Question 757·Medium·Math

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The linear function $g$ satisfies $g(-2)=-7$ and $g(4)=5$ . Which equation defines $g$ ?

A linear function has a constant rate of change, called its slope. Two given input-output pairs determine both that slope and the output when the input is $0$ . Put the pairs in a Desmos table and fit $y_1\sim mx_1+b$ to find the rule. Don't treat either given output as the value at $x=0$ unless its input is actually $0$ .

Hints

Hint 1
In $g(-2)=-7$ , the number inside the parentheses is the input, and $-7$ is the output. That gives a point on the line. What point does the other function value give?
Hint 2
A line can be written $g(x)=mx+b$ , where $m$ is its slope and $b$ is its output when $x=0$ . Put both points in a Desmos table, then fit one rule to both rows.

Step-by-step

Fit the two points in Desmos
Step 1
Turn the function values into points
$g(-2)=-7$ means input $-2$ gives output $-7$ , so the line passes through $(-2,-7)$ . Likewise, $g(4)=5$ gives $(4,5)$ . The input comes first in each point.
Step 2
Find the slope and intercept
Enter those points in a Desmos table, then type $y_1\sim mx_1+b$ . The $\sim$ tells Desmos to fit a line; $m$ is its slope, or output change per $1$ input, and $b$ is its output when $x=0$ . Under PARAMETERS, Desmos shows $m=2$ and $b=-3$ . Both rows must use the same slope and intercept. Neither given output is $b$ , since neither input is $0$ .
Step 3
Write and check the rule
Use the fitted values to type $g(x)=mx+b$ , then type $g(-2)$ and $g(4)$ . Desmos returns $-7$ and $5$ , matching both given outputs. With $m=2$ and $b=-3$ , the equation is $g(x)=2x-3$ . Choice C.

Question 757·Medium·Math

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Step-by-step

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Strategy

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Step-by-step

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