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This module discusses the graphing of exponential curves.

By plotting points, you can discover that the graph of y = 2 x size 12{y=2 rSup { size 8{x} } } {} looks like this:

Graph
y = 2 x size 12{y=2 rSup { size 8{x} } } {}

A few points to notice about this graph.

  • It goes through the point ( 0,1 ) size 12{ \( 0,1 \) } {} because 2 0 = 1 size 12{2 rSup { size 8{0} } =1} {} .
  • It never dips below the x size 12{x} {} -axis. The domain is unlimited, but the range is y>0. (*Think about our definitions of exponents: whether x size 12{x} {} is positive or negative, integer or fraction, 2 x size 12{2 rSup { size 8{x} } } {} is always positive.)
  • Every time you move one unit to the right, the graph height doubles. For instance, 2 5 size 12{2 rSup { size 8{5} } } {} is twice 2 4 size 12{2 rSup { size 8{4} } } {} , because it multiplies by one more 2. So as you move to the right, the y size 12{y} {} -values start looking like 8, 16, 32, 64, 128, and so on, going up more and more sharply.
  • Conversely, every time you move one unit to the left, the graph height drops in half. So as you move to the left, the y size 12{y} {} -values start looking like 1 2 size 12{ { {1} over {2} } } {} , 1 4 size 12{ { {1} over {4} } } {} , 1 8 size 12{ { {1} over {8} } } {} , and so on, falling closer and closer to 0.

What would the graph of y = 3 x size 12{y=3 rSup { size 8{x} } } {} look like? Of course, it would also go through ( 0,1 ) size 12{ \( 0,1 \) } {} because 3 0 = 1 size 12{3 rSup { size 8{0} } =1} {} . With each step to the right, it would triple ; with each step to the left, it would drop in a third . So the overall shape would look similar, but the rise (on the right) and the drop (on the left) would be faster.

Two overlapping exponential graph that intersect at (0,1)
y = 2 x size 12{y=2 rSup { size 8{x} } } {} in thin line; y = 3 x size 12{y=2 rSup { size 8{x} } } {} in thick line; They cross at ( 0,1 ) size 12{ \( 0,1 \) } {}

As you might guess, graphs such as 5 x size 12{5 rSup { size 8{x} } } {} and 10 x size 12{"10" rSup { size 8{x} } } {} all have this same characteristic shape. In fact, any graph a x size 12{a rSup { size 8{x} } } {} where a > 1 size 12{a>1} {} will look basically the same: starting at ( 0,1 ) size 12{ \( 0,1 \) } {} it will rise more and more sharply on the right, and drop toward zero on the left. This type of graph models exponential growth —functions that keep multiplying by the same number. A common example, which you work through in the text, is compound interest from a bank.

The opposite graph is 1 2 x size 12{ left ( { {1} over {2} } right ) rSup { size 8{x} } } {} .

Exponential graph with rising sharply to the left and drops towards zero towards the right.
y = 1 2 x size 12{y= left ( { {1} over {2} } right ) rSup { size 8{x} } } {}

Each time you move to the right on this graph, it multiplies by 1 2 size 12{ { {1} over {2} } } {} : in other words, it divides by 2, heading closer to zero the further you go. This kind of equation is used to model functions that keep dividing by the same number; for instance, radioactive decay. You will also be working through examples like this one.

Of course, all the permutations from the first chapter on “functions” apply to these graphs just as they apply to any graph. A particularly interesting example is 2 x size 12{2 rSup { size 8{ - x} } } {} . Remember that when you replace x size 12{x} {} with x size 12{ - x} {} , f ( 3 ) size 12{f \( 3 \) } {} becomes the old f ( 3 ) size 12{f \( - 3 \) } {} and vice-versa; in other words, the graph flips around the y size 12{y} {} -axis. If you take the graph of 2 x size 12{2 rSup { size 8{x} } } {} and permute it in this way, you get a familiar shape:

The graph flips around the y-axis
y = 2 x size 12{y=2 rSup { size 8{ - x} } } {}

Yes, it’s 1 2 x size 12{ left ( { {1} over {2} } right ) rSup { size 8{x} } } {} in a new disguise!

Why did it happen that way? Consider that 1 2 x = 1 x 2 x size 12{ left ( { {1} over {2} } right ) rSup { size 8{x} } = { {1 rSup { size 8{x} } } over {2 rSup { size 8{x} } } } } {} . But 1 x size 12{1 rSup { size 8{x} } } {} is just 1 (in other words, 1 to the anything is 1), so 1 2 x = 1 2 x size 12{ left ( { {1} over {2} } right ) rSup { size 8{x} } = { {1} over {2 rSup { size 8{x} } } } } {} . But negative exponents go in the denominator: 1 2 x size 12{ { {1} over {2 rSup { size 8{x} } } } } {} is the same thing as 2 x size 12{2 rSup { size 8{ - x} } } {} ! So we arrive at: 1 2 x = 2 x size 12{ left ( { {1} over {2} } right ) rSup { size 8{x} } =2 rSup { size 8{ - x} } } {} . The two functions are the same, so their graphs are of course the same.

Another fun pair of permutations is:

y = 2 2 x size 12{y=2 cdot 2 rSup { size 8{x} } } {} Looks just like y = 2 x size 12{y=2 rSup { size 8{x} } } {} but vertically stretched: all y­-values double

y = 2 x + 1 size 12{y=2 rSup { size 8{x+1} } } {} Looks just like y = 2 x size 12{y=2 rSup { size 8{x} } } {} but horizontally shifted: moves 1 to the left

If you permute 2 x size 12{2 rSup { size 8{x} } } {} in these two ways, you will find that they create the same graph.

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Source:  OpenStax, Math 1508 (lecture) readings in precalculus. OpenStax CNX. Aug 24, 2011 Download for free at http://cnx.org/content/col11354/1.1
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