Question

which graph best represents $y = -4(x + 3) - 2$?
11.

Explanation:

Step1: Simplify the Equation

First, simplify the given equation \( y = -4(x + 3) - 2 \). Using the distributive property, we get \( y = -4x - 12 - 2 \), which simplifies to \( y = -4x - 14 \). This is in slope - intercept form (\( y=mx + b \)), where the slope \( m=-4 \) (negative, so the line should be decreasing from left to right) and the y - intercept \( b=-14 \)? Wait, no, wait. Wait, let's re - expand: \( y=-4(x + 3)-2=-4x-12 - 2=-4x-14 \)? Wait, that can't be right. Wait, no, maybe I made a mistake. Wait, \( y=-4(x + 3)-2=-4x-12-2=-4x - 14 \). But let's check the vertex form. Wait, the original equation is in point - slope form? Wait, no, \( y=-4(x + 3)-2 \) can be seen as a linear equation. The slope is - 4, and to find the y - intercept, set \( x = 0 \): \( y=-4(0 + 3)-2=-12-2=-14 \). To find the x - intercept, set \( y = 0 \): \( 0=-4(x + 3)-2\Rightarrow4(x + 3)=-2\Rightarrow x + 3=-\frac{2}{4}=-\frac{1}{2}\Rightarrow x=-\frac{1}{2}-3=-\frac{7}{2}=-3.5 \).

But maybe a better approach: let's find two points. When \( x=-3 \), \( y=-4(0)-2=-2 \). So the point \( (-3,-2) \) is on the line. The slope is - 4, which means for every 1 unit we move to the right (increase x by 1), we move down 4 units (decrease y by 4).

Now, let's analyze the slope: a negative slope means the line is decreasing. So we can eliminate graphs with positive slopes (like F and G, since their lines are increasing from left to right). Now we are left with H and J.

Now, let's check the y - intercept or the point. Let's take the point \( x = 0 \), \( y=-4(0 + 3)-2=-14 \)? Wait, that seems very low. Wait, maybe I misread the equation. Wait, the equation is \( y=-4(x + 3)-2 \). Let's check the value when \( x=-3 \), \( y=-2 \). Let's check the slope. The slope is - 4, which is a steep negative slope.

Looking at the graphs:

- Graph F: The line is increasing (positive slope), so eliminate F.
- Graph G: The line is increasing (positive slope), so eliminate G.
- Now between H and J. Let's check the y - intercept. Wait, maybe I made a mistake in the expansion. Wait, \( y=-4(x + 3)-2=-4x-12 - 2=-4x-14 \). But if we look at the graphs, the y - intercepts of H and J: Let's check the y - intercept. For graph H: Let's see, when \( x = 0 \), what's y? For graph J: Let's see the position. Wait, maybe the original equation was \( y=-4(x + 3)-2 \), let's rewrite it as \( y=-4x-14 \). The slope is - 4, which is a very steep negative slope.

Looking at the graphs, graph J has a steeper negative slope and the correct y - intercept? Wait, no, maybe I messed up the expansion. Wait, no, \( y=-4(x + 3)-2=-4x-12-2=-4x - 14 \). But let's check the x - intercept: \( 0=-4x-14\Rightarrow4x=-14\Rightarrow x =-\frac{14}{4}=-\frac{7}{2}=-3.5 \).

Now, let's look at the graphs:

- Graph H: Let's see the slope. If we take two points, the slope seems less steep (closer to - 1 or - 2).
- Graph J: The slope is steeper (since the slope is - 4). Also, the y - intercept of J: when \( x = 0 \), \( y=-14 \)? But the graphs seem to have y - intercepts around - 7 or something? Wait, maybe I made a mistake in the equation. Wait, wait, \( y=-4(x + 3)-2=-4x-12 - 2=-4x-14 \). Wait, that's correct. But maybe the graphs are drawn with a different scale. Wait, let's check the point \( x=-3 \), \( y=-2 \). So the point \( (-3,-2) \) is on the line. Let's see which graph passes through \( (-3,-2) \) and has a slope of - 4.

Graph J: Let's check the point \( x=-3 \). If \( x=-3 \), \( y=-4(-3)-14 = 12-14=-2 \). Yes, that's correct. And the slope is - 4, which is a steep negative slope. Graph H: Let's check the slope. If we take two points, the slope is not - 4. So the graph that best represents \( y=-4(x + 3)-2 \) is J.

Step2: Analyze the Slope and Intercept

1. **Slope Analysis**: The slope of the line \( y=-4(x + 3)-2 \) (or \( y=-4x-14 \)) is \( m = - 4 \), which is a negative number. This means the line should be decreasing (going down from left to right). So we can eliminate graphs with positive slopes (F and G, as their lines are increasing from left to right).
2. **Y - intercept and Point Check**: When \( x=-3 \), \( y=-2 \) (by substituting \( x = - 3 \) into the equation \( y=-4(x + 3)-2 \)). Also, the slope of - 4 means that for every 1 unit increase in \( x \), \( y \) decreases by 4 units. Graph J has a steep negative slope (consistent with \( m=-4 \)) and passes through the point \( (-3,-2) \) (or has the correct slope - intercept relationship). Graph H has a less steep slope, so it does not match the slope of - 4.

Answer:

J