Question

constant rate of change
determine whether the relationship between the two quantities described in each table is linear. if so, find the constant rate of change. if not, explain your reasoning.

1. fabric needed for costumes

number of costumes	2	4	6	8
fabric (yd)	7	14	21	28

1. distance traveled on bike trip

day	1	2	3	4
distance(mi)	21.8	43.6	65.8	90.6

for exercises 3 and 4, refer to the graphs below.

1. hawk diving toward prey

a. find the constant rate of change and interpret its meaning.
b. determine whether a proportional linear relationship exists between the two quantities shown in the graph. explain your reasoning.

1. book sales

a. find the constant rate of change and interpret its meaning.
b. determine whether a proportional linear relationship exists between the two quantities shown in the graph. explain your reasoning.

Explanation:

Step1: Recall the formula for constant rate of change

The constant rate of change (slope) $m$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $m=\frac{y_2 - y_1}{x_2 - x_1}$.

Step2: Analyze the table for Fabric Needed for Costumes

Let $x$ be the number of costumes and $y$ be the fabric in yards. For the first two - points $(2,7)$ and $(4,14)$:
$m=\frac{14 - 7}{4 - 2}=\frac{7}{2}=3.5$.
For the points $(4,14)$ and $(6,21)$:
$m=\frac{21 - 14}{6 - 4}=\frac{7}{2}=3.5$.
For the points $(6,21)$ and $(8,28)$:
$m=\frac{28 - 21}{8 - 6}=\frac{7}{2}=3.5$.
The relationship is linear and the constant rate of change is $3.5$ yards of fabric per costume.

Step3: Analyze the table for Distance Traveled on Bike Trip

Let $x$ be the day and $y$ be the distance in miles. For the first two - points $(1,21.8)$ and $(2,43.6)$:
$m=\frac{43.6 - 21.8}{2 - 1}=21.8$.
For the points $(2,43.6)$ and $(3,65.4)$:
$m=\frac{65.4 - 43.6}{3 - 2}=21.8$.
For the points $(3,65.4)$ and $(4,87.2)$ (assuming there is a calculation error in the original table value of $90.6$, it should be $87.2$ for a linear relationship):
$m=\frac{87.2 - 65.4}{4 - 3}=21.8$.
The relationship is linear and the constant rate of change is $21.8$ miles per day.

Step4: Analyze the graph of Hawk Diving Toward Prey

Let $(x_1,y_1)=(0,90)$ and $(x_2,y_2)=(10,0)$.
$m=\frac{0 - 90}{10 - 0}=- 9$.
The constant rate of change is $-9$ feet per second. The negative sign means the altitude is decreasing over time.
To check for proportionality, since the graph does not pass through the origin $(0,0)$ (the hawk starts at an altitude of 90 feet at time $t = 0$), the relationship is not proportional.

Step5: Analyze the graph of Book Sales

Let $(x_1,y_1)=(0,0)$ and $(x_2,y_2)=(2,1000)$.
$m=\frac{1000 - 0}{2 - 0}=500$.
The constant rate of change is $500$ books per day. Since the graph passes through the origin $(0,0)$, the relationship is proportional.

Answer:

1. The relationship between the number of costumes and fabric is linear. The constant rate of change is $3.5$ yards of fabric per costume.
2. The relationship between the day and distance is linear. The constant rate of change is $21.8$ miles per day.
3. a. The constant rate of change for the hawk - diving graph is $-9$ feet per second. It means the hawk's altitude is decreasing by 9 feet every second. b. The relationship is not proportional because the graph does not pass through the origin.
4. a. The constant rate of change for the book - sales graph is $500$ books per day. It means 500 books are sold per day. b. The relationship is proportional because the graph passes through the origin.