Question

question 25 of 40
a student was given two data sets, set a and set b. which of the following statements is true?
set a

x	0	1	2	3	4
y	10	20	30	40	50

set b

x	0	1	2	3	4
y	2	10	50	250	1250

a. set a is a linear function and the values increase at a slower rate than set b.
b. set a is a linear function and the values increase at a faster rate than set b.
c. set b is an exponential function and the values increase at the same rate as set a.
d. set b is an exponential function and the values increase at a slower rate than set a.

Explanation:

Step1: Determine the function type of Set A

For Set A, when \(x\) increases by 1 each time (\(x = 0,1,2,3,4\)), \(y\) increases by 10 each time (\(y=10,20,30,40,50\)). The rate of change \(\frac{\Delta y}{\Delta x}=\frac{20 - 10}{1-0}=10\). The equation of the line is \(y=10x + 10\) (using the slope - intercept form \(y=mx + b\), where \(m = 10\) and \(b = 10\) when \(x = 0,y=10\)), so Set A is a linear function.

Step2: Determine the function type of Set B

For Set B, when \(x = 0,y = 2\); when \(x=1,y = 10\) (a factor of 5 increase since \(\frac{10}{2}=5\)); when \(x = 2,y=50\) (a factor of 5 increase from \(y = 10\) since \(\frac{50}{10}=5\)); when \(x=3,y = 250\) (a factor of 5 increase from \(y = 50\) since \(\frac{250}{50}=5\)); when \(x = 4,y=1250\) (a factor of 5 increase from \(y = 250\) since \(\frac{1250}{250}=5\)). The equation of the function is \(y=2\times5^{x}\), so Set B is an exponential function.

Step3: Compare the rates of increase

The linear function in Set A has a constant rate of change of 10. The exponential function in Set B has a multiplicative rate of change of 5, and as \(x\) gets larger, the values of the exponential function in Set B will grow much faster than the linear function in Set A.

Answer:

A. Set A is a linear function and the values increase at a slower rate than Set B.