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Algebra / Linear functions Difficulty: Medium

Caleb used juice to make popsicles. The function $f (x) = - 5 x + 30$ approximates the volume, in fluid ounces, of juice Caleb had remaining after making $x$ popsicles. Which statement is the best interpretation of the y-intercept of the graph of $y = f (x)$ in the xy-plane in this context?

Caleb used approximately $5$ fluid ounces of juice for each popsicle.

Caleb had approximately $5$ fluid ounces of juice when he began to make the popsicles.

Caleb had approximately $30$ fluid ounces of juice when he began to make the popsicles.

Caleb used approximately $30$ fluid ounces of juice for each popsicle.

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Explanation

Choice C is correct. An equation that defines a linear function $f$ can be written in the form $f (x) = m x + b$ , where $m$ represents the slope and $b$ represents the y-intercept, $left parenthesis 0 comma b right parenthesis$ , of the line of $y = f (x)$ in the xy-plane. The function $f (x) = - 5 x + 30$ is linear. Therefore, the graph of the given function $y = f (x)$ in the xy-plane has a y-intercept of $left parenthesis 0 comma 30 right parenthesis$ . It’s given that $f (x)$ gives the approximate volume, in fluid ounces, of juice Caleb had remaining after making $x$ popsicles. It follows that the y-intercept of $left parenthesis 0 comma 30 right parenthesis$ means that Caleb had approximately $30$ fluid ounces of juice remaining after making $0$ popsicles. In other words, Caleb had approximately $30$ fluid ounces of juice when he began to make the popsicles.

Choice A is incorrect. This is an interpretation of the slope, rather than the y-intercept, of the graph of $y = f (x)$ in the xy-plane.

Choice B is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.