Question

the graph of an equation is sketched in the figure. describe five ordered - pair solutions of this equation by using a table. (simplify your answers.)

x	y
- 4	\square
- 2	\square
0	\square
2	\square
4	\square

Explanation:

Step1: Analyze the graph

The graph is a straight line. Let's find the equation of the line. From the graph, we can see two points: when \( x = -2 \), \( y = -1 \) (wait, no, let's check the grid. Wait, looking at the graph, when \( x = -4 \), let's see the y-coordinate. Wait, the line passes through, for example, when \( x = -4 \), let's check the grid. Wait, the line: let's find two points. Let's take \( x = -2 \), \( y = -1 \)? No, wait the grid has each square as 1 unit. Wait, looking at the graph, when \( x = -4 \), the y-coordinate: let's see the line. Wait, the line goes through (-4, 0)? Wait no, let's check the points. Wait, when \( x = -2 \), \( y = -1 \)? No, maybe I made a mistake. Wait, let's look at the graph again. The line: when \( x = -4 \), let's see the vertical line at \( x = -4 \), intersects the line at y=0? Wait no, the line is decreasing. Let's find the slope. Let's take two points: ( -2, -1 )? No, wait, when \( x = 0 \), what's y? Wait, the line crosses the y-axis at (0, -2)? Wait no, looking at the graph, when \( x = -2 \), \( y = -1 \)? No, maybe the line is \( y = -\frac{1}{2}x - 2 \)? Wait, let's check. If \( x = -4 \), then \( y = -\frac{1}{2}(-4) - 2 = 2 - 2 = 0 \). If \( x = -2 \), \( y = -\frac{1}{2}(-2) - 2 = 1 - 2 = -1 \)? No, that doesn't match. Wait, maybe the line is \( y = -\frac{1}{2}x - 1 \)? No, let's check the points. Wait, when \( x = -4 \), the line is at y=0? Wait, the grid: each square is 1 unit. So when \( x = -4 \), moving up from x=-4, the line is at y=0? Wait, no, the line is blue. Let's see: when \( x = -4 \), the y-coordinate is 0? Wait, then when \( x = -2 \), y = -1? No, wait, maybe the line is \( y = -\frac{1}{2}x \). Wait, no. Wait, let's take \( x = -4 \), y=0; \( x = -2 \), y=-1; \( x = 0 \), y=-2; \( x = 2 \), y=-3; \( x = 4 \), y=-4. Wait, that makes sense. Let's check: slope between (-4, 0) and (-2, -1) is \( \frac{-1 - 0}{-2 - (-4)} = \frac{-1}{2} = -\frac{1}{2} \). Then the equation is \( y - 0 = -\frac{1}{2}(x + 4) \), so \( y = -\frac{1}{2}x - 2 \)? Wait, no, if \( x = -4 \), \( y = -\frac{1}{2}(-4) - 2 = 2 - 2 = 0 \), correct. \( x = -2 \), \( y = -\frac{1}{2}(-2) - 2 = 1 - 2 = -1 \)? No, that's not. Wait, maybe the equation is \( y = -\frac{1}{2}x \). Wait, if \( x = -4 \), \( y = 2 \)? No, that's not. Wait, I think I messed up the graph. Wait, the graph: the line is going from the top left (high x negative, high y positive) to bottom right (high x positive, low y negative). Wait, when \( x = -4 \), let's see the y-coordinate: looking at the graph, the line at \( x = -4 \) is at y=0? Wait, no, the line is blue, and when \( x = -4 \), the vertical line x=-4 intersects the blue line at y=0? Then \( x = -2 \), y=-1? \( x = 0 \), y=-2? \( x = 2 \), y=-3? \( x = 4 \), y=-4? Let's check the slope: from (-4, 0) to (-2, -1): slope is (-1 - 0)/(-2 - (-4)) = -1/2. From (-2, -1) to (0, -2): slope is (-2 - (-1))/(0 - (-2)) = -1/2. So the slope is -1/2, and the y-intercept: when x=0, y=-2. So the equation is \( y = -\frac{1}{2}x - 2 \)? Wait, no, when x=0, y=-2? But in the table, x=0 is a row. Wait, the table has x=-4, -2, 0, 2, 4. Let's compute y for each x using the equation \( y = -\frac{1}{2}x - 1 \)? No, wait, let's check the graph again. Wait, maybe the line passes through (-4, 0), (-2, -1), (0, -2), (2, -3), (4, -4). Let's verify:

For x = -4: y = -\frac{1}{2}(-4) - 2 = 2 - 2 = 0. Correct.

For x = -2: y = -\frac{1}{2}(-2) - 2 = 1 - 2 = -1. Correct.

For x = 0: y = -\frac{1}{2}(0) - 2 = -2. Correct.

For x = 2: y = -\frac{1}{2}(2) - 2 = -1 - 2 = -3. Correct.

For x = 4: y = -\frac{1}{2}(4) - 2 = -2 - 2 = -4. Correct.

Yes, that makes sense. So the equation is \( y = -\frac{1}{2}x - 2 \). Wait, no, when x=-4, y=0: -\frac{1}{2}*(-4) -2 = 2 -2 =0. Correct. x=-2: -\frac{1}{2}*(-2) -2=1-2=-1. Correct. x=0: -2. Correct. x=2: -3. Correct. x=4: -4. Correct.

Step2: Fill the table

Now, fill the table with these y-values.

For x = -4: y = 0

For x = -2: y = -1

For x = 0: y = -2

For x = 2: y = -3

For x = 4: y = -4

Wait, but let's check the graph again. When x=-4, the line is at y=0? Yes, the vertical line x=-4 intersects the blue line at y=0. Then x=-2: y=-1, x=0: y=-2, x=2: y=-3, x=4: y=-4. That matches the slope of -1/2.

So the table should be:

x | y

-4 | 0

-2 | -1

0 | -2

2 | -3

4 | -4

Answer:

x	y
-2	-1
0	-2
2	-3
4	-4

So the filled table is:

For x=-4, y=0; x=-2, y=-1; x=0, y=-2; x=2, y=-3; x=4, y=-4.