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GRAPHS OF EXPONENTIAL FUNCTIONS
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By Nancy Marcus
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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.
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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.
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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.
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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.
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In the following examples, we will solve the equations algebraically
and graphically.
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Example 16: **Solve the equation
for x algebraically and graphically.

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Algebraically:**

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

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

3. Solve for using the quadratic formula.

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

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

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

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Graphically - Method 1:**

1. Graph 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.

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Graphically - Method 2:**

1. Subtract 2 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 = 2.106204.

4. The solution is x = 2.1062404.

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

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