Question

1. determine if the table describes a linear relationship or exponential relationship:

x	-1	0	1	2	3	4
y	\\(\frac{1}{6}\\)	1	6	36	216	1,296

a. it is exponential growth with factor of 6
b. it is linear because the average rate of change is 6
c. it is linear because the average rate of change is -6
d. it is exponential decay by a factor of -6

1. use the graph below to find the average rate of change for hour 3 to 4 given the distance is in miles and time is in hours

a. 3 mph
b. 2 mph
c. 5 mph
d. 1 mph
graph: x-axis (time) from 0 to 5, y-axis (distance) from 0 to 16. points at (1, ~3), (2, 8), (3, 8), (4, 10), (5, ~14)

Explanation:

Question 9

Step1: Check linearity (rate of change)

For linear, the rate of change (slope) should be constant. Calculate the difference between consecutive \( y \)-values:
From \( x=-1 \) to \( x=0 \): \( 1 - \frac{1}{6} = \frac{5}{6} \)
From \( x=0 \) to \( x=1 \): \( 6 - 1 = 5 \)
These differences are not constant, so it is not linear.

Step2: Check exponential (ratio of consecutive \( y \)-values)

Calculate the ratio \( \frac{y_{n+1}}{y_n} \):
\( \frac{1}{\frac{1}{6}} = 6 \), \( \frac{6}{1} = 6 \), \( \frac{36}{6} = 6 \), \( \frac{216}{36} = 6 \), \( \frac{1296}{216} = 6 \).
The ratio is constant (6) and greater than 1, so it is exponential growth.

The average rate of change (slope) between \( x=3 \) and \( x=4 \) is calculated as \( \frac{\Delta y}{\Delta x} \).
From the graph: At \( x=3 \), \( y=8 \); at \( x=4 \), \( y=10 \).
\( \Delta y = 10 - 8 = 2 \), \( \Delta x = 4 - 3 = 1 \).
Thus, average rate of change \( = \frac{2}{1} = 2 \) mph.

Answer:

a. It is exponential growth with factor of 6

Question 10