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Sagot :
To determine which statement is true about the graph of the given equation:
[tex]\[ y + 4 = 4(x + 1) \][/tex]
we start by converting the equation into the slope-intercept form [tex]\( y = mx + b \)[/tex].
1. First, distribute the 4 on the right-hand side:
[tex]\[ y + 4 = 4x + 4 \][/tex]
2. Next, isolate [tex]\( y \)[/tex] by subtracting 4 from both sides:
[tex]\[ y = 4x + 4 - 4 \][/tex]
[tex]\[ y = 4x \][/tex]
Now, the equation is in the slope-intercept form [tex]\( y = 4x \)[/tex]. This means the line has a slope of 4 and a y-intercept of 0. We will verify which set of points lies on this line.
### Checking each option:
- Option A: (1, -4) and (0, 0)
- Plugging [tex]\( x = 1 \)[/tex] into [tex]\( y = 4x \)[/tex]:
[tex]\[ y = 4(1) = 4 \][/tex]
This does not match with the point (1, -4).
- Plugging [tex]\( x = 0 \)[/tex] into [tex]\( y = 4x \)[/tex]:
[tex]\[ y = 4(0) = 0 \][/tex]
This matches with the point (0, 0), but since both points must satisfy the equation, this option is incorrect.
- Option B: (1, 4) and (2, 8)
- Plugging [tex]\( x = 1 \)[/tex] into [tex]\( y = 4x \)[/tex]:
[tex]\[ y = 4(1) = 4 \][/tex]
This matches with the point (1, 4).
- Plugging [tex]\( x = 2 \)[/tex] into [tex]\( y = 4x \)[/tex]:
[tex]\[ y = 4(2) = 8 \][/tex]
This matches with the point (2, 8), so this option is correct.
- Option C: (-4, -1) and (-3, -3)
- Plugging [tex]\( x = -4 \)[/tex] into [tex]\( y = 4x \)[/tex]:
[tex]\[ y = 4(-4) = -16 \][/tex]
This does not match with the point (-4, -1).
- Plugging [tex]\( x = -3 \)[/tex] into [tex]\( y = 4x \)[/tex]:
[tex]\[ y = 4(-3) = -12 \][/tex]
This does not match with the point (-3, -3), so this option is incorrect.
- Option D: (4, 1) and (5, 5)
- Plugging [tex]\( x = 4 \)[/tex] into [tex]\( y = 4x \)[/tex]:
[tex]\[ y = 4(4) = 16 \][/tex]
This does not match with the point (4, 1).
- Plugging [tex]\( x = 5 \)[/tex] into [tex]\( y = 4x \)[/tex]:
[tex]\[ y = 4(5) = 20 \][/tex]
This does not match with the point (5, 5), so this option is incorrect.
Since the correct option must satisfy the equation [tex]\( y = 4x \)[/tex] for both points given, the true statement about the graph is:
B. The graph is a line that goes through the points (1,4) and (2,8).
[tex]\[ y + 4 = 4(x + 1) \][/tex]
we start by converting the equation into the slope-intercept form [tex]\( y = mx + b \)[/tex].
1. First, distribute the 4 on the right-hand side:
[tex]\[ y + 4 = 4x + 4 \][/tex]
2. Next, isolate [tex]\( y \)[/tex] by subtracting 4 from both sides:
[tex]\[ y = 4x + 4 - 4 \][/tex]
[tex]\[ y = 4x \][/tex]
Now, the equation is in the slope-intercept form [tex]\( y = 4x \)[/tex]. This means the line has a slope of 4 and a y-intercept of 0. We will verify which set of points lies on this line.
### Checking each option:
- Option A: (1, -4) and (0, 0)
- Plugging [tex]\( x = 1 \)[/tex] into [tex]\( y = 4x \)[/tex]:
[tex]\[ y = 4(1) = 4 \][/tex]
This does not match with the point (1, -4).
- Plugging [tex]\( x = 0 \)[/tex] into [tex]\( y = 4x \)[/tex]:
[tex]\[ y = 4(0) = 0 \][/tex]
This matches with the point (0, 0), but since both points must satisfy the equation, this option is incorrect.
- Option B: (1, 4) and (2, 8)
- Plugging [tex]\( x = 1 \)[/tex] into [tex]\( y = 4x \)[/tex]:
[tex]\[ y = 4(1) = 4 \][/tex]
This matches with the point (1, 4).
- Plugging [tex]\( x = 2 \)[/tex] into [tex]\( y = 4x \)[/tex]:
[tex]\[ y = 4(2) = 8 \][/tex]
This matches with the point (2, 8), so this option is correct.
- Option C: (-4, -1) and (-3, -3)
- Plugging [tex]\( x = -4 \)[/tex] into [tex]\( y = 4x \)[/tex]:
[tex]\[ y = 4(-4) = -16 \][/tex]
This does not match with the point (-4, -1).
- Plugging [tex]\( x = -3 \)[/tex] into [tex]\( y = 4x \)[/tex]:
[tex]\[ y = 4(-3) = -12 \][/tex]
This does not match with the point (-3, -3), so this option is incorrect.
- Option D: (4, 1) and (5, 5)
- Plugging [tex]\( x = 4 \)[/tex] into [tex]\( y = 4x \)[/tex]:
[tex]\[ y = 4(4) = 16 \][/tex]
This does not match with the point (4, 1).
- Plugging [tex]\( x = 5 \)[/tex] into [tex]\( y = 4x \)[/tex]:
[tex]\[ y = 4(5) = 20 \][/tex]
This does not match with the point (5, 5), so this option is incorrect.
Since the correct option must satisfy the equation [tex]\( y = 4x \)[/tex] for both points given, the true statement about the graph is:
B. The graph is a line that goes through the points (1,4) and (2,8).
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