Graphing exponential functions on the SAT

SAT Subscore: Passport to Advanced Mathematics

Exponential function is a function with a positive constant other than 1 raised to an exponent that includes a variable.

The basic form of the exponential function is f(x)=b^x (b is the base and x is the exponent).
The base b is always positive (b>0) and not equal to one (b≠1).
For example: the function f(x)=3^x is an exponential function where the base is a constant b=3 and the exponent is the variable x.

The y axis intercept of the basic exponential function graph f(x)=b^xis equal to 1 for all values of b.

An exponential function slope is always increasing or always decreasing. The slope form depends on the value of the base b. All graphs of exponential functions are curved.

The ends of the exponential function graph: The graph of a basic exponential function f(x)=b^x has a horizontal asymptote on one of its ends (positive x axis or negative x axis). The other end of the function approaches infinity. The end behavior depends on the value of the base b.

We can shift an exponential function graph by adding a constant to the function or by multiplying the exponential term by a coefficient, so that the basic function f(x)=b^x will become f(x)=a*b^x+d.

Continue reading this page for detailed explanations and examples.

Two types of exponential functions

The form of the graph of the basic exponential function f(x)=b^x depends on the size of the base b:

If b>1 then the slope of the graph is always positive (the graph is increasing), as x increases the graph approaches infinity and as x decreases the graph approaches zero (horizontal asymptote at x=0).

If 0<b<1 then the slope of the graph is always negative (the graph is decreasing), as x increases the graph approaches zero (horizontal asymptote at x=0) and as x decreases the graph approaches infinity.

Continue reading this page for detailed explanations and examples.

The features of exponential functions

Y intercept of an exponential function

Y intercept this is the point where the function crosses the y axis.

To find the y intercept, we set x=0 and evaluate the function.

Note that any number raised to a power of 0 is equal to 1, therefore in the basic function f(x)=b^x the intercept is equal to 1 for all values of b.

Consider the following example:

Find the y intercepts of a functions f(x)=3^x+1 and f(x)=3^x-5.

f(x)=3^x+1
f(0)=3⁰+1=1+1=2

f(x)=3^x-5
f(0)=3^x-5=1-5=-4

The slope of an exponential function

An exponential function slope is always increasing or always decreasing, therefore, to find the slope of the function we only need to evaluate 2 points from the function.

Let’s mark point 1 (x₁,y₁) and point 2 (x₂, y₂).

If the y value increases when the x value increases (exponential grows), then the slope of the function is positive, and the function graph is always increasing (meaning that x₂>x₁ and y₂>y₁).

If the y value decreases when the x value increases (exponential decay), then the slope of the function is negative, and the function graph is always decreasing (meaning that x₂>x₁ and y₂<y₁).

Consider the following example:

Determine if the slope of the function f(x)=3^x+1 is negative or positive.

f(x)=3^x+1
f(0)=3⁰+1=1+1=2
f(1)=3¹+1=3+1=4

The points are (0,2) and (1,4). We got a bigger y for a bigger x, therefore the slope of the function is positive.

Consider the following example:

Determine if the slope of the function f(x)=0.5^x+1 is negative or positive.

f(x)=0.5^x+1
f(0)=0.5⁰+1=1+1=2
f(1)=0.5¹+1=1.5

The points are (0,2) and (1,1.5). We got a smaller y for a bigger x, therefore the slope of the function is negative.

The graphs below show the functions f(x)=0.5^x+1 and f(x)=3^x+1.

The points x+0 and x=1 that were used to find the slopes are marked in red on the graphs.

The horizontal asymptote- the end behavior

Asymptote is a line that the graph approaches but never touches.

Horizontal asymptote is a horizontal line that the graph approaches when x gets very large or very small.

The graph of a basic exponential function f(x)=b^x has a horizontal asymptote on one of its ends (positive x axis or negative x axis):

If b>1 then the function f(x)=b^x has a horizontal asymptote at the negative end of the x axis.
When we raise a base that is bigger than 1 to a high negative exponent the output is a very small number, therefore the output of the function f(x)=b^x is very close to zero.
For example:
Given the function f(x)=b^x
if x=-4 and b=3 we get f(x)=3^-4=1/3⁴=0.012
if x=-10 and b=3 we get f(x)=3^-10=1/3¹⁰=0.00002

If 0<b<1 then the function f(x)=b^x has a horizontal asymptote at the positive end of the x axis.
When we raise a base that is smaller than 1 to a high positive exponent the output is a very small number, therefore the output of the function f(x)=b^x is very close to zero.
For example:
Given the function f(x)=b^x
if x=4 and b=0.5 we get f(x)=0.5⁴=0.5*0.5*0.5*0.5=0.063
if x=10 and b=0.5 we get f(x)=0.5¹⁰=0.5*0.5*0.5*0.5…*0.5=0.00098

Consider the following example:

Determine the asymptotes of the functions f(x)=3^x and f(x)=0.5^x.

The graphs below show the functions f(x)=0.5^x and f(x)=3^x.
The table near the graphs shows different (x,y) values that were taken to plot the graphs.
The horizontal asymptote x=0 is marked in red on the graph and inside the table.

For the graph f(x)=3^x small x values will get an output that is close to zero, therefore the function has a horizontal asymptote at x=0.

For the graph f(x)=0.5^x big x values will get an output that is close to zero, therefore the function has a horizontal asymptote at x=0.

Horizontal asymptote of a function with a constant term f(x)=bx+d

In this type of functions, a constant term d is added to the basic function f(x)=b^x so we get a function form of f(x)=b^x+d.

When the function has a constant term, the asymptote will approach the constant instead of zero.

