4.2ExponentialFunctions
The majority of our exploration up to this point has centered around linear functions. Recall that linear
functions have a constant rate of change, or constant difference. Let’s look at a table of a linear function compared
to another table.
1
2
Constant
difference
+2
2
4
+2
3
6
+2
4
8
+2
5
10
F
No constant
difference
Constant
growth factor
1
2
+2
2
4
×2
+4
3
8
×2
+8
4
16
×2
5
32
+16
Notice that the table of is linear with a constant difference of 2. The function F does not have a
constant difference, but it does grow by a constant growth factor. We can tell it is a factor instead of a difference
because it is multiplied instead of added. We should be able to easily see that the equation of () is () = 2.
Looking at F(), since the constant growth factor is 2, we are multiplying by 2 each time we increase by
one. This means we have the equation F() 21 or two to the power. This is an exponential function. In fact,
any function that grows by a constant factor, meaning it is of the form () < 1 for some non-negative number
<, is called an exponential function.
Evaluation and Graphing of Exponentials
Evaluation of exponential functions works just like any other function. We simply substitute in the input
and find the output. For example, consider the following:
188
1
G() = 21
() ACB
F() 21…
ℎ() 31 − 2
G(2) 2
(3) ACB
F(−3) 3…C…
ℎ(−3) 3…C − 2
G(2) 4
(3) s
F(−3) C
C
{
ℎ(−3) −1 s
Since we can evaluate these functions so readily, we can also easily graph each one as follows. Notice
that the domain in each case is any real number, but the ranges vary.
1
G() = 21
G()
F
−2
0.25
() = ACB
−1
0.5
0
1
1
2
2
4
−2
9
4?<FG: 0, ∞
4?<FG: 0, ∞
F 21…
31 2
2
0.0625
1
0.125
4?<FG: 0, ∞
0
0.25
1
0.5
2
1
2
1. 8±
−1
3
0
1
1
0. 3±
2
0. 1±
1
1. 6±
0
1
1
1
2
7
4?<FG: 2, ∞
Notice that using the standard inputs of the interval !2,2" gives some odd values. Since exponential
growth is so rapid, it quickly leads us to either very large or very small numbers.
189
Average Rate of Change and End Behavior
We already know how to find the average rate of change over an interval, but now let’s relate that to
something called “end behavior” of a function. End behavior is talking about what happens to a function as it goes
off to infinity to the right of the graph and negative infinity to the left of the graph. (That’s not the most academic
explanation, but it will suffice for 8th grade!)
Let’s look at the function () = 21 + 4. Its table and graph look like the following:
()
−2
4.25
−1
4.5
0
5
1
6
2
8
Average Rate of Change on !2, 1": 0.25
Average Rate of Change on !1,0": 0.5
Average Rate of Change on !0,1": 1
Average Rate of Change on !1,2": 2
Average Rate of Change on !2,3": 4
Notice that the average rate of change is continually increasing as we move further to the right on the
graph. We would say that the end behavior of this function on the right is that it is going to infinity. In other words,
it’s just going to keep going up and up.
Also we can see that the average rate of change is getting closer and closer to zero as we look further to
the left on the graph. We would say that the end behavior of this function on the left is that it is getting closer and
closer to the value of 4. It will keep getting closer and closer to 4 but never actually have the value of 4. We call
this an asymptote, and we might represent it on the graph by drawing a dotted line at a height of 4 as follows:
Since 21 can never actually equal zero (it will keep being a smaller and
smaller fraction), then 21 4 will never actually equal four. No matter
how negative an input is, say negative one trillion, we’ll still have a fraction plus
four.
190
Lesson 4.2
Evaluate the following using the given functions.
() = 9
8 n() = …
() = 9 ∗ AB
1. (−2)
2. (−1)
3. (0)
4. (1)
5. F(−1)
6. F(0)
7. F(1)
8. F(2)
9. (2)
10. (3)
11. (4)
12. (5
Graph the following exponential functions.
13. () = 21
14. () = 0.51
1
15. ACB
191
16. 21…C
17. 21 3
18. 21 3
19. 21˜C
20. 21 21. 21
192
Find the average rate of change over the interval !−, ".
22. () = 21˜
()
23. () = 21˜ + 3
()
24. () = 21˜ − 3
()
25. () = 21…
()
26. () = 21
()
27. () = 21˜
()
193
Find the average rate of change over the given intervals of the following function.
1
1
1
28. () = on !−4, −2"
29. () = on !−2,2"
30. () = on !2,4"
31. () = 21 on !−4, −2"
()
32. () = 21 on !−2,2"
()
33. () = 21 on !2,4"
()
()
194
()
()