Download 7.2 Identifying Exponential Functions

Identifying Exponential Functions (7.2)
Identifying Exponential Functions
In the previous section you worked with several examples of exponential functions. You
probably noticed that they all had a very basic function formula. Here it is, written formally:
Definition - Exponential Function: An exponential function has the form f(x) = a · bx (b > 0).
The number “a” is called the initial value or the vertical scale factor. The number “b” is called
the base of the exponential function.
The Add-Multiply Property of Exponential Functions
 In the table of an exponential function, if the x-values go up by 1’s, the y-values are
repeatedly multiplied by b.
 A table can be recognized as representing an exponential function if whenever a fixed number
is repeatedly added to x, the y-values are repeatedly multiplied by some number (example:
a table where every time that 2 is added to x, y is multiplied by 9).
 Compare this “add-multiply” property of exponential function tables with the “add-add”
property of linear function tables. (For linear functions, when 1 is added to x, the slope m is
added to y.)
Finding an exponential function that exactly fits a table
Here is how to find a function formula f(x) = a · bx when given a table (or some points or
function values):



At a minimum, two points are needed to be able to find the function formula.
a is always equal to f(0). [Reason: f(0) = a · b0 = a · 1 = a.]
The “base” b can be found as a ratio using any two points whose x-values are 1 apart; for
example:
b = ff ((10)) , or b = ff ((12)) , or b = ff (( 23)) , etc. [Why is this true?]

If the above facts aren’t enough to find a and b, here is a process that will work when given
any two points:
 Write the equation f(x) = a · bx twice.
 Substitute the values of x and f(x) for one point into the first equation; do the same for the
second point and the second equation.
 Now you have two equations with two unknowns a and b. Solve for a and b using
substitution. [Other equation solving methods we’ve studied, such as elimination or
matrix row operations, cannot be used here because they work only for linear equations.]
 Finally, write f(x) = a · bx and put in the a and b values you’ve found.
 Notice that this is basically the procedure for solving a system of equations by
substitution, which we studied earlier in the year.
Finding an exponential function that approximately fits a table
 Finding the “best fitting” exponential function for a table is called exponential regression.
 Your calculator can perform exponential regression. The method is almost identical to linear
or quadratic regression. (The only difference is: on the CALC menu choose ExpReg instead
of LinReg or QuadReg)
Identifying Exponential Functions (7.2)
Example 1: Find the equation for an exponential function that passes through the points (2,10)
and (5,80).
Way 1:
Use the add-multiply property to find b.
x
y
2 10
3 10b
4 10b2
5 80 = 10b3
Way 2:
Use substitution into f(x) = a · bx.
(2,10): 10 = a · b2, or a = 10/ b2.
(5,80): 80 = a · b5, or a = 80/ b5.
Setting a = a gives:
10/ b2 = 80/ b5
b3 = 8
b = 2.
Since 80 = 10b3, 8 = b3 or b = 2.
Use the add-multiply property again to find a.
x
y
0 a = 10/b2
1 10/b
2 10
We found that b = 2, so
Since a = 10/b2, a = 10/4 or a = 5/2.
Back-substitute into either equation
for a:
a = 10/ b2 = 10/4 = 5/2
The equation is f ( x)  52  (2) x
The equation is f ( x)  52  (2) x
Example 2: Write an exponential function that matches the table at right.
Find b using the add-multiply property or substitution like in Example 1.
12b3 = 48, so b3 = 4 or b = 3 4 ≈ 1.587.
Find a using the add-multiply property or substitution like in Example 1.
a = 12/ 3 4 ≈ 7.560.
The equation is y = 12/ 3 4 · ( 3 4 )x or y ≈ 7.560 · (1.587)x
x
1
4
7
10
y
12
48
192
768
Identifying Exponential Functions (7.2)
Exercises
1. Assume that f(x) is an exponential function. First, use the given values to find the equation
for the exponential function. Second, use your equation to evaluate f(10).
a. f(0) = 4; f(1) = 9
so, f(10) = ____
b.
f(0) = 4; f(2) = 9
so, f(10) = ____
2. For all parts of this problem, let f(x) = 2x.
a. Show that f(x + 1) = 2 f(x).
b. Complete this sentence with a transformation description, based on the equation from
part a:
“Translating the graph of f(x) to the left by 1 is equivalent to _____________________.”
c. Make a carefully drawn graph of f(x). Locate some points
that confirm the transformation description you wrote in
part b.
Identifying Exponential Functions (7.2)
3. Answer the questions that follow about these tables:
i.
ii.
x
0
1
2
3
y
8
4
2
1
iii.
x
0
2
4
6
y
¼
1
4
16
iv.
x
2
4
6
8
y
3
18
33
48
x
2
4
6
8
y
3
18
54
162
a. Decide whether each table above represents an exponential function, a linear function, or
neither. Explain how you make each decision.
b. For each function above that you identified as exponential, find an a · bx formula
(don’t use a calculator, and try to do any arithmetic in your head).
c. For each function above that you identified as linear, find a linear function formula
(don’t use a calculator, and try to do any arithmetic in your head).
Identifying Exponential Functions (7.2)
4. Suppose f(x) is an exponential function with f(2) = 100 and f(5) = 60.
a. Without finding a general function formula for f(x), find the values of f(8) and f(11).
Hint: Use the add-multiply property.
b. Find a function formula for f(x) using a method where you set up two f(x) = a · bx
equations, then solve the system of equations to find a and b.
5. Here are some values of a function g(x) that is approximately, but not exactly, exponential:
g(0) = 4, g(1) = 7, g(2) = 15, g(3) = 40. Using your calculator, find the exponential function
that best fits function g(x). Graph both the data points and the exponential function together
on your calculator screen, then sketch the screen.
6. The standard form for an exponential function formula is a · bx. Here are some exponential
functions that are not given in standard form. Rewrite the formulas in standard form.
a. f(x) = 5 · 3(2x).
b. g(x) = 4 · 10(x – 1).
7. a. Solve the equation 2x = 8 in your head.
b. Solve the equation 2x = 7 using any method you like.
c. h(x) = 2(1 – 3x)