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 16: Solve the equation tex2html_wrap_inline44 for x algebraically and graphically.


1. Subtract 2 from both sides of the equation so that one side of the equation is equal to zero tex2html_wrap_inline46 .

2. If you rewrite tex2html_wrap_inline46 as tex2html_wrap_inline50 , you will see that the equation is a quadratic in tex2html_wrap_inline52 This means you can solve for tex2html_wrap_inline54 using the quadratic formula.

3. Solve for tex2html_wrap_inline54 using the quadratic formula. tex2html_wrap_inline58

4. Then tex2html_wrap_inline60 and tex2html_wrap_inline62 . There is no value of x such that tex2html_wrap_inline54 will be a negative answer. Therefore, only tex2html_wrap_inline60 is valid.

5. Recall that we are not solving for tex2html_wrap_inline54 , we are solving for x.

6. Take the natural logarithm of both sides. The actual value of x is the tex2html_wrap_inline70 or 2.106204.

Graphically - Method 1:

1. Graph tex2html_wrap_inline72 and graph g(x)=2 on the same coordinate axis and find the point(s), if any, of intersection.

2. The graphs intersect at one point. (2.106204, 2). The solution is the value of x, or x = 2.106204.

Graphically - Method 2:

1. Subtract 2 from both sides of the equation tex2html_wrap_inline44 to have tex2html_wrap_inline46 .

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 tex2html_wrap_inline80 .

3. The graph of f(x) crosses the x-axis at x = 2.106204.

4. The solution is x = 2.1062404.

If you would like to review graphs of logarithmic functions, click on Example.

[Exponential Rules] [Logarithms]

[Algebra] [Trigonometry ] [Complex Variables]

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Author: Nancy Marcus

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