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.

\

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

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:

• The independent variable is Speed \(s\) in mi/hr.
• The dependent variable is Stopping Distance \(d\) in ft.
• The domain is the set of input values: \(\{10, 20, 30, 40, 50, 60, 70\}\).
• The range is the set of output values: \(\{15, 39, 77, 119, 170, 223, 301\}\).

Analyze the graph of the function

We examine the given data points to find the matching graph:

• At \(s = 10\), \(d = 15\).
• At \(s = 40\), \(d = 119\).
• At \(s = 70\), \(d = 301\).
• The graph must have Speed on the horizontal axis (from 0 to 70) and Distance on the vertical axis (from 0 to 350).
• Graph C correctly plots Speed on the horizontal axis and Distance on the vertical axis, showing a curve that bends upward (increasing at an increasing rate).

Describe the function behavior

We calculate the successive differences in stopping distance for each 10 mi/hr increase in speed:

• From 10 to 20: \(39 - 15 = 24\) ft
• From 20 to 30: \(77 - 39 = 38\) ft
• From 30 to 40: \(119 - 77 = 42\) ft
• From 40 to 50: \(170 - 119 = 51\) ft
• From 50 to 60: \(223 - 170 = 53\) ft
• From 60 to 70: \(301 - 223 = 78\) ft

Since these differences are positive and generally increasing, the rate at which stopping distance increases grows faster as speed increases. This matches option C.

Answer:

a. Variables, Domain, and Range:

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

b. Graph Selection:
The correct graph is C (which plots Speed on the horizontal axis, Distance on the vertical axis, and shows an upward-curving trend).

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