1. 10.3 Graphing Exponential Functions

2. Example: Graphing an Exponential Function with b > 1
Graph f(x) = 2x by hand.

3. Solution
First, list input-output pairs of the function f in a table. Note that as the value of x increases by 1, the value of y is multiplied by 2 (the base).

4. Solution
Next, plot the solutions from the table and sketch an increasing curve that contains the plotted points.

5. Solution
We can set up a window to verify our graph on a graphing calculator.

6. Exponential Curve
The graph of an exponential function is called an exponential curve.

7. Base Multiplier Property
For an exponential function of the form y = abx, if the value of the independent variable increases by 1, the value of the dependent variable is multiplied by b.

8. Increasing or Decreasing Property
Let f(x) = abx, where a > 0. Then,
If b > 1, then the function f is increasing. We say the function grows exponentially.
If 0 < b < 1, then the function f is decreasing. We say the function decays exponentially.

9. y-Intercept of an Exponential Function
For an exponential function of the form
y = abx,
the y-intercept is (0, a).

10. y-Intercept of an Exponential Function
Warning
For an exponential function of the form y = bx (rather than y = abx), the y-intercept is not (0, b). By writing y = bx = 1bx, we see the y-intercept is (0, 1).

11. Example: Intercepts and Graph of an Exponential Function
Let
1. Find the y-intercept of the graph of f.
2. Find the x-intercept of the graph of f.
3. Graph f by hand.

12. Solution 1. Since is of the form f(x) = abx, the
y-intercept is (0, a), or (0, 6).

13. Solution
2. By the base multiplier property, as the value of x increases by 1, the value of y is multiplied by one half. Values are shown in the table below.

14. Solution
2. When we halve a number, it becomes smaller. But no number of halvings will give a result that is zero. So, as x grows large, y will become extremely close to, but never equal, 0. Likewise, the graph of f gets arbitrarily close to, but never reaches, the x-axis. In this case, we call the x-axis a horizontal asymptote. We conclude that the function f has no x-intercepts.

15. Solution
3. Plot the points from the table and sketch a decreasing exponential curve that contains the points.

16. Reflection Property
The graphs of f(x) = –abx and g(x) = abx are reflections of each other across the x-axis.

17. Horizontal Asymptote
For all exponential functions, the x-axis is a horizontal asymptote.

18. Domain and Range of an Exponential Function
The domain of any exponential function f(x) = abx is the set of real numbers.
The range of an exponential function f(x) = abx is the set of all positive real numbers if a > 0, and the range is the set of all negative real numbers if a < 0.

19. Example: Finding Values of a Function from Its Graph
The graph of an exponential function f is shown below.
1. Find f(2).
2. Find x when f(x) = 2.
3. Find x when f(x) = 0.

20. Solution
1. The blue arrows show that the input x = 2 leads to the output y = 8. We conclude that f(2) = 8.
2. The red arrows show that the output y = 2 originates from the input x = –2. So, x = –2 when f(x) = 2.
3. Recall that the graph of an exponential functions gets close to, but never reaches, the x-axis. So, there is no value of x where f(x) = 0.