x}, we get:

x	- 3	- 2	- 1	0	1	2	3
y	27	9	3	1	\dfrac{1}{3}	\dfrac{1}{9}	\dfrac{1}{27}

Find the horizontal asymptote of the curve y=3^{-x}.

Worked Solution

Create a strategy

Find the value y approaches for large values of x.

Apply the idea

Refer to the table of values in part (b). Notice that when substituting larger and larger positive values of x into y=3^{-x}, the values of y decreases and approach 0. This means that the curve approaches y=0.

So the horizontal asymptote of the curve y=3^{-x} is y=0.

Hence plot the curve y=3^{-x}.

Worked Solution

Create a strategy

Plot some of the points in the table of values and draw the curve passing through each plotted point.

Apply the idea

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Some of the ordered pairs of points to be plotted on the coordinate plane are (-1,3), (0,1), and (3,\dfrac{1}{3}).

This curve of the equation y = 3^{-x} must pass through each of the plotted points and approaches the horizontal asymptote y=0.

Is the function y = 3^{-x}, an increasing or decreasing function?

Worked Solution

Create a strategy

Look at the trend of the graph as x increases.

Apply the idea

As we move along the x-axis from left to right, the graph trends downwards. So this means that as x increases, the y-values decrease.

So the function y = 3^{-x} is a decreasing function

Example 3

Consider the graph of y=3^{x} below.

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How do we shift the graph of y = 3^{x} to get the graph of y = 3^{x}-4?

Move the graph 4 units to the right.

Move the graph downwards by 4 units.

Move the graph 4 units to the left.

Move the graph upwards by 4 units.

Worked Solution

Create a strategy

Use the fact that to vertically translate y=3^{x} by k units we use the equation y=3^{x} + k.

Apply the idea

The given equation y = 3^{x}-4 means that k=-4. So the graph of y = 3^{x} is vertically translated downwards by 4 to get y = 3^{x}-4.

Option B is the correct answer.

Plot y = 3^{x}-4.

Worked Solution

Create a strategy

Sketch the original graph then translate it including its asymptote.

Apply the idea

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To translate the asymptote 4 units down, we move the asymptote from y=0 to y=-5.

The y-intercept (0,1) is translated downwards to(0,-4), along with the rest of the points on the graph of y = 2^{x}.

The shifted exponential graph is the lower graph shown here, labelled y = 2^{x}-5.

Idea summary

Exponential graphs can be transformed in the following ways (starting with the exponential graph defined by y=2^{x}):

Vertically translated by k units: y=2^{x} + k
Horizontally translated by h units: y=2^{x-h}
Vertically scaled by a scale factor of a: y=a(2^{x})
Vertically reflected about the x-axis: y=-2^{x}
Horizontally reflected about the y-axis: y=2^{-x}

Outcomes

ACMNA239

Explore the connection between algebraic and graphical representations of relations such as simple quadratics, circles and exponentials using digital technology as appropriate

4.03 Exponential graphs

Ideas

Features of exponential graphs

Examples

Example 1

Transformations of exponential graphs

Exploration

Examples

Example 2

Example 3

Outcomes

ACMNA239

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