A linear function has a slope of and satisfies . What is the value of ?

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SAT Linear Functions Question 77 | Free SAT Prep | Aniko

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Question 77·Medium·Linear Functions

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A linear function $p$ has a slope of $-4$ and satisfies $p(-2)=7$ . What is the value of $p(5)$ ?

For linear function questions where you know the slope and one function value, first interpret the function value as a point on the line, like $p(a)=b$ meaning $(a,b)$ . Then compare the given $x$ to the target $x$ to find $\Delta x$ , multiply by the slope to get $\Delta y$ , and adjust the known $y$ accordingly. This avoids extra algebra; you do not need to write the full equation of the line unless that feels more comfortable, but always keep track of signs when computing changes.

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Interpret p(-2) = 7

Think about what $p(-2)=7$ means on a graph. What point $(x,y)$ does this give you on the line?

Use the meaning of slope

Slope is the change in $y$ over the change in $x$ . If the slope is $-4$ , what does that tell you about how $y$ changes when $x$ increases by $1$ ?

Compare x = −2 and x = 5

How much does $x$ change when you go from $-2$ to $5$ ? Multiply that change by the slope to find how much $y$ changes, then apply that to the starting value $7$ .

Desmos Guide

Use Desmos to compute p(5) from the slope and a point

In Desmos, enter the expression 7 + (-4)*(5 - (-2)). This uses the starting value $7$ at $x=-2$ , the slope $-4$ , and the change in $x$ from $-2$ to $5$ . The numeric result shown is the value of $p(5)$ .

Step-by-step Explanation

Translate the given information

The statement $p(-2)=7$ means that the point $(-2, 7)$ lies on the line representing the function $p$ .

We also know the slope of this line is $-4$ . Slope is the change in y divided by the change in x.

Find how much x changes from −2 to 5

We want to find $p(5)$ , which corresponds to the point $(5, p(5))$ .

Compute the change in $x$ from $-2$ to $5$ :

\Delta x = 5 - (-2) = 7

So $x$ increases by $7$ units.

Use the slope to find the change in y and compute p(5)

A slope of $-4$ means that for every increase of $1$ in $x$ , $y$ decreases by $4$ .

If $x$ increases by $7$ , then the change in $y$ is:

\Delta y = -4 \cdot 7 = -28

We start at $y = 7$ when $x = -2$ , so at $x = 5$ :

p(5) = 7 + (-28) = 7 - 28 = -21

So the value of $p(5)$ is $-21$ .

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Step-by-step Explanation

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Step-by-step Explanation