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Exponential Curves

Module by: Kenny M. Felder. E-mail the author

Summary: This module discusses the graphing of exponential curves.

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

Figure 1: y = 2 x y = 2 x size 12{y=2 rSup { size 8{x} } } {}
Graph

A few points to notice about this graph.

  • It goes through the point (0,1)(0,1) size 12{ \( 0,1 \) } {} because 20=120=1 size 12{2 rSup { size 8{0} } =1} {}.
  • It never dips below the xx size 12{x} {}-axis. The domain is unlimited, but the range is y>0. (*Think about our definitions of exponents: whether xx size 12{x} {} is positive or negative, integer or fraction, 2x2x 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, 2525 size 12{2 rSup { size 8{5} } } {} is twice 2424 size 12{2 rSup { size 8{4} } } {}, because it multiplies by one more 2. So as you move to the right, the yy 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 yy size 12{y} {}-values start looking like 1212 size 12{ { {1} over {2} } } {}, 1414 size 12{ { {1} over {4} } } {}, 1818 size 12{ { {1} over {8} } } {}, and so on, falling closer and closer to 0.

What would the graph of y=3xy=3x size 12{y=3 rSup { size 8{x} } } {} look like? Of course, it would also go through (0,1)(0,1) size 12{ \( 0,1 \) } {} because 30=130=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.

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

As you might guess, graphs such as 5x5x size 12{5 rSup { size 8{x} } } {} and 10x10x size 12{"10" rSup { size 8{x} } } {} all have this same characteristic shape. In fact, any graph axax size 12{a rSup { size 8{x} } } {} where a>1a>1 size 12{a>1} {} will look basically the same: starting at (0,1)(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 12x12x size 12{ left ( { {1} over {2} } right ) rSup { size 8{x} } } {}.

Figure 3: y = 1 2 x y = 1 2 x size 12{y= left ( { {1} over {2} } right ) rSup { size 8{x} } } {}
Exponential graph with rising sharply to the left and drops towards zero towards the right.

Each time you move to the right on this graph, it multiplies by 1212 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−x2−x size 12{2 rSup { size 8{ - x} } } {}. Remember that when you replace xx size 12{x} {} with −x−x size 12{ - x} {}, f(3)f(3) size 12{f \( 3 \) } {} becomes the old f(−3)f(−3) size 12{f \( - 3 \) } {} and vice-versa; in other words, the graph flips around the yy size 12{y} {}-axis. If you take the graph of 2x2x size 12{2 rSup { size 8{x} } } {} and permute it in this way, you get a familiar shape:

Figure 4: y = 2 − x y = 2 − x size 12{y=2 rSup { size 8{ - x} } } {}
The graph flips around the y-axis

Yes, it’s 12x12x size 12{ left ( { {1} over {2} } right ) rSup { size 8{x} } } {} in a new disguise!

Why did it happen that way? Consider that 12x=1x2x12x=1x2x size 12{ left ( { {1} over {2} } right ) rSup { size 8{x} } = { {1 rSup { size 8{x} } } over {2 rSup { size 8{x} } } } } {}. But 1x1x size 12{1 rSup { size 8{x} } } {} is just 1 (in other words, 1 to the anything is 1), so 12x=12x12x=12x 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: 12x12x size 12{ { {1} over {2 rSup { size 8{x} } } } } {} is the same thing as 2−x2−x size 12{2 rSup { size 8{ - x} } } {}! So we arrive at: 12x=2−x12x=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 y = 2 ⋅ 2 x size 12{y=2 cdot 2 rSup { size 8{x} } } {} Looks just like y = 2 x y = 2 x size 12{y=2 rSup { size 8{x} } } {} but vertically stretched: all y­-values double

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

If you permute 2x2x size 12{2 rSup { size 8{x} } } {} in these two ways, you will find that they create the same graph.

Figure 5: y = 2 ⋅ 2 x y = 2 ⋅ 2 x size 12{y=2 cdot 2 rSup { size 8{x} } } {} aka y = 2 x + 1 y = 2 x + 1 size 12{y=2 rSup { size 8{x+1} } } {}
Graph of 2 to the x. Shifted one to the left

Once again, this is predictable from the rules of exponents: 2⋅2x=21⋅2x=2x+12⋅2x=21⋅2x=2x+1 size 12{2 cdot 2 rSup { size 8{x} } =2 rSup { size 8{1} } cdot 2 rSup { size 8{x} } =2 rSup { size 8{x+1} } } {}

Using exponential functions to model behavior

In the first chapter, we talked about linear functions as functions that add the same amount every time. For instance, y=3x+4y=3x+4 size 12{y=3x+4} {} models a function that starts at 4; every time you increase xx size 12{x} {} by 1, you add 3 to yy size 12{y} {}.

