Graphs of Exponential Functions
Learning Outcomes
- Determine whether an exponential function and its associated graph represents growth or decay.
- Sketch a graph of an exponential function.
- Graph exponential functions shifted horizontally or vertically and write the associated equation.
- Graph a stretched or compressed exponential function.
- Graph a reflected exponential function.
- Write the equation of an exponential function that has been transformed.
Characteristics of Graphs of Exponential Functions
Before we begin graphing, it is helpful to review the behavior of exponential growth. Recall the table of values for a function of the form whose base is greater than one. We’ll use the function . Observe how the output values in the table below change as the input increases by 1.x | –3 | –2 | –1 | 0 | 1 | 2 | 3 |
1 | 2 | 4 | 8 |
- the output values are positive for all values of x
- as x increases, the output values increase without bound
- as x decreases, the output values grow smaller, approaching zero
x | –3 | –2 | –1 | 0 | 1 | 2 | 3 |
8 | 4 | 2 | 1 |
- the output values are positive for all values of x
- as x increases, the output values grow smaller, approaching zero
- as x decreases, the output values grow without bound
A General Note: Characteristics of the Graph of the Parent Function
An exponential function with the form , , , has these characteristics:- one-to-one function
- horizontal asymptote:
- domain:
- range:
- x-intercept: none
- y-intercept:
- increasing if
- decreasing if
How To: Given an exponential function of the form , graph the function
- Create a table of points.
- Plot at least 3 point from the table including the y-intercept .
- Draw a smooth curve through the points.
- State the domain, , the range, , and the horizontal asymptote, .
Example: Sketching the Graph of an Exponential Function of the Form
Sketch a graph of . State the domain, range, and asymptote.Answer: Before graphing, identify the behavior and create a table of points for the graph.
- Since b = 0.25 is between zero and one, we know the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote y = 0.
- Create a table of points.
x –3 –2 –1 0 1 2 3 64 16 4 1 0.25 0.0625 0.015625 - Plot the y-intercept, , along with two other points. We can use and .
Try It
Sketch the graph of . State the domain, range, and asymptote.Answer: The domain is , the range is , and the horizontal asymptote is .
Graphing Exponential Functions Using Transformations
Transformations of exponential graphs behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations—shifts, reflections, stretches, and compressions—to the parent function without loss of shape. For instance, just as the quadratic function maintains its parabolic shape when shifted, reflected, stretched, or compressed, the exponential function also maintains its general shape regardless of the transformations applied.Graphing a Vertical Shift
The first transformation occurs when we add a constant d to the parent function giving us a vertical shift d units in the same direction as the sign. For example, if we begin by graphing a parent function, , we can then graph two vertical shifts alongside it using : the upward shift, and the downward shift, . Both vertical shifts are shown in the figure below. Observe the results of shifting vertically:- The domain remains unchanged.
- When the function is shifted up 3 units giving :
- The y-intercept shifts up 3 units to .
- The asymptote shifts up 3 units to .
- The range becomes .
- When the function is shifted down 3 units giving :
- The y-intercept shifts down 3 units to .
- The asymptote also shifts down 3 units to .
- The range becomes .
Graphing a Horizontal Shift
The next transformation occurs when we add a constant c to the input of the parent function giving us a horizontal shift c units in the opposite direction of the sign. For example, if we begin by graphing the parent function , we can then graph two horizontal shifts alongside it using : the shift left, , and the shift right, . Both horizontal shifts are shown in the graph below. Observe the results of shifting horizontally:- The domain, , remains unchanged.
- The asymptote, , remains unchanged.
- The y-intercept shifts such that:
- When the function is shifted left 3 units to , the y-intercept becomes . This is because , so the initial value of the function is 8.
- When the function is shifted right 3 units to , the y-intercept becomes . Again, see that , so the initial value of the function is .
A General Note: Shifts of the Parent Function
For any constants c and d, the function shifts the parent function- shifts the parent function vertically d units, in the same direction as the sign of d.
- shifts the parent function horizontally c units, in the opposite direction as the sign of c.
- has a y-intercept of .
- has a horizontal asymptote of y = d.
- has a range of .
- has a domain of which remains unchanged.
How To: Given an exponential function with the form , graph the translation
- Draw the horizontal asymptote y = d.
- Shift the graph of left c units if c is positive and right units if c is negative.
- Shift the graph of up d units if d is positive and down d units if d is negative.
- State the domain, , the range, , and the horizontal asymptote .
Example: Graphing a Shift of an Exponential Function
Graph . State the domain, range, and asymptote.Answer: We have an exponential equation of the form , with , , and . Draw the horizontal asymptote , so draw . Identify the shift; it is .
Stretching, Compressing, or Reflecting an Exponential Function
While horizontal and vertical shifts involve adding constants to the input or to the function itself, a stretch or compression occurs when we multiply the parent function by a constant . For example, if we begin by graphing the parent function , we can then graph the stretch, using , to get and the compression, using , to get .A General Note: Stretches and Compressions of the Parent Function
The function- is stretched vertically by a factor of a if .
- is compressed vertically by a factor of a if .
