In the -plane, the graph of a linear function has a slope of and passes through the point . Which equation defines ?

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SAT Linear Functions Question 31 | Free SAT Prep | Aniko

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Question 31·Easy·Linear Functions

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In the $xy$ -plane, the graph of a linear function $g$ has a slope of $-4$ and passes through the point $(0,7)$ . Which equation defines $g$ ?

For linear-function questions, quickly rewrite the information in slope-intercept form $y = mx + b$ : the slope in the problem is $m$ , and any point with $x=0$ directly gives you $b$ . Once you know $m$ and $b$ , plug them into the form and then match the resulting equation to the answer choices; if needed, double-check by plugging in the given point to see which option gives the correct $y$ -value.

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Think about the general form of a line

Write down the slope-intercept form of a linear equation, where you clearly label which part is the slope and which part is the y-intercept.

Use the point with x = 0

If a line passes through $(0,7)$ , what does that tell you about the y-intercept $b$ in the equation $y = mx + b$ ?

Combine slope and intercept

You know the slope is $-4$ and the y-intercept is 7. Plug these into $y = mx + b$ , then see which option matches that equation.

Desmos Guide

Graph all four choices

In four separate lines, type y = 7x - 4, y = -4x + 7, y = -7x + 4, and y = 4x - 7 so you can see all four lines on the same coordinate plane.

Check which line goes through (0,7)

Click or tap on the point where each line crosses the y-axis (where $x=0$ ). Look at the y-coordinate and see which line has a point at $(0,7)$ .

Confirm the slope is -4

For the line that passes through $(0,7)$ , pick two points on that line (for example, where $x=0$ and where $x=1$ ) and see how $y$ changes when $x$ increases by 1. The correct graph will have $y$ decrease by 4 when $x$ increases by 1, showing a slope of $-4$ ; match that graph to its equation from the choices.

Step-by-step Explanation

Recall slope-intercept form

A linear function can be written in slope-intercept form as

y = mx + b

where $m$ is the slope and $b$ is the y-intercept (the value of $y$ when $x=0$ ). Since $g$ is linear, it can be written this way.

Use the point (0,7) to find the y-intercept

The graph passes through $(0,7)$ . On a graph, the point with $x=0$ lies on the y-axis, so its $y$ -value is the y-intercept.

That means $b = 7$ , so the function has the form

y = mx + 7.

Use the given slope

We are told the slope of $g$ is $-4$ . In the equation $y = mx + b$ , the slope is $m$ , the coefficient of $x$ . Substitute $m=-4$ and $b=7$ into $y = mx + b$ to obtain the equation; we will express it in function notation and select the matching choice in the next step.

Match with the answer choices

We find that the equation is $y = -4x + 7$ , which in function notation is $g(x) = -4x + 7$ . Among the choices, this corresponds to option B.

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Step-by-step Explanation

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Step-by-step Explanation