Question 804·Medium·Math

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A line is shown on the graph.

Which choice gives an equation of the line?

A graph with two marked points gives you enough information to find a line’s slope and intercept. Divide the change in $y$ by the change in $x$ , then put one point into $y=mx+b$ and use a Desmos regression to find $b$ . A point’s $y$ -value is the intercept only when its $x$ -value is $0$ . Keep the slope fraction in the right order.

Hints

Hint 1
A slope compares the change in $y$ with the change in $x$ . Move from one marked point to the other: how far do you go right, and how far do you go down? Down counts as negative.
Hint 2
The y-intercept $b$ is the height where $x=0$ . Neither marked point has $x=0$ , so neither marked $y$ -value is $b$ . Put the slope and one marked point into $y=mx+b$ .
Hint 3
A Desmos regression can find the unknown $b$ : type the equation from your point with $\sim$ instead of $=$ . Read $b$ under PARAMETERS, then match both the slope and intercept to a choice.

Step-by-step

Find the slope, then the intercept
Step 1
Read the changes between the marked points
From $(-1,5)$ to $(3,-1)$ , you move $4$ units right and $6$ units down. Use the marked coordinates, not how steep the line looks.
Step 2
Find the slope
The slope is the vertical change divided by the horizontal change. Down is negative, so divide the $-6$ change by the $4$ change:
$m=\frac{-6}{4}.$
Reduce the fraction:
$m=-\frac32.$
Don’t flip the fraction: the horizontal change goes on the bottom.
Step 3
Use a point to find the intercept
The y-intercept $b$ is $y$ when $x=0$ . The marked point $(-1,5)$ is not on the $y$ -axis, so $5$ is not $b$ . Substitute that point and the slope into $y=mx+b$ :
$5=-\frac32(-1)+b.$
Step 4
Solve for the intercept in Desmos
Type $5\sim-\frac32(-1)+b$ . The $\sim$ tells Desmos to fit the unknown $b$ ; it shows $b=3.5$ under PARAMETERS. Since $3.5=\frac72$ , that’s the line’s intercept.
Step 5
Match both parts of the equation
In slope-intercept form $y=mx+b$ , the slope multiplies $x$ . Match both $m=-\frac32$ and $b=\frac72$ :
$y=-\frac32x+\frac72.$
That’s the equation of the graphed line. Choice A.

Question 804·Medium·Math

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