Page 743 - Algebra 1

The common ratio of a geometric sequence is comparable to the base of an exponential function. For any exponential function, as the x-values change by a constant amount, the y-values change by a constant factor. For f (x) = 4(2)x, as each x-value increases by 1, each y-value increases by a factor of 2.
Change: +1
Change: ×2
The base 2 of the exponential function f(x) = 4(2)x is the common ratio of
the sequence 2, 4, 8, 16, 32, ....
Identifying an Exponential Function
Determine if each set of ordered pairs satisfies an exponential function. Explain your answer.
a. ⎧⎨(0, -3), (-2, -_1 ), (1, -9), (-1, -1)⎬⎫ ⎩3⎭
SOLUTION
Reading Math
In the expression f(x) = 4(2)x, 2 is the base and x is the exponent.
x
-1
0
1
2
3
f (x)
2
4
8
16
32
Example
2
Arrange the ordered pairs so that the x-values are increasing. ⎧⎨(-2, -_1), (-1, -1), (0, -3), (1, -9)⎬⎫
⎩3⎭
The x-values increase by the constant amount of 1.
Divide each y-value by the y-value before it. -1÷-_1 =-1×-3=3
-9 ÷ -3 = 3
Because each ratio is the same, 3, the base b = 3. The set of ordered pairs
satisfies an exponential function.
b. {(6, 150), (4, 100), (8, 200), (2, 50)} SOLUTION
Arrange the ordered pairs so that the x-values are increasing. {(2, 50), (4, 100), (6, 150), (8, 200)}
The x-values increase by the constant amount of 2.
Divide each y-value by the y-value before it.
100 ÷ 50 = 2 1 5 0 ÷ 1 0 0 = 1 _1
2
2 0 0 ÷ 1 5 0 = 1 _1 3
Because the ratios are not the same, the ordered pairs do not satisfy an exponential function.
3
-3 ÷ -1 = 3
Math Reasoning
Analyze What type of function do the ordered pairs in Example 2b satisfy, and why?
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