Question 407·Easy·Algebra

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The linear function $g$ passes through $(-2,4)$ and $(4,-2)$ . Which of the following defines $g$ ?

For a linear function given by two points, find the slope, then use either point to find where the line crosses the vertical axis. Desmos can evaluate the slope fraction. Subtract the coordinates in the same order on top and bottom. Don't treat a point's output as the intercept unless its input is $0$ .

Hints

Hint 1
The slope is the change in output divided by the change in input. From the first point to the second, which output goes first in the subtraction? Use the same point first when subtracting the inputs.
Hint 2
In slope-intercept form, $g(x)=mx+b$ , the slope is $m$ and the output at $x=0$ is $b$ . Neither given input is $0$ , so put one point into the equation to find $b$ .

Step-by-step

Find the slope, then the intercept
Step 1
Find the slope from the two points
The slope is the change in output divided by the change in input. Go from $(-2,4)$ to $(4,-2)$ , subtracting in the same order for both changes. Type $\frac{-2-4}{4-(-2)}$ into Desmos; it shows $-1$ . So the output falls by $1$ each time the input rises by $1$ .
Step 2
Use a point to make an equation
The point $(-2,4)$ means $g(-2)=4$ . In slope-intercept form, $g(x)=mx+b$ , where $m$ is the slope and $b$ is the output at $x=0$ . Substitute the slope and both coordinates of the point:
$4=-1(-2)+b.$
Step 3
Multiply the two negatives
Multiply $-1$ by $-2$ . Two negatives make a positive:
$4=2+b.$
Step 4
Find the intercept
Subtract $2$ from both sides to isolate the intercept $b$ :
$b=2.$
Step 5
Write the function
Put the slope $-1$ and intercept $2$ into $g(x)=mx+b$ : $g(x)=-x+2$ . This is the linear function through the two given points. Choice D.

Question 407·Easy·Algebra

Strategy

Hints

Step-by-step

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Strategy

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Step-by-step

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