Last modified on November 28th, 2024

chapter outline

 

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 = 5x 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) = 5x

Step 1: Choosing Values of x

xf(x)
-20.04
-10.2
01
15
225

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

How to Graph Exponents

The graph shows that y = 5x 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)

Example 1: Graph y = 2x 

Choosing the values for x,

xy
-20.25
-10.5
01
12
24

After plotting the points, we get the required graph.

Graphing Exponents Example 1

The graph shows that y = 2x 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 Behavior: 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

xf(x)
-15
01
10.2
20.04
30.008

Step 2: Plotting the Points

Now, plotting the points in the graph, we get

Graphing Negative Exponents

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)

Example 2: Graph y = 2-x 

Choosing the values for x,

xy
-20.25
-10.5
01
12
24

After plotting the points, we get the required graph.

Graphing Exponents Example 2

The graph shows that y = 2-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 slope of the above graph is negative.
  • Domain: The domain consists of all real numbers. Thus, it is written in interval notation as (-∞, ∞)
  • End Behavior: 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)

Transformations

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

Vertical Shift

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

  • f(x) = bx + d for the upward shift
  • f(x) = bx – d for the downward shift
Vertical Shift of Exponent

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

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

Horizontal Shift

For the parent function f(x) = bx, 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) = bx + d for the left shift
  • f(x) = bx – d for the right shift
Horizontal Shift of Exponent

The domain (-∞, ∞) remains unchanged. 

Stretching and Compressing

Stretching or compressing occurs when we multiply the parent function f(x) = bx by a constant |a| > 0

  • The function will become f(x) = abx, where a > 0 after stretching
  • The function will become f(x) = ${\dfrac{1}{a}b^{x}}$, where a > 0 after compressing
Stretching or Compressing of Exponent

Here, 

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

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

Reflecting

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

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

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

Example 3: Given the exponential function f(x) = 3x, describe the transformations for the following functions and sketch their graphs:

a) g(x) = 3x + 5 

b) h(x) = 3x – 4

c) j(x) = -3x + 5

d) k(x) = 3 · 3x – 2 

Given the exponential function is f(x) = 3x 

a) As we know, the horizontal shift of the parent function f(x) = bx gives the translated equations:

  • f(x) = bx + d for the left shift
  • f(x) = bx – d for the right shift

Here, g(x) = 3x + 5 represents the horizontally shifted function 5 units to the left relative to the parent function f(x) = 3x  

After plotting them, we get the required graph.

Graphing Exponents Example 3a

b) As we know, the vertical shift to the parent function f(x) = bx gives the translated equations:

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

Here, h(x) = 3x – 4 represents the vertically shifted function 4 units downward relative to the parent function f(x) = 3x  

After plotting them, we get the required graph.

Graphing Exponents Example 3b

c) As we know, the vertical shift to the parent function f(x) = bx gives the translated equations:

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

Also, if we multiply the parent function f(x) = bx by –1, it is reflected along the x-axis.

Here, j(x) = -3x + 5 has a negative sign in front of 3x and is in the form j(x) = -bx + d

Thus, j(x) = -3x + 5 is a reflection of the parent function f(x) = 3x along the x-axis. It is then vertically shifted up by 5 units.

After plotting them, we get the required graph.

Graphing Exponents Example 3c

d) As we know, the horizontal shift to the parent function f(x) = bx gives the translated equations:

  • f(x) = bx + d for the left shift
  • f(x) = bx – d for the right shift

Also, if we multiply the parent function by a constant a (here, a > 0),  it stretches f(x), and the translated equation becomes f(x) = abx 

Here, k(x) = 3 · 3x – 2 vertically stretches the graph, making it steeper than the original function f(x) = 3x. It is then horizontally shifted 2 units to the right relative to f(x).

Graphing Exponents Example 3d

Last modified on November 28th, 2024