Question 875·Medium·Math

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A scatterplot shows the relationship between a car’s speed $x$ , in miles per hour (mph), and the braking distance $y$ , in feet, on a dry road. A line of best fit is drawn on the scatterplot.

Which choice is an equation of the line of best fit shown?

A drawn line of best fit follows the trend of scattered measurements; it doesn't have to touch every dot. Read two clear grid crossings on the line, then divide the change in braking distance by the change in speed to find its slope. Use one crossing to find the intercept. Choose points on the drawn line, not two measured dots.

Hints

Hint 1
An observed point is a measured dot, while the fitted line gives a predicted braking distance. Read grid intersections on the line, even if no dot sits at those places.
Hint 2
The slope compares the change in braking distance with the matching change in speed. Subtract the coordinates of your two line points in the same order. Which change goes on top?
Hint 3
In $y=mx+b$ , the intercept $b$ is the predicted distance when $x=0$ . Use the slope and one point on the line to find it. A point's height is not the intercept unless its speed is $0$ .

Step-by-step

Read two points on the fitted line
Step 1
Read the line, not the dots
The drawn line passes through the grid crossings $(20,50)$ and $(50,140)$ . That means it predicts $50$ feet at $20$ mph and $140$ feet at $50$ mph. Use the line crossings, not nearby measured dots, because a best-fit line can miss the dots.
Step 2
Find the slope from the two crossings
The slope is the change in braking distance divided by the change in speed. Type $\frac{140-50}{50-20}$ , subtracting in the same order on top and bottom. Desmos shows $3$ : the line rises $90$ feet over a $30$ -mph increase, or $3$ feet per mph.
Step 3
Use a line point to write the intercept equation
A line has slope-intercept form $y=mx+b$ , where $b$ is its predicted distance when $x=0$ . Put the slope $3$ and the line point $(20,50)$ into that form:
$50=3(20)+b.$
The $50$ is not the intercept: it belongs to a speed of $20$ , not $0$ .
Step 4
Find the intercept
Type $50\sim3(20)+b$ . The tilde tells Desmos to find the unknown intercept that makes the equation work. Under PARAMETERS, Desmos shows $b=-10$ .
Step 5
Write the equation of the fitted line
Put $m=3$ and $b=-10$ into $y=mx+b$ :
$y=3x-10.$
So the line models braking distance in feet from speed in mph with $y=3x-10$ . Choice A.

Question 875·Medium·Math

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