GRAPHS OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS


GRAPHS OF EXPONENTIAL FUNCTIONS

By Nancy Marcus

In this section we will illustrate, interpret, and discuss the graphs of exponential and logarithmic functions. We will also illustrate how you can use graphs to HELP you solve exponential and logarithmic problems and check your solutions.

Graphs of Functions in General:

Recall that graphs are made up of points that are plotted on rectangular coordinate systems. The points consist of two parts: (independent variable, dependent variable). The dependent variable is so named because it's value and behavior depend on the value and behavior of the independent variable. For purposes of discussion, we will refer to the independent variable as x and the dependent variable as y or f(x). In reality, you could label them anything: (p, q), (a, b), (d, c), etc.

Reminder: Remember that when we talk about the function, the function value, the value of the function, y or f(x), we are talking about the value and behavior of the y part of the point (x, y) in the full set of the points that form the graph..

Exponential Graphs:

Once you know the shape of an exponential graph , you can shift it vertically or horizontally, stretch it, shrink it, reflect it, check answers with it, and most important interpret the graph.

Example 1:

The function tex2html_wrap_inline18 is always positive. There is simply no value of x that will cause the value of tex2html_wrap_inline20 to be negative. What does this mean in terms of a graph? It means that the entire graph of the function tex2html_wrap_inline18 is located in quadrants I and II.

Graph the function tex2html_wrap_inline18 . Notice that the graph never crosses the x-axis. Why is that so? It is because there is no value of x that will cause the value of f(x) in the formula tex2html_wrap_inline18 to equal 0.

Notice that the graph crosses the y-axis at 1. Why is that so? The value of x is always zero on the y-axis. Substitute 0 for x in the equation tex2html_wrap_inline18 : tex2html_wrap_inline30 . This translates to the point (0, 1).

Notice on the graph that, as the value of x increases, the value of f(x) also increases. This means that the function is an increasing function. Recall that an increasing function is a one-to-one-function, and a one-to-one function has a unique inverse.

The inverse of an exponential function is a logarithmic function and the inverse of a logarithmic function is an exponential function.

Notice also on the graph that as x gets larger and larger, the function value of f(x) is increasing more and more dramatically. This is why the function is called an exponential function.

If you are interested in reviewing the graphs of exponential functions, examples and problems, click on Exponential.

Logarithmic Graphs:

Once you know the shape of a logarithmic graph , you can shift it vertically or horizontally, stretch it, shrink it, reflect it, check answers with it, and most important interpret the graph.

Example 2:

Graph the function tex2html_wrap_inline32 . Notice that the graph of this function is located entirely in quadrants I and IV. Notice also that the graph never touches the y-axis.

What does that mean? It means that the value of x (domain of the function f(x) in the equation tex2html_wrap_inline32 is always positive. Why is this so? Recall that the equation tex2html_wrap_inline32 can be rewritten as the exponential function tex2html_wrap_inline38 . There is no value of f(x) that can cause the value of x to be negative or zero.

The graph of tex2html_wrap_inline32 will never cross the y-axis because x can never equal 0. The graph will always cross the x-axis at 1.

Notice on the graph that, as x increases, the f(x) also increases. This means that the function is an increasing function. Recall that an increasing function is a one-to-one-function, and a one-to-one function has a unique inverse.

Notice on the graph that the increase in the value of the function is most dramatic between 0 and 1. After x = 1, as x gets larger and larger, the increasing function values begin to slow down (the increase get smaller and smaller as x gets larger and larger).

Notice on the graph that the function values are positive for x's that are greater than 1 and negative for x's less than 1.

If you are interested in reviewing the graphs of logarithmic functions, click on Logarithmic.

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Author: Nancy Marcus

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