Exponential functions are conceptually very analogous: they multiply by the same amount every time. For instance, y=4×3xy=4×3x size 12{y=4 times 3 rSup { size 8{x} } } {} models a function that starts at 4; every time you increase xx size 12{x} {} by 1, you multiplyyy size 12{y} {} by 3.

Linear functions can go down, as well as up, by having negative slopes: y=−3x+4y=−3x+4 size 12{y= - 3x+4} {} starts at 4 and subtracts 3 every time. Exponential functions can go down, as well as up, by having fractional bases: y=4×(13)xy=4×(13)x size 12{y=4 times \( { {1} over {3} } \) rSup { size 8{x} } } {} starts at 4 and divides by 3 every time.

Exponential functions often defy intuition, because they grow much faster than people expect.

Modeling exponential functions

Your father’s house was worth $100,000 when he bought it in 1981. Assuming that it increases in value by 8%8% size 12{8%} {} every year, what was the house worth in the year 2001? (*Before you work through the math, you may want to make an intuitive guess as to what you think the house is worth. Then, after we crunch the numbers, you can check to see how close you got.)

Often, the best way to approach this kind of problem is to begin by making a chart, to get a sense of the growth pattern.

Table 1
Year Increase in Value Value
1981 N/A 100,000
1982 8%8% size 12{8%} {} of 100,000 = 8,000 108,000
1983 8%8% size 12{8%} {} of 108,000 = 8,640 116,640
1984 8%8% size 12{8%} {} of 116,640 = 9,331 125,971

Before you go farther, make sure you understand where the numbers on that chart come from. It’s OK to use a calculator. But if you blindly follow the numbers without understanding the calculations, the whole rest of this section will be lost on you.

In order to find the pattern, look at the “Value” column and ask: what is happening to these numbers every time? Of course, we are adding 8%8% size 12{8%} {} each time, but what does that really mean? With a little thought—or by looking at the numbers—you should be able to convince yourself that the numbers are multiplying by 1.08 each time. That’s why this is an exponential function: the value of the house multiplies by 1.08 every year.

So let’s make that chart again, in light of this new insight. Note that I can now skip the middle column and go straight to the answer we want. More importantly, note that I am not going to use my calculator this time—I don’t want to multiply all those 1.08s, I just want to note each time that the answer is 1.08 times the previous answer.

Table 2
Year House Value
1981 100,000
1982 100 , 000 × 1 . 08 100 , 000 × 1 . 08 size 12{"100","000" times 1 "." "08"} {}
1983 100 , 000 × 1 . 08 2 100 , 000 × 1 . 08 2 size 12{"100","000" times 1 "." "08" rSup { size 8{2} } } {}
1984 100 , 000 × 1 . 08 3 100 , 000 × 1 . 08 3 size 12{"100","000" times 1 "." "08" rSup { size 8{3} } } {}
1985 100 , 000 × 1 . 08 4 100 , 000 × 1 . 08 4 size 12{"100","000" times 1 "." "08" rSup { size 8{4} } } {}
y y size 12{y} {} 100 , 000 × 1 . 08 something 100 , 000 × 1 . 08 something size 12{"100","000" times 1 "." "08" rSup { size 8{"something"} } } {}

If you are not clear where those numbers came from, think again about the conclusion we reached earlier: each year, the value multiplies by 1.08. So if the house is worth 100,000×1.082100,000×1.082 size 12{"100","000" times 1 "." "08" rSup { size 8{2} } } {}in 1983, then its value in 1984 is 100,000×1.082×1.08100,000×1.082×1.08 size 12{ left ("100","000" times 1 "." "08" rSup { size 8{2} } right ) times 1 "." "08"} {}, which is 100,000×1.083100,000×1.083 size 12{"100","000" times 1 "." "08" rSup { size 8{3} } } {}.

Once we write it this way, the pattern is clear. I have expressed that pattern by adding the last row, the value of the house in any year yy size 12{y} {}. And what is the mystery exponent? We see that the exponent is 1 in 1982, 2 in 1983, 3 in 1984, and so on. In the year yy size 12{y} {}, the exponent is y−1981y−1981 size 12{y - "1981"} {}.

So we have our house value function:

v ( y ) = 100 , 000 × 1 . 08 y − 1981 v ( y ) = 100 , 000 × 1 . 08 y − 1981 size 12{v \( y \) ="100","000" times 1 "." "08" rSup { size 8{y - "1981"} } } {}
(1)

That is the pattern we needed in order to answer the question. So in the year 2001, the value of the house is 100,000×1.0820100,000×1.0820 size 12{"100","000" times 1 "." "08" rSup { size 8{"20"} } } {}. Bringing the calculator back, we find that the value of the house is now $466,095 and change.

Wow! The house is over four times its original value! That’s what I mean about exponential functions growing faster than you expect: they start out slow, but given time, they explode. This is also a practical life lesson about the importance of saving money early in life—a lesson that many people don’t realize until it’s too late.

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