The graph of the linear function f in the xy-plane passes... | SAT Linear functions Question, Answer Explained

Math · AlgebraHard

The graph of the linear function $f$ in the $xy$ -plane passes through the points $(-3, 10)$ and $(5, -6)$ . For what value of $x$ does $f(x) = x$ ?

$-\frac{4}{3}$

$\frac{4}{3}$

$2$

$4$

Show answer and explanation

Answer: B

The slope is $\frac{-6 - 10}{5 - (-3)} = \frac{-16}{8} = -2$ . Using $(-3, 10)$ : $10 = -2(-3) + b$ , so $b = 4$ and $f(x) = -2x + 4$ . Setting $f(x) = x$ gives $-2x + 4 = x$ , so $4 = 3x$ and $x = \frac{4}{3}$ .

Choice C solves $f(x) = 0$ , which gives the $x$ -intercept instead of the point where $f(x)$ equals $x$ .

Why the other choices are wrong

A. This makes a sign error when moving terms. From -2x + 4 = x, adding 2x gives 4 = 3x, so x is positive.
C. x = 2 solves f(x) = 0, which gives the x-intercept. The question asks where f(x) equals x.
D. 4 is the y-intercept of the graph, f(0), not the value of x where f(x) = x.

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