Graphing Exponential Functions: Examples (page 3 of 4)

Sections: Introductory concepts, Step-by-step graphing instructions, Worked examples

• Graph y = 2^x + 4

This is the standard exponential, except that the "+ 4" pushes the graph up so it is four units higher than usual.

First I compute some points:
Then I plot those points:
Student very often only compute y-values for x-values that are close to zero. Then either they have no idea where the graph goes on the left-hand side, and leave it hanging there:		incorrect graph
...or else they take the graph down to the x-axis, as is usual for the standard exponential graph:		incorrect graph
But this isn't the standard exponential graph; it is the standard exponential graph raised by four units. When x is negative, y = 2x + 4 won't be very close to zero; instead, it will be very close to 4, because the values will be "a teensy-tiny little number, plus four", which works out to be a teensy-tiny bit more than four.      Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved
To help me with my graph, and to indicate that I know that y = 2x + 4 never goes below (or even touches, for that matter) the line y = 4, I will drawn a dashed line at y = 4:		drawing an asymptote can be helpful
This dashed-in line, indicating where the graph goes as x heads off to the side, is called a "horizontal asymptote", or just an "asymptote". It is not required that you draw it in, but it can be helpful, and can point out to your teacher on the test that you do know what you're doing.
Then I'll draw the exponential:		graph of y = 2x + 4

• Graph y = 5^–x

I need to remember that the "negative" exponent reverses the location (along the x-axis) in which the power on 5 is negative. When the x-values are negative (that is, when I'm on the left-hand side of the graph), the value of –x will be positive, so the graph will grow quickly on the left-hand side. On the other hand, when the x-values are positive (that is, on the right-hand side of the graph), the value of –x will be negative, so the graph will stay very close to the x-axis.

In other words, the standard values are reversed:

Then y = 5–x graphs as:

Any graph that looks like the above (big on the left and crawling along the x-axis on the right) displays exponential decay, rather than exponential growth. For a graph to display exponential decay, either the exponent is "negative" or else the base is between 0 and 1. You should expect to need to be able to identify the type of exponential equation from the graph. The first two worked examples displayed exponential growth; the last example above displays exponential decay; and the following displays exponential growth again.

• Graph y = 2^{(x + 3)}

This is not the same as "2^x + 3". In "2^x + 3", the standard exponential is shifted up three units. In this case, the shift in "inside" the exponential. Instead of the "+ 3" shifting the "2^x" up by three, the "+ 3" shifts the "2^x" over sideways by three. The only question is: shifts sideways which way, left or right? The way I keep it straight is to consider one of the basic points on any exponential. When the power is zero, the exponential is 1.  For "2^{(x + 3)}", when is the power zero? When x + 3 = 0, so x = –3. That is, the basic plot point (0, 1) has been shifted to the point (–3, 1), so the graph has been shifted three points to the left:

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