10.3 Graphing Exponential Functions

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

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).

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

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

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

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.

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.

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

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).

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.

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

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.

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.

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

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

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

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.

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.

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.