Answer :
To determine which equation represents the table, we need to test each equation with the given [tex]\(x\)[/tex] values and see which one produces the corresponding [tex]\(y\)[/tex] values from the table.
The table provides the following [tex]\(x\)[/tex] and [tex]\(y\)[/tex] pairs:
- For [tex]\(x = -4\)[/tex], [tex]\(y = -3\)[/tex]
- For [tex]\(x = 0\)[/tex], [tex]\(y = -2\)[/tex]
- For [tex]\(x = 4\)[/tex], [tex]\(y = -1\)[/tex]
Let's test each given equation with these [tex]\(x\)[/tex] values and calculate the corresponding [tex]\(y\)[/tex] values.
### Testing Equation [tex]\( y = \frac{1}{4} x - 2 \)[/tex]
1. When [tex]\( x = -4 \)[/tex]:
[tex]\[ y = \frac{1}{4} (-4) - 2 = -1 - 2 = -3 \][/tex]
Matches the table value.
2. When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = \frac{1}{4} (0) - 2 = 0 - 2 = -2 \][/tex]
Matches the table value.
3. When [tex]\( x = 4 \)[/tex]:
[tex]\[ y = \frac{1}{4} (4) - 2 = 1 - 2 = -1 \][/tex]
Matches the table value.
This equation matches all the [tex]\( y \)[/tex] values in the table.
### Testing Equation [tex]\( y = -\frac{1}{4} x + 4 \)[/tex]
1. When [tex]\( x = -4 \)[/tex]:
[tex]\[ y = -\frac{1}{4} (-4) + 4 = 1 + 4 = 5 \][/tex]
Does not match the table value.
2. When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -\frac{1}{4} (0) + 4 = 0 + 4 = 4 \][/tex]
Does not match the table value.
3. When [tex]\( x = 4 \)[/tex]:
[tex]\[ y = -\frac{1}{4} (4) + 4 = -1 + 4 = 3 \][/tex]
Does not match the table value.
This equation does not match the [tex]\( y \)[/tex] values in the table.
### Testing Equation [tex]\( y = 4 x - 2 \)[/tex]
1. When [tex]\( x = -4 \)[/tex]:
[tex]\[ y = 4 (-4) - 2 = -16 - 2 = -18 \][/tex]
Does not match the table value.
2. When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 4 (0) - 2 = 0 - 2 = -2 \][/tex]
Matches the table value.
3. When [tex]\( x = 4 \)[/tex]:
[tex]\[ y = 4 (4) - 2 = 16 - 2 = 14 \][/tex]
Does not match the table value.
This equation does not match the [tex]\( y \)[/tex] values in the table.
### Testing Equation [tex]\( y = -4 x + 3 \)[/tex]
1. When [tex]\( x = -4 \)[/tex]:
[tex]\[ y = -4 (-4) + 3 = 16 + 3 = 19 \][/tex]
Does not match the table value.
2. When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -4 (0) + 3 = 0 + 3 = 3 \][/tex]
Does not match the table value.
3. When [tex]\( x = 4 \)[/tex]:
[tex]\[ y = -4 (4) + 3 = -16 + 3 = -13 \][/tex]
Does not match the table value.
This equation does not match the [tex]\( y \)[/tex] values in the table.
After testing all the given equations, we can conclude that the equation [tex]\( y = \frac{1}{4} x - 2 \)[/tex] is the one that accurately represents the relationship expressed in the table.
The table provides the following [tex]\(x\)[/tex] and [tex]\(y\)[/tex] pairs:
- For [tex]\(x = -4\)[/tex], [tex]\(y = -3\)[/tex]
- For [tex]\(x = 0\)[/tex], [tex]\(y = -2\)[/tex]
- For [tex]\(x = 4\)[/tex], [tex]\(y = -1\)[/tex]
Let's test each given equation with these [tex]\(x\)[/tex] values and calculate the corresponding [tex]\(y\)[/tex] values.
### Testing Equation [tex]\( y = \frac{1}{4} x - 2 \)[/tex]
1. When [tex]\( x = -4 \)[/tex]:
[tex]\[ y = \frac{1}{4} (-4) - 2 = -1 - 2 = -3 \][/tex]
Matches the table value.
2. When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = \frac{1}{4} (0) - 2 = 0 - 2 = -2 \][/tex]
Matches the table value.
3. When [tex]\( x = 4 \)[/tex]:
[tex]\[ y = \frac{1}{4} (4) - 2 = 1 - 2 = -1 \][/tex]
Matches the table value.
This equation matches all the [tex]\( y \)[/tex] values in the table.
### Testing Equation [tex]\( y = -\frac{1}{4} x + 4 \)[/tex]
1. When [tex]\( x = -4 \)[/tex]:
[tex]\[ y = -\frac{1}{4} (-4) + 4 = 1 + 4 = 5 \][/tex]
Does not match the table value.
2. When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -\frac{1}{4} (0) + 4 = 0 + 4 = 4 \][/tex]
Does not match the table value.
3. When [tex]\( x = 4 \)[/tex]:
[tex]\[ y = -\frac{1}{4} (4) + 4 = -1 + 4 = 3 \][/tex]
Does not match the table value.
This equation does not match the [tex]\( y \)[/tex] values in the table.
### Testing Equation [tex]\( y = 4 x - 2 \)[/tex]
1. When [tex]\( x = -4 \)[/tex]:
[tex]\[ y = 4 (-4) - 2 = -16 - 2 = -18 \][/tex]
Does not match the table value.
2. When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 4 (0) - 2 = 0 - 2 = -2 \][/tex]
Matches the table value.
3. When [tex]\( x = 4 \)[/tex]:
[tex]\[ y = 4 (4) - 2 = 16 - 2 = 14 \][/tex]
Does not match the table value.
This equation does not match the [tex]\( y \)[/tex] values in the table.
### Testing Equation [tex]\( y = -4 x + 3 \)[/tex]
1. When [tex]\( x = -4 \)[/tex]:
[tex]\[ y = -4 (-4) + 3 = 16 + 3 = 19 \][/tex]
Does not match the table value.
2. When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -4 (0) + 3 = 0 + 3 = 3 \][/tex]
Does not match the table value.
3. When [tex]\( x = 4 \)[/tex]:
[tex]\[ y = -4 (4) + 3 = -16 + 3 = -13 \][/tex]
Does not match the table value.
This equation does not match the [tex]\( y \)[/tex] values in the table.
After testing all the given equations, we can conclude that the equation [tex]\( y = \frac{1}{4} x - 2 \)[/tex] is the one that accurately represents the relationship expressed in the table.
