Question

robert pays for his family to go to the arcade. he pays an entrance fee for his group and an additional amount per game that his family plays as shown in the graph. use the graph to answer 5 - 7.

1. find the slope and interpret its meaning.

the slope is \\( \frac{1}{4} \\)

1. find the y - intercept and interpret its meaning.

the y - intercept is 12

1. write an equation to represent the relationship between x, the number of games and y, the total cost.

\\( y = \frac{1}{4}x + 12 \\)

Explanation:

Step1: Find the slope

The formula for slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's take two points \((0,12)\) and \((4,15)\).
\(m=\frac{15 - 12}{4-0}=\frac{3}{4}\). The slope \(\frac{3}{4}\) means that for each additional game played (\(x\) increases by \(1\)), the total cost (\(y\)) increases by \(\frac{3}{4}\) dollars.

Step2: Find the \(y -\)intercept

The \(y -\)intercept is the value of \(y\) when \(x = 0\). From the graph, when \(x=0\) (no games played), \(y = 12\). The \(y -\)intercept \(12\) represents the entrance fee (the cost when no games are played).

Step3: Write the equation

The equation of a line is \(y=mx + b\), where \(m\) is the slope and \(b\) is the \(y -\)intercept. Substituting \(m=\frac{3}{4}\) and \(b = 12\), we get \(y=\frac{3}{4}x+12\).

Answer:

1. The slope is \(\frac{3}{4}\), meaning the cost increases by \(\frac{3}{4}\) dollars per game.
2. The \(y -\)intercept is \(12\), meaning the entrance fee is \(12\) dollars.
3. The equation is \(y=\frac{3}{4}x + 12\).