Question

1. the graph below shows the number of miles a car has driven over time.

a. is the number of miles driven proportional to the amount of time? how do you know?
b. what does the point (2, 100) mean in the context of the situation?
c. what does the point (5, 250) mean in the context of the situation?
d. what does the point (1, 50) mean in the context of the situation?

Explanation:

a.

Step1: Check proportionality conditions

A relationship is proportional if the graph is a straight - line passing through the origin. The given graph is a straight line and passes through \((0,0)\).

Step2: Calculate the unit rate

For \((1,50)\), the rate is \(\frac{y}{x}=\frac{50}{1} = 50\). For \((2,100)\), \(\frac{y}{x}=\frac{100}{2}=50\). For \((3,150)\), \(\frac{y}{x}=\frac{150}{3}=50\). For \((4,200)\), \(\frac{y}{x}=\frac{200}{4}=50\). For \((5,250)\), \(\frac{y}{x}=\frac{250}{5}=50\). The ratio \(\frac{y}{x}\) (miles per hour) is constant.

b.

Step1: Interpret the coordinates

In the context of \(x\) (hours) and \(y\) (miles), for the point \((x = 2,y = 100)\), the \(x\) - value represents time and the \(y\) - value represents distance.

c.

Step1: Interpret the coordinates

For the point \((x = 5,y = 250)\), the \(x\) - value represents time and the \(y\) - value represents distance.

d.

Step1: Interpret the coordinates

For the point \((x = 1,y = 50)\), the \(x\) - value represents time and the \(y\) - value represents distance.

Answer:

a. Yes, the number of miles driven is proportional to the amount of time. The graph is a straight line passing through the origin and the ratio \(\frac{y}{x}\) (miles per hour) is constant (\(50\)).
b. The point \((2,100)\) means that in \(2\) hours, the car has driven \(100\) miles.
c. The point \((5,250)\) means that in \(5\) hours, the car has driven \(250\) miles.
d. The point \((1,50)\) means that in \(1\) hour, the car has driven \(50\) miles.