Question

the following data table represents a function.
a. identify the independent and dependent variables, and describe the domain and range.
b. make a clear graph of the function.
c. describe the function in words.

\\(\

$$\begin{array}{cc} \\text{speed (mi/hr)} & \\text{stopping distance (reaction plus braking in ft)} \\\\ 10 & 15 \\\\ 20 & 39 \\\\ 30 & 77 \\\\ 40 & 119 \\\\ 50 & 170 \\\\ 60 & 223 \\\\ 70 & 301 \\end{array}$$

\\)

c. choose the correct description below.
a. the function shows a constant rate of increase in stopping distance relative to speed.
b. the rate at which the speed increases grows faster as stopping distance increases.
c. the rate at which the stopping distance increases grows faster as speed increases.
d. the rate at which the stopping distance decreases grows faster as speed increases.

Explanation:

Identify variables and domain/range

Using the Independent and Dependent Variables and Domain and Range knowledge points

• Independent variable: Speed \(s\) (in mi/hr), as stopping distance depends on speed.
• Dependent variable: Stopping Distance \(d\) (in ft).
• Domain: \(\{10, 20, 30, 40, 50, 60, 70\}\)
• Range: \(\{15, 39, 77, 119, 170, 223, 301\}\)

Graph the function

Using the Function Representation knowledge point

• Plotting the points: \((10, 15)\), \((20, 39)\), \((30, 77)\), \((40, 119)\), \((50, 170)\), \((60, 223)\), \((70, 301)\).
• The horizontal axis represents Speed (mi/hr) and the vertical axis represents Distance (ft).
• The curve starts shallow and curves upward more steeply.
• Graph C shows the correct axes labels (Distance on vertical axis, Speed on horizontal axis) and the correct upward-curving shape.

Analyze the rate of change

Using the Rate of Change Analysis knowledge point

• First differences in stopping distance:
• \(39 - 15 = 24\)
• \(77 - 39 = 38\)
• \(119 - 77 = 42\)
• \(170 - 119 = 51\)
• \(223 - 170 = 53\)
• \(301 - 223 = 78\)
• Since the differences are positive and generally increasing, the rate at which stopping distance increases grows faster as speed increases. This matches option C.

Answer:

Question a

• Independent Variable: Speed (mi/hr)
• Dependent Variable: Stopping Distance (ft)
• Domain: \(\{10, 20, 30, 40, 50, 60, 70\}\)
• Range: \(\{15, 39, 77, 119, 170, 223, 301\}\)

Question b

The correct graph is C (which has Distance on the vertical axis, Speed on the horizontal axis, and curves upward increasingly steeply).

Question c

The correct description is C: The rate at which the stopping distance increases grows faster as speed increases.