- has a y-intercept is .
- has a horizontal asymptote of , range of , and domain of which are all unchanged from the parent function.
Example: Graphing the Stretch of an Exponential Function
Sketch a graph of . State the domain, range, and asymptote.Answer: Before graphing, identify the behavior and key points on the graph.
- Since is between zero and one, the left tail of the graph will increase without bound as x decreases, and the right tail will approach the x-axis as x increases.
- Since a = 4, the graph of will be stretched vertically by a factor of 4.
- Create a table of points:
x –3 –2 –1 0 1 2 3 32 16 8 4 2 1 0.5 - Plot the y-intercept, , along with two other points. We can use and .
- Draw a smooth curve connecting the points.
Try It
Sketch the graph of . State the domain, range, and asymptote.Answer: The domain is ; the range is ; the horizontal asymptote is .
Graphing Reflections
In addition to shifting, compressing, and stretching a graph, we can also reflect it about the x-axis or the y-axis. When we multiply the parent function by –1, we get a reflection about the x-axis. When we multiply the input by –1, we get a reflection about the y-axis. For example, if we begin by graphing the parent function , we can then graph the two reflections alongside it. The reflection about the x-axis, , and the reflection about the y-axis, , are both shown below.A General Note: Reflecting the Parent Function
The function- reflects the parent function about the x-axis.
- has a y-intercept of .
- has a range of .
- has a horizontal asymptote of and domain of which are unchanged from the parent function.
- reflects the parent function about the y-axis.
- has a y-intercept of , a horizontal asymptote at , a range of , and a domain of which are unchanged from the parent function.
Example: Writing and Graphing the Reflection of an Exponential Function
Find and graph the equation for a function, , that reflects about the x-axis. State its domain, range, and asymptote.Answer: Since we want to reflect the parent function about the x-axis, we multiply by –1 to get . Next we create a table of points.
–3 | –2 | –1 | 0 | 1 | 2 | 3 | |
–64 | –16 | –4 | –1 | –0.25 | –0.0625 | –0.0156 |
Summarizing Transformations of the Exponential Function
Now that we have worked with each type of translation for the exponential function, we can summarize them to arrive at the general equation for transforming exponential functions.Transformations of the Parent Function | |
---|---|
Transformation | Form |
Shift
|
|
Stretch and Compression
|
|
Reflection about the x-axis | |
Reflection about the y-axis | |
General equation for all transformations |
A General Note: Transformations of Exponential Functions
A transformation of an exponential function has the form , where the parent function, , , is- shifted horizontally c units to the left.
- stretched vertically by a factor of if .
- compressed vertically by a factor of if .
- shifted vertically d units.
- reflected about the x-axis when a < 0.
Example: Writing a Function from a Description
Write the equation for the function described below. Give the horizontal asymptote, domain, and range.- is vertically stretched by a factor of 2, reflected across the y-axis, and then shifted up 4 units.
Answer: We want to find an equation of the general form . We use the description provided to find a, b, c, and d.
- We are given the parent function , so b = e.
- The function is stretched by a factor of 2, so a = 2.
- The function is reflected about the y-axis. We replace x with –x to get: .
- The graph is shifted vertically 4 units, so d = 4.
The domain is ; the range is ; the horizontal asymptote is .
Try It
Write the equation for the function described below. Give the horizontal asymptote, the domain, and the range.- is compressed vertically by a factor of , reflected across the x-axis, and then shifted down 2 units.
Answer: ; the domain is ; the range is ; the horizontal asymptote is .
Using a Graph to Approximate a Solution to an Exponential Equation
Graphing can help you confirm or find the solution to an exponential equation. For example, can be solved to find the specific value for x that makes it a true statement. Graphing along with in the same window, the point(s) of intersection if any represent the solutions of the equation. To use a calculator to solve this, press [Y=] and enter next to Y1=. Then enter 42 next to Y2=. For a window, use the values –3 to 3 for and –5 to 55 for.Press [GRAPH]. The graphs should intersect somewhere near. For a better approximation, press [2ND] then [CALC]. Select [5: intersect] and press [ENTER] three times. The x-coordinate of the point of intersection is displayed as 2.1661943. (Your answer may be different if you use a different window or use a different value for Guess?) To the nearest thousandth,x≈2.166.Try It
Solve graphically. Round to the nearest thousandth.Answer:
Key Equations
General Form for the Transformation of the Parent Function |
Key Concepts
- The graph of the function has a y-intercept at , domain of , range of , and horizontal asymptote of .
- If , the function is increasing. The left tail of the graph will approach the asymptote , and the right tail will increase without bound.
- If 0 < b < 1, the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote .
- The equation represents a vertical shift of the parent function .
- The equation represents a horizontal shift of the parent function .
- The equation , where , represents a vertical stretch if or compression if of the parent function .
- When the parent function is multiplied by –1, the result, , is a reflection about the x-axis. When the input is multiplied by –1, the result, , is a reflection about the y-axis.
- All transformations of the exponential function can be summarized by the general equation .
- Using the general equation , we can write the equation of a function given its description.
- Approximate solutions of the equation can be found using a graphing calculator.