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Algebra / Linear functions Difficulty: Hard

The graph of the linear function $y = f (x) + 19$ is shown. If $c$ and $d$ are positive constants, which equation could define $f$ ?

$f (x) = - d - c x$

$f (x) = d - c x$

$f (x) = - d + c x$

$f (x) = d + c x$

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Explanation

Choice A is correct. It’s given that the graph of the linear function $y = f (x) + 19$ is shown. This means that the graph of $y = f (x) + 19$ can be translated down $19$ units to create the graph of $y = f (x)$ and the y-coordinate of every point on the graph of $y = f (x) + 19$ can be decreased by $19$ to find the resulting point on the graph of $y = f (x)$ . The y-intercept of the graph of $y = f (x) + 19$ is $left parenthesis 0 comma 3 right parenthesis$ . Translating the graph of $y = f (x) + 19$ down $19$ units results in a y-intercept of the graph of $y = f (x)$ at the point $left parenthesis 0 comma 3 minus 19 right parenthesis$ , or $left parenthesis 0 comma negative 16 right parenthesis$ . The graph of $y = f (x) + 19$ slants down from left to right, so the slope of the graph is negative. The translation of a linear graph changes its position, but does not change its slope. It follows that the slope of the graph of $y = f (x)$ is also negative. The equation of a linear function $f$ can be written in the form $f (x) = b + m x$ , where $b$ is the y-coordinate of the y-intercept and $m$ is the slope of the graph of $y = f (x)$ . It's given that $c$ and $d$ are positive constants. Since the y-coordinate of the y-intercept and the slope of the graph of $y = f (x)$ are both negative, it follows that $f (x) = - d - c x$ could define $f$ .

Choice B is incorrect. This could define a linear function where its graph has a positive, not negative, y-intercept.

Choice C is incorrect. This could define a linear function where its graph has a positive, not negative, slope.

Choice D is incorrect. This could define a linear function where its graph has a positive, not negative, y-intercept and a positive, not negative, slope.