Question

9 the cost of hiring a birthday party service is shown below.
a) what is the initial value of the graph?
b) what does the initial value represent?
c) calculate the rate of change?
d) what does the rate of change represent?
e) write the equation for this graph.
f) how much would it cost if you invited 100 guests?
g) how many guests did you invite if it costs $1000?
birthdays r us

Explanation:

Step1: Identify initial value

The initial value is the y-intercept, where the number of guests (x) = 0. From the graph, this is $10.00.

Step2: Explain initial value meaning

The initial value is the fixed cost when no guests are invited.

Step3: Calculate rate of change

Pick two points: (0, 10) and (10, 15). Rate of change = $\frac{\text{Change in Cost}}{\text{Change in Guests}} = \frac{15-10}{10-0} = \frac{5}{10} = 0.5$

Step4: Explain rate of change meaning

The rate of change is the cost per additional guest.

Step5: Write linear equation

Use slope-intercept form $y = mx + b$, where $m=0.5$, $b=10$. Equation: $y = 0.5x + 10$

Step6: Calculate cost for 100 guests

Substitute $x=100$ into the equation: $y = 0.5(100) + 10 = 50 + 10 = 60$

Step7: Find guests for $1000 cost

Set $y=1000$, solve for $x$:
$1000 = 0.5x + 10$
$0.5x = 1000 - 10 = 990$
$x = \frac{990}{0.5} = 1980$

Answer:

a) $\$10.00$
b) It represents the fixed base cost of the party service (cost with 0 guests).
c) $\$0.50$ per guest
d) It represents the additional cost for each extra guest invited.
e) $y = 0.5x + 10$ (where $y$ = total cost, $x$ = number of guests)
f) $\$60.00$
g) 1980 guests