GRAPHS OF EXPONENTIAL FUNCTIONS
By Nancy Marcus
In this section we will illustrate, interpret, and discuss the graphs of exponential functions. We will also illustrate how you can use graphs to HELP you solve exponential problems.
Solving an equation from a graph. When we solve an equation algebraically, we set the equation equal to zero and find those values that cause the equation to equal zero. When we solve an equation graphically, we look for points of intersection.
If there is one equation, we look to see where the graph crosses the x-axis. The x-intercepts are the solutions to the equation. The x-intercepts are those values of x that cause the function value to be zero.
If you are solving two equations graphically, you have two options. You can graph both equations and determine the value of x at the point(s) of intersection. You can also create a new graph by subtracting one function from another, graph the new function, and find the x-intercepts.
In the following examples, we will solve the equations algebraically and graphically.
Example 15: Solve the equation for x algebraically and graphically.
1.Take the natural logarithm of both sdies of the equation. The left side of the equation can be simplified to x. The actual value of x is and the approximate value of x is 1.609438.
Graphically - Method 1:
1.Graph and graph g(x)=5 on the same coordinate axis and find the point(s), if any, of intersection.
2.The graph intersection at one point. (1.609438, 5). The solution is the value of x, or x = 1.609438.
Graphically - Method 2:
1.Subtract 5 from both sides of the equation to have .
2.Call the left side of the equation f(x) and graph f(x). Call the right side of the equation h(x) and graph h(x). Since the right side of the equation is nothing more than the x-axis, you will be looking for the x intercepts on the graphs of .
3.The graph of f(x) crosses the x-axis at x = 1.609438.
If you would like to review another example, click on Example.
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Author: Nancy Marcus