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Algebra / Linear functions Difficulty: Medium

The function $h$ is defined by $h (x) = 4 x + 28$ . The graph of $y = h (x)$ in the xy-plane has an x-intercept at $left parenthesis a comma 0 right parenthesis$ and a y-intercept at $left parenthesis 0 comma b right parenthesis$ , where $a$ and $b$ are constants. What is the value of $a + b$ ?

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Explanation

Choice A is correct. The x-intercept of a graph in the xy-plane is the point on the graph where $y equals 0$ . It's given that function $h$ is defined by $h (x) = 4 x + 28$ . Therefore, the equation representing the graph of $y = h (x)$ is $y = 4 x + 28$ . Substituting $0$ for $y$ in the equation $y = 4 x + 28$ yields $0 = 4 x + 28$ . Subtracting $28$ from both sides of this equation yields $- 28 = 4 x$ . Dividing both sides of this equation by $4$ yields $- 7 = x$ . Therefore, the x-intercept of the graph of $y = h (x)$ in the xy-plane is $left parenthesis negative 7 comma 0 right parenthesis$ . It's given that the x-intercept of the graph of $y = h (x)$ is $left parenthesis a comma 0 right parenthesis$ . Therefore, $a = - 7$ . The y-intercept of a graph in the xy-plane is the point on the graph where $x equals 0$ . Substituting $0$ for $x$ in the equation $y = 4 x + 28$ yields $y = 4 (0) + 28$ , or $y equals 28$ . Therefore, the y-intercept of the graph of $y = h (x)$ in the xy-plane is $left parenthesis 0 comma 28 right parenthesis$ . It's given that the y-intercept of the graph of $y = h (x)$ is $left parenthesis 0 comma b right parenthesis$ . Therefore, $b equals 28$ . If $a = - 7$ and $b equals 28$ , then the value of $a + b$ is $- 7 + 28$ , or $21$ .

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

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

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