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Advanced Math / Nonlinear functions Difficulty: Medium

$f (x) = (x + 6) (x + 5) (x - 4)$

The function $f$ is given. Which table of values represents $y = f (x) - 3$ ?

$x$	$y$
$negative 6$	$negative 9$
$negative 5$	$negative 8$
$4$	$1$

$x$	$y$
$negative 6$	$negative 3$
$negative 5$	$negative 3$
$4$	$negative 3$

$x$	$y$
$negative 6$	$negative 3$
$negative 5$	$negative 2$
$4$	$7$

$x$	$y$
$negative 6$	$3$
$negative 5$	$3$
$4$	$3$

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Explanation

Choice B is correct. It’s given that $f (x) = (x + 6) (x + 5) (x - 4)$ and $y = f (x) - 3$ . Substituting $(x + 6) (x + 5) (x - 4)$ for $f (x)$ in the equation $y = f (x) - 3$ yields $y = (x + 6) (x + 5) (x - 4) - 3$ . Substituting $negative 6$ for $x$ in this equation yields $y = (- 6 + 6) (- 6 + 5) (- 6 - 4) - 3$ , or $y = - 3$ . Substituting $negative 5$ for $x$ in the equation $y = (x + 6) (x + 5) (x - 4) - 3$ yields $y = (- 5 + 6) (- 5 + 5) (- 5 - 4) - 3$ , or $y = - 3$ . Substituting $4$ for $x$ in the equation $y = (x + 6) (x + 5) (x - 4) - 3$ yields $y = (4 + 6) (4 + 5) (4 - 4) - 3$ , or $y = - 3$ . Therefore, when $x = - 6$ then $y = - 3$ , when $x = - 5$ then $y = - 3$ , and when $x equals 4$ then $y = - 3$ . Thus, the table of values in choice B represents $y = f (x) - 3$ .

Choice A is incorrect. This table represents $y = x - 3$ rather than $y = f (x) - 3$ .

Choice C is incorrect. This table represents $y = x + 3$ rather than $y = f (x) - 3$ .

Choice D is incorrect. This table represents $y = f (x) + 3$ rather than $y = f (x) - 3$ .