Question

1. use the graph below to determine the correct average rate of change:

(graph of candle length (inches) vs time (minutes) with points and a line)
a. the rate of change of the candle is it increases 1 inch every 10 min
b. the rate of change of the candle is it increases 10 inches every 1 min
c. the rate of change of the candle is it decreases 1 inch every 10 min
d. the rate of change of the candle is it decreases 10 inch every 1 min

1. find and describe the average rate of change for four consecutive pairs of values in the table.

x	-3	-2	-1	0	1	2
y	11	7	3	-1	-5	-9

a. -4, linear
b. -4, exponential decay
c. 4, linear
d. 4, exponential decay

Explanation:

Question 7

Step1: Identify two points on the graph

From the graph, we can see that when \( x = 0 \) (time = 0 minutes), \( y = 10 \) (candle length = 10 inches). When \( x = 100 \) (time = 100 minutes), \( y = 0 \) (candle length = 0 inches).

Step2: Calculate the rate of change

The formula for the rate of change (slope) is \( \frac{\Delta y}{\Delta x} \), where \( \Delta y = y_2 - y_1 \) and \( \Delta x = x_2 - x_1 \).
Substituting the values: \( \Delta y = 0 - 10 = - 10 \) (negative because the length is decreasing) and \( \Delta x = 100 - 0 = 100 \).
So the rate of change is \( \frac{-10}{100}=-\frac{1}{10} \) inches per minute. This means the candle decreases 1 inch every 10 minutes.

Step3: Analyze the options

• Option a: Says the candle increases, which is wrong as the length is decreasing.
• Option b: Says it increases 10 inches every minute, which is wrong.
• Option c: Says it decreases 1 inch every 10 minutes, which matches our calculation.
• Option d: Says it decreases 10 inches every minute, which is wrong.

Step1: Recall the formula for average rate of change

The average rate of change between two points \( (x_1,y_1) \) and \( (x_2,y_2) \) is \( \frac{y_2 - y_1}{x_2 - x_1} \).

Step2: Calculate the rate of change for consecutive pairs

• Between \( x=-3,y = 11 \) and \( x=-2,y = 7 \): \( \frac{7 - 11}{-2-(-3)}=\frac{-4}{1}=-4 \)
• Between \( x=-2,y = 7 \) and \( x=-1,y = 3 \): \( \frac{3 - 7}{-1-(-2)}=\frac{-4}{1}=-4 \)
• Between \( x=-1,y = 3 \) and \( x = 0,y=-1 \): \( \frac{-1 - 3}{0-(-1)}=\frac{-4}{1}=-4 \)
• Between \( x = 0,y=-1 \) and \( x = 1,y=-5 \): \( \frac{-5-(-1)}{1 - 0}=\frac{-4}{1}=-4 \)

Step3: Determine the type of function

Since the average rate of change is constant (-4) for all consecutive pairs, the function is linear.

Answer:

c. The rate of change of the candle is it decreases 1 inch every 10 min

Question 8