Linear functions SAT question

Math · Algebra · Hard · Answer it here, explanation included.

$x$	$f (x)$
$1$	$- 64$
$2$	$0$
$3$	$64$

For the linear function $f$ , the table shows three values of $x$ and their corresponding values of $f (x)$ . Function $f$ is defined by $f (x) = a x + b$ , where $a$ and $b$ are constants. What is the value of $a - b$ ?

ExplanationShow

Choice D is correct. The table gives that $f (x) = 0$ when $x = 2$ . Substituting 0 for $f (x)$ and $2$ for $x$ into the equation $f (x) = a x + b$ yields $0 = 2 a + b$ . Subtracting $2 a$ from both sides of this equation yields $b = - 2 a$ . The table gives that $f (x) = - 64$ when $x = 1$ . Substituting $- 2 a$ for $b$ , $- 64$ for $f (x)$ , and $1$ for $x$ into the equation $f (x) = a x + b$ yields $- 64 = a (1) + (- 2 a)$ . Combining like terms yields $- 64 = - a$ , or $a = 64$ . Since $b = - 2 a$ , substituting $64$ for $a$ into this equation gives $b = (- 2) (64)$ , which yields $b = - 128$ . Thus, the value of $a - b$ can be written as $64 - (- 128)$ , which is $192$ .

Choice A is incorrect. This is the value of $a + b$ , not $a - b$ .

Choice B is incorrect. This is the value of $a - 2$ , not $a - b$ .

Choice C is incorrect. This is the value of $2 a$ , not $a - b$ .

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