Question

6.

x	2	6	10	14
y	8	28	48	68

7.

x	-6	4	9	20
y	5	10	12.5	18

1. an interior designer charges customers an initial consultation fee plus an hourly rate. the table shows the linear relationship between x, the number of hours, and y, the total cost of hiring the designer.

hours	0	2	8	15
total cost ($)	125	235	565	950

a. find the rate of change.
b. what does the rate of change represent in the context of the situation?

Explanation:

Problem 6

Step1: Check x,y differences

$\Delta x = 6-2=4$, $\Delta y=28-8=20$

Step2: Calculate slope (rate of change)

$\text{Rate of change} = \frac{\Delta y}{\Delta x} = \frac{20}{4}=5$

Step3: Verify linearity

Check other pairs: $\frac{48-28}{10-6}=\frac{20}{4}=5$, $\frac{68-48}{14-10}=\frac{20}{4}=5$

Step1: Check x,y differences

$\Delta x=4-(-6)=10$, $\Delta y=10-5=5$; $\Delta x=9-4=5$, $\Delta y=12.5-10=2.5$

Step2: Calculate slope consistency

$\frac{5}{10}=0.5$, $\frac{2.5}{5}=0.5$, $\frac{18-12.5}{20-9}=\frac{5.5}{11}=0.5$

Step3: Find y-intercept

Use $y=mx+b$: $5=0.5(-6)+b \to 5=-3+b \to b=8$

Part a:

Step1: Pick two data points

Use $(0,125)$ and $(2,235)$

Step2: Calculate rate of change

$\text{Rate of change} = \frac{235-125}{2-0} = \frac{110}{2}=55$

Step3: Verify with other points

$\frac{565-235}{8-2}=\frac{330}{6}=55$, $\frac{950-565}{15-8}=\frac{385}{7}=55$

Part b:

Step1: Interpret rate of change

The rate of change links hours to total cost.

Answer:

The linear relationship has a rate of change of 5, and the equation is $y=5x-2$

---

Problem 7