For example: b values of 0<b<1:
if x=4, b=0.5 and c=5 we get f(x)=0.5⁴+5=5.063
if x=10, b=0.5 and c=-4 we get f(x)=0.5¹⁰-4=-4.00098

For example: b values of b>1:
if x=-4, b=3 and c=5 we get f(x)=3^-4+5=1/3⁴+5=5.012
if x=-10, b=3 and c=-4 we get f(x)=3^-10-4=1/3¹⁰-4=-4.00002

Consider the following example:

Determine the asymptotes of the functions f(x)=3^x+1 and f(x)=0.5^x+1.

The graphs below show the functions f(x)=0.5^x+1 and f(x)=3^x+1.
The table near the graphs shows different (x,y) values that were taken to plot the graphs.

The horizontal asymptote x=1 is marked in red on the graph and inside the table.

For the graph f(x)=3^x+1 small x values will get an output that is close to zero plus 1, therefore the function has a horizontal asymptote at x=1.

For the graph f(x)=0.5^x+1 big x values will get an output that is close to zero plus 1, therefore the function has a horizontal asymptote at x=1.

Graphing an exponential function steps

An exponential function f(x)=b^x behavior is divided into 2 areas by the y axis: a positive x area and a negative x area. Therefore, to graph an exponential function, we need to include a point from each area.

The steps for graphing an exponential function:
Step 1: Plotting points:
Evaluate and plot 3 points- its y intercept, a point with a positive x value (like x=1) and a point with a negative x value (like x=-1).
Step 2: Sketching a curve:
Sketch a curve between the 3 points and extend it on both sides. One end will approach a horizontal asymptote of zero along the x axis (if the graph is f(x)=b^x+d the asymptote will be x=k instead of x=0). The other and will approach infinity along the y axis.

The graphs below show the functions f(x)=0.5^x and f(x)=3^x.
The table near the graphs shows different (x,y) values that were taken to plot the graphs.
The 3 points from the different areas that were needed to plot the graphs (x=0, x=3 and x=-3) are marked in red and blue in the table and on the graphs.

Both functions f(x)=3^x and f(x)=0.5^xhave a y intercept at a point (1,0).

The green function f(x)=3^x where b>1 has big y outputs at positive x values (approaching to infinity) and small y outputs at negative x values (approaching to zero).

The orange function f(x)=0.5^x where 0<b<1 has big y outputs at negative x values (approaching to infinity) and small y outputs at positive x values (approaching to zero).

Shifting an exponential function

We can shift an exponential function by adding a constant to the function or by multiplying the exponential term by a coefficient.

Shifting an exponential function- adding a constant term

To shift an exponential function up we need to add a constant term to the basic function f(x)=b^x getting f(x)=b^x+d.

To shift an exponential function down we need to subtract a constant term from the basic function f(x)=b^x getting f(x)=b^x-d.

The shifting results:
The y axis intercept shifts up or down d units.
The asymptote shifts up or down d units.
The other end of the function that approaches infinity remains unchanged.

In the basic function f(x)=b^x the horizontal asymptote is x=0 and the y intercept is y=1.
In the function f(x)=b^x+d the horizontal asymptote will be x=d and the y intercept will be y=1+d.
In the function f(x)=b^x-d the horizontal asymptote will be x=-d and the y intercept will be y=1-d.

The graphs below show the functions f(x)=0.5^x , f(x)=0.5^x+10, f(x)=3^x and f(x)=3^x+10.
The table near the graphs shows different (x,y) values that were taken to plot the graphs.

The y axis intersection points are marked in red in the table and on the graph. The y axis intersection point before shifting was y=1 and after the shifting it became y=d+1=10+1=11.

Every point on both functions shifted up by the size of the constant d (d=10), you can see the change comparing the columns of the tables. Two points from each function are marked in blue in the tables and on the graphs as an example.

The asymptotes of both functions shifted up by the size of the constant d (d=10), they are marked in red in the table and on the graphs.

Shifting an exponential function- multiplying by a coefficient

We can shift an exponential function by multiplying the exponential term by a coefficient so that the function f(x)=b^x will become f(x)=a*b^x.

The shifting results:
The y axis intercept will become a*1=a.
The asymptote remains unchanged (x=0).
The other end of the function that approaches infinity remains unchanged.

The graphs below show the functions f(x)=0.5^x , f(x)=5*0.5^x, f(x)=3^x and f(x)=5*3^x.
The table near the graphs shows different (x,y) values that were taken to plot the graphs.

The y axis intersection points are marked in red in the table and on the graph. The y axis intersection point before shifting was y=1 and after the shifting it became y=a*1=5.

Note that the shifting size of each point is different, since bigger x values result in a bigger output and therefore a bigger shifting, you can see the change comparing the columns of the tables.

The asymptotes of both functions remained unchanged, they are marked in black in the tables and on the graphs.

Shifting an exponential function- adding a constant term and multiplying by a coefficient

The basic function f(x)=b^x will become f(x)=a*b^x+d.

The shifting results:
The y axis intercept will become a*1+d=a+d.
The asymptote will be x=d.
The other end of the function that approaches infinity remains unchanged.

The graphs below show the functions f(x)=0.5^x , f(x)=5*0.5^x+10, f(x)=3^x and f(x)=5*3^x+10.
The table near the graphs shows different (x,y) values that were taken to plot the graphs.

The y axis intersection points are marked in red in the table and on the graphs. The y axis intersection point before shifting was y=1 and after the shifting it became y=a*1+d=5*1+10=15.

Note that the shifting size of each point is different, since bigger x values result in a bigger output and therefore a bigger shifting, you can see the change comparing the columns of the tables.

The asymptotes of both functions shifted up by the size of the constant d (d=10), they are marked in black in the table and on the graphs.