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Algebra / Linear functions Difficulty: Medium

$h$ $h$	$f (h)$ $f (h)$
$1$ $1$	$155$ $155$
$3$ $3$	$285$ $285$

The table gives the number of hours, $h$ , of labor and a plumber’s total charge $f (h)$ , in dollars, for two different jobs.

There is a linear relationship between $h$ and $f (h)$ . Which equation represents this relationship?

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Explanation

Choice C is correct. It's given that there is a linear relationship between a plumber's hours of labor, $h$ , and the plumber's total charge $f (h)$ , in dollars. It follows that the relationship can be represented by an equation of the form $f (h) = m h + b$ , where $m$ is the rate of change of the function $f$ and $b$ is a constant. The rate of change of $f$ can be calculated by dividing the difference in two values of $f (h)$ by the difference in the corresponding values of $h$ . Based on the values given in the table, the rate of change of $f$ is $\frac{285 - 155}{3 - 1}$ , or $65$ . Substituting $65$ for $m$ in the equation $f (h) = m h + b$ yields $f (h) = 65 h + b$ . The value of $b$ can be found by substituting any value of $h$ and its corresponding value of $f (h)$ for $h$ and $f (h)$ , respectively, in this equation. Substituting $1$ for $h$ and $155$ for $f (h)$ yields $155 = 65 (1) + b$ , or $155 = 65 + b$ . Subtracting $65$ from both sides of this equation yields $90 equals b$ . Substituting $90$ for $b$ in the equation $f (h) = 65 h + b$ yields $f (h) = 65 h + 90$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.