Question

the function h is defined by $h(x) = 4x + 28$. the graph of $y = h(x)$ in the $xy$-plane has an $x$-intercept at $(a, 0)$ and a $y$-intercept at $(0, b)$, where $a$ and $b$ are constants. what is the value of $a + b$?
a 21
b 28
c 32
d 35

Explanation:

Step1: Find x-intercept (a)

To find the x-intercept, set \( y = 0 \) (since the x-intercept is at \( (a, 0) \)) in the function \( h(x)=4x + 28 \). So we have the equation \( 0 = 4a+28 \).
Subtract 28 from both sides: \( - 28=4a \).
Divide both sides by 4: \( a=\frac{-28}{4}=-7 \).

Step2: Find y-intercept (b)

To find the y-intercept, set \( x = 0 \) (since the y-intercept is at \( (0, b) \)) in the function \( h(x)=4x + 28 \). So we substitute \( x = 0 \) into the function: \( h(0)=4(0)+28=28 \). Thus, \( b = 28 \).

Step3: Calculate \( a + b \)

Now we substitute the values of \( a=-7 \) and \( b = 28 \) into \( a + b \). So \( a + b=-7+28 = 21 \)? Wait, no, wait, I made a mistake. Wait, let's re - check step 1. Wait, the function is \( h(x)=4x + 28 \), x - intercept is when \( y = 0 \), so \( 0=4x + 28 \), then \( 4x=-28 \), so \( x=-7 \)? But that would make \( a=-7 \), and \( b = 28 \), so \( a + b=-7 + 28=21 \)? But wait, the options have 21 as option A. Wait, but let's check again. Wait, maybe I misread the function. Wait, the function is \( h(x)=4x + 28 \), yes. So x - intercept: \( y = 0\), so \( 0 = 4x+28\Rightarrow4x=-28\Rightarrow x=-7\), so \( a=-7 \). Y - intercept: \( x = 0\), so \( y=4(0)+28 = 28\), so \( b = 28 \). Then \( a + b=-7 + 28=21 \). Wait, but let's check the options. Option A is 21. But wait, maybe I made a mistake in the sign? Wait, no, the x - intercept is at \( (a,0) \), so when \( y = 0 \), solving \( 4x+28 = 0\) gives \( x=-7 \), so \( a=-7 \), \( b = 28 \), so \( a + b=21 \). So the answer should be A. 21? Wait, but let's check again. Wait, maybe the function is \( h(x)=4x + 28 \), maybe it's a typo? Wait, no, the problem says "the graph of \( y = h(x) \)", so \( y=4x + 28 \). So x - intercept: \( y = 0\), \( 0=4x + 28\Rightarrow x=-7 \), y - intercept: \( x = 0\), \( y = 28 \). Then \( a=-7 \), \( b = 28 \), \( a + b=21 \). So the answer is 21, which is option A.

Wait, but wait, maybe I made a mistake in the sign of a. Wait, the x - intercept is at \( (a,0) \), so \( a\) is the x - coordinate, which is - 7, and \( b\) is 28, so \( a + b=-7+28 = 21 \). So the correct answer is 21, which is option A.

Answer:

A. 21