The linear function satisfies and . Which equation defines ?

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SAT Linear Functions Question 35 | Free SAT Prep | Aniko

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Question 35·Medium·Linear Functions

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The linear function $f$ satisfies $f(-3)=8$ and $f(5)=-4$ . Which equation defines $f$ ?

For linear-function questions where you are given two function values, immediately think of them as points on a line. Use the slope formula $m = (y_2 - y_1)/(x_2 - x_1)$ to get the slope, then plug this into $y = mx + b$ and use one point to solve for $b$ . On multiple-choice items, you can save time by first computing the slope to eliminate any options with the wrong slope, then quickly plug one of the given $x$ -values into the remaining choices to see which one produces the correct $y$ -value.

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Turn the function values into coordinates

When you see $f(a)=b$ , think of it as the point $(a,b)$ on the graph. What two points do $f(-3)=8$ and $f(5)=-4$ give you?

Use the two points to find the slope

Use the slope formula $m = (y_2 - y_1)/(x_2 - x_1)$ with your two points. Be careful with the signs when subtracting the $x$ - and $y$ -values.

Use $y = mx + b$ to find the intercept

Once you know the slope $m$ , write $f(x) = mx + b$ , plug in the $x$ - and $y$ -values from one of the points, and solve for $b$ .

Check which option fits both conditions

After you find $m$ and $b$ , compare your equation to the answer choices, or quickly plug $x=-3$ and $x=5$ into any remaining options to see which one gives $8$ and $-4$ .

Desmos Guide

Plot the two given points

Create a table in Desmos. In the $x$ column enter $-3$ and $5$ , and in the $y$ column enter $8$ and $-4$ . You will see the points $(-3,8)$ and $(5,-4)$ on the graph.

Have Desmos compute the slope

In a new expression line, type m = (-4 - 8) / (5 - (-3)). Desmos will display the numerical value of the slope $m$ for the line through the two points.

Find the intercept and match an option

In another line, type b = 8 - m * (-3) to compute the $y$ -intercept using the point $(-3,8)$ . Then type f(x) = m*x + b to graph the line. Use the values of $m$ and $b$ Desmos shows to write the equation of the line and select the answer choice whose equation matches it exactly.

Alternative: Test each answer choice directly

Type each option into Desmos (for example, y = -2/3 x + 7/2, etc.). For each graph, click at $x=-3$ and $x=5$ (or type those $x$ -values) and check the corresponding $y$ -values. The correct choice is the one whose line passes through both points $(-3,8)$ and $(5,-4)$ .

Step-by-step Explanation

Translate function values into points

Because $f$ is linear, its graph is a straight line. The information

$f(-3)=8$ means the point $(-3,8)$ is on the line.
$f(5)=-4$ means the point $(5,-4)$ is on the line.

So the line passes through $(-3,8)$ and $(5,-4)$ .

Find the slope of the line

Use the slope formula with the two points $(-3,8)$ and $(5,-4)$ :

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - 8}{5 - (-3)} = \frac{-12}{8} = -\frac{3}{2}.

So the slope of the line is $m = -\tfrac{3}{2}$ .

Use slope-intercept form to find the intercept

Write the equation in slope-intercept form $f(x) = mx + b$ using $m = -\tfrac{3}{2}$ :

f(x) = -\frac{3}{2}x + b.

Now plug in one of the known points, for example $(-3,8)$ , to solve for $b$ :

8 = -\frac{3}{2}(-3) + b = \frac{9}{2} + b.

Convert $8$ to halves: $8 = \tfrac{16}{2}$ . Then

\frac{16}{2} = \frac{9}{2} + b \Rightarrow b = \frac{16}{2} - \frac{9}{2} = \frac{7}{2}.

So the $y$ -intercept is $b = \tfrac{7}{2}$ .

Write the final equation and match the choice

Substitute $m = -\tfrac{3}{2}$ and $b = \tfrac{7}{2}$ back into the equation:

f(x) = -\frac{3}{2}x + \frac{7}{2}.

This matches answer choice D) $f(x) = -\tfrac{3}{2}x + \tfrac{7}{2}$ .

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation