9.02 Exponential graphs
Graphs of exponential equations such as $$y=3x have the $$x in the power. These types of functions are called exponential functions.
The exponential graph defined by $$y=2x
Features of exponential graphs
Like lines, exponential graphs will always have a $$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 $$y=2x is $$(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-intercepts. For example, the graph of $$y=2x has no $$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 $$y=2x is $$y=0
Translate exponential graphs up and down the y-axis
An exponential graph can be vertically translated (moved up and down) by increasing or decreasing the $$y-values by a constant number. So to move $$y=2x up by $$k units we use the function $$y=2x+k.
Vertically translating up by $$2 ($$y=2x+2) and down by $$2 ($$y=2x−2
Reflect exponential graphs about the x or y-axis
We can vertically reflect an exponential graph about the $$x-axis by taking the negative of the $$y-values. So to reflect $$y=2x about the $$x-axis gives us $$y=−2x.
We can similarly horizontally reflect an exponential graph about the $$y-axis by taking the negative of the $$x-values. So to reflect $$y=2x about the $$y-axis gives us $$y=2−x. Note that this is the same function as $$y=(12)x . Why?
Reflecting the exponential graph about the $$y-axis ($$y=2−x) and about the $$x-axis ($$y=−2x)
Exponential graphs have a $$y-intercept and can have $$0 or $$1$$x-intercepts, depending on the solutions to the exponential equation.
Exponential graphs have a horizontal asymptote which is the horizontal line that the graph approaches but does not intersect.
Exponential graphs can be transformed in a number of ways including the following (starting with the exponential graph defined by $$y=2x):
- Vertically translated by $$k units: $$y=2x+k
- Vertically reflected about the $$x-axis: $$y=−2x
- Horizontally reflected about the $$y-axis: $$y=2−x
Practice questions
Question 1
Consider the equation $$y=4x.
Complete the table of values.
$$x $$−3 $$−2 $$−1 $$0 $$1 $$2 $$3 $$y $$ $$ $$ $$ $$ $$ $$ Using some of these points, graph the equation $$y=4x on the number plane.
Loading Graph...Which of the options completes the statement?
As $$x increases, the $$y-values
increase
Adecrease
Bstay the same
CWhich of the options completes the statement?
As $$x decreases, the $$y-values
increase
Adecrease
Bstay the same
CWhich 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.
AThe curve never crosses the $$x-axis.
BThe curve crosses the $$x-axis at exactly one point on the graph shown.
CAt what value of $$y does the graph cross the $$y-axis?
Question 2
Consider the function $$y=3−x :
Find the $$y-value of the $$y-intercept of the curve $$y=3−x.
Fill in the table of values for $$y=3−x.
$$x $$−3 $$−2 $$−1 $$0 $$1 $$2 $$3 $$y $$ $$ $$ $$ $$ $$ $$ Find the horizontal asymptote of the curve $$y=3−x.
Hence plot the curve $$y=3−x.
Loading Graph...Is the function $$y=3−x, an increasing or decreasing function?
Increasing function
ADecreasing function
B
Question 3
Consider the graph of $$y=2x below.
How do we shift the graph of $$y=2x to get the graph of $$y=2x−5?
Move the graph upwards by $$5 units.
AMove the graph downwards by $$5 units.
BMove the graph $$5 units to the left.
CMove the graph $$5 units to the right.
DThen, plot $$y=2x−5.
The graph of $$y=2x is shown for reference.Loading Graph...