Graphing Exponents

An exponent refers to the power a number is raised to. For example, the number 5 raised to the power of 3 is 5³, which means 5 × 5 × 5 = 125

In an exponential function, the output depends on an expression where the variable appears in the exponent. For example, y = 5^x is an exponential function as the variable x is in the exponent to base 5. 

Exponential functions are nonlinear because their slopes are always changing. They appear as curves rather than straight lines on their graphs.

A graph of an exponential function will always reach positive or negative infinity on one end and approach but never reach a horizontal line on the other. The horizontal line that the graph approaches but never reaches is called the horizontal asymptote.

An exponential function can be of two types. It is either always increasing or always decreasing. Accordingly, they are called Exponential Growth Function and Exponential Decay Function, respectively.

Exponential Growth Functions 

If f(1) > f(0), then the slope of the graph is positive and is thus an exponential growth function.

Let us graph the exponential function f(x) = 5^x

Step 1: Choosing Values of x

x	f(x)
-2	0.04
-1	0.2
0	1
1	5
2	25

The output values in the table change as the input increases by 1.

Step 2: Plotting the Points

Now, plotting the points in the graph, we get

The graph shows that y = 5^x is an increasing function, which means it increases as x increases and is thus an exponential growth function.

Slope

Since f(1) > f(0), the slope of the above graph is positive.

Domain

The domain consists of all real numbers. Thus, it is written in interval notation as (-∞, ∞)

End Behaviour

The value of y on the right end of the graph approaches infinity. Thus, the range is (0, ∞)

Horizontal Asymptote

The horizontal asymptote of the function is the x-axis (i.e., y = 0)

Exponential Decay Functions

If f(1) < f(0), then the slope of the graph is negative and is thus an exponential decay function.

Let us graph the exponential function f(x) = 5^-x

Step 1: Choosing Values of x

x	f(x)
-1	5
0	1
1	0.2
2	0.04
3	0.008

Step 2: Plotting the Points

Now, plotting the points in the graph, we get

The graph shows that y = 5^-x is a decreasing function, which means it decreases as x increases and is thus an exponential decay function.  

Slope

Since f(1) < f(0), the above graph is negative.

Domain

The domain consists of all real numbers. Thus, it is written in interval notation as (-∞, ∞)

End Behaviour

The value of y on the left end of the graph approaches infinity. Thus, the range is (0, ∞)

Horizontal Asymptote

The horizontal asymptote of the function is the x-axis (i.e., y = 0)

Using Transformations

As with other parent functions, exponential graphs can be transformed by shifting, reflecting, stretching, and compressing relative to the parent function f(x) = b^x 

Vertical Shift

For the parent function f(x) = b^x, vertical translation shifts the graph up or down by adding or subtracting a constant d to the equation. The translated equation becomes 

• f(x) = b^x + d for the upward shift
• f(x) = b^x – d for the downward shift

Here, the parent function f(x) = 5^x is shifted 3 units up and 3 units down. The translated equation becomes f(x) = 5^x + 3 and f(x) = 5^x – 3 respectively. 

The domain (-∞, ∞) remains unchanged. The range for f(x) = 5^x + 3 is (3, ∞) and the range for f(x) = 5^x – 3 is (-3, ∞)

Horizontal Shift

For the parent function f(x) = b^x, horizontal translation shifts the graph to the left or right by adding or subtracting a constant d to the exponent. The translated equation becomes 

• f(x) = b^{x + d} for the left shift
• f(x) = b^{x – d} for the right shift

The domain (-∞, ∞) remains unchanged. 

Stretching or Compressing

A stretch or compression occurs when we multiply the parent function f(x) = b^x by a constant |a| > 0

• The function will become f(x) = ab^x, where a > 0 after stretching
• The function will become f(x) = ${\dfrac{1}{a}b^{x}}$, where a > 0 after compressing

Here, 

f(x) = 2(5^x) stretches the parent function to f(x) = 5^x 

f(x) = ${\dfrac{1}{2}\left( 5^{x}\right)}$ compresses the parent function to f(x) = 5^x 

Reflecting

Exponential functions can also be reflected along the x-axis or the y-axis. 

• When we multiply the parent function f(x) = b^x by –1, we get a reflection along the x-axis. 
• When we multiply the exponent of the parent function f(x) = b^x by –1, we get a reflection along the y-axis.

Here, the parent function f(x) = 2^x is reflected along the x and the y-axis.