The function is linear with slope and satisfies . Which of the following defines ?

Solution available with step-by-step explanation

SAT Linear Functions Question 76 | Free SAT Prep | Aniko

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Question 76·Easy·Linear Functions

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The function $g$ is linear with slope $-2$ and satisfies $g(2)=-1$ . Which of the following defines $g$ ?

For linear function questions, first match the given slope by looking at the coefficient of $x$ in slope-intercept form $y=mx+b$ and eliminating any answer choices with the wrong slope. Then use any given point, like $g(2)=-1$ , by substituting the $x$ -value into the remaining choices to see which one produces the correct $y$ -value. This “eliminate by slope, then plug in the point” approach is fast and reduces mistakes on the SAT.

Hints

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Think about slope in the equation

In a linear equation written as $y=mx+b$ , which part represents the slope? Use this to decide which answer choices even have the correct slope of $-2$ .

Eliminate choices with the wrong slope

Look at the coefficient of $x$ in each answer option. Cross out any choices that do not have a coefficient of $-2$ .

Use the given point to test the remaining options

For the options that are left, plug in $x=2$ and see what $g(2)$ equals. Which option gives an output of $-1$ ?

Desmos Guide

Enter the answer choices as separate functions

Type each option into Desmos exactly as written: y = -2x - 3, y = 2x + 3, y = 2x - 3, and y = -2x + 3. Each will appear as a different line on the graph.

Check the slopes visually or numerically

Look at the lines and note which ones slope downward from left to right (negative slope) and which slope upward (positive slope). Only the lines with negative slope could match a slope of $-2$ .

Verify which line goes through the point (2, -1)

In Desmos, click on or add the point $(2,-1)$ (you can type (2,-1) as a point). See which of the lines passes exactly through this point. The equation of that line is the function that correctly defines $g$ .

Step-by-step Explanation

Use the given slope to narrow the choices

A linear function in slope-intercept form looks like $y=mx+b$ , where $m$ is the slope.

Here, the slope is $-2$ , so we need $m=-2$ .
That means we are only interested in answer choices where the coefficient of $x$ is $-2$ .

From the options:

$g(x)=-2x-3$ has slope $-2$ .
$g(x)=-2x+3$ has slope $-2$ .
The other two choices have slope $2$ , so they cannot be correct.

Use the point $g(2)=-1$ to test the remaining options

We know the function passes through the point $(2,-1)$ , which means when $x=2$ , the output (or $y$ -value) must be $-1$ .

Take each remaining option and substitute $x=2$ :

For $g(x)=-2x-3$ , compute $g(2)$ .
For $g(x)=-2x+3$ , compute $g(2)$ .

Only one of these will give $-1$ .

Compute and identify the correct function

Now calculate each value:

For $g(x)=-2x-3$ : $g(2)=-2(2)-3=-4-3=-7$ , which is not $-1$ .
For $g(x)=-2x+3$ : $g(2)=-2(2)+3=-4+3=-1$ , which matches $g(2)=-1$ .

So the function that fits both the slope and the point is $g(x)=-2x+3$ .

Strategy

Hints

Desmos Guide

Step-by-step Explanation

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Strategy

Hints

Desmos Guide

Step-by-step Explanation