Question
Which graph represents the function y – 3 = y minus 3 equals StartFraction 3 Over 2 EndFraction left-parenthesis x minus 4 right-parenthesis.(x – 4)?
A coordinate grid with a line passing through the points (0, negative 3) and (2, 0).
A coordinate grid with a line passing through the points (negative 2, 0) and (0, negative 3).
A coordinate grid with a line passing through the points (negative 4, negative 3) and (2, 1).
A coordinate grid with a line passing through the points (negative 1, 1) and (1, 1).
A coordinate grid with a line passing through the points (0, negative 3) and (2, 0).
A coordinate grid with a line passing through the points (negative 2, 0) and (0, negative 3).
A coordinate grid with a line passing through the points (negative 4, negative 3) and (2, 1).
A coordinate grid with a line passing through the points (negative 1, 1) and (1, 1).
Answers
GPT-4o mini
To determine which graph represents the function given by the equation \( y - 3 = \frac{3}{2}(x - 4) \), we can first rewrite the equation in slope-intercept form (y = mx + b), where \( m \) is the slope and \( b \) is the y-intercept.
1. Start with the equation:
\[
y - 3 = \frac{3}{2}(x - 4)
\]
2. Distribute \( \frac{3}{2} \):
\[
y - 3 = \frac{3}{2}x - 6
\]
3. Add 3 to both sides to solve for \( y \):
\[
y = \frac{3}{2}x - 6 + 3
\]
\[
y = \frac{3}{2}x - 3
\]
Now, we see that the slope \( m \) is \( \frac{3}{2} \) and the y-intercept \( b \) is \( -3 \). This means the line crosses the y-axis at \( (0, -3) \).
Next, we will check if the points mentioned in the choices can confirm this information:
- The graph should have a y-intercept at (0, -3) and be consistent with a slope of \( \frac{3}{2} \).
Now, let's examine the options:
1. **First Option**: A line passing through the points (0, -3) and (2, 0):
- The point (0, -3) is the y-intercept.
- The line rises from (0, -3) to (2, 0), which means it has a positive slope.
- It shows a rise of 3 (from -3 to 0) for a run of 2 (from 0 to 2), which is a slope of \( \frac{3}{2} \).
2. **Second Option**: A line passing through (−2, 0) and (0, −3):
- The line crosses y at (0, -3), but it decreases more than it rises when calculating based on these two points.
3. **Third Option**: A line passing through (−4, −3) and (2, 1):
- The crossing point is not (0, -3). Plus, the rise is not consistent with our slope.
4. **Fourth Option**: A line passing through (−1, 1) and (1, 1):
- This is flat, with no y-intercept at (0, -3) and does not match.
From this analysis, the correct representation for the function \( y - 3 = \frac{3}{2}(x - 4) \) is the **first option**, which has the points (0, -3) and (2, 0). It has the correct slope and intercept.
1. Start with the equation:
\[
y - 3 = \frac{3}{2}(x - 4)
\]
2. Distribute \( \frac{3}{2} \):
\[
y - 3 = \frac{3}{2}x - 6
\]
3. Add 3 to both sides to solve for \( y \):
\[
y = \frac{3}{2}x - 6 + 3
\]
\[
y = \frac{3}{2}x - 3
\]
Now, we see that the slope \( m \) is \( \frac{3}{2} \) and the y-intercept \( b \) is \( -3 \). This means the line crosses the y-axis at \( (0, -3) \).
Next, we will check if the points mentioned in the choices can confirm this information:
- The graph should have a y-intercept at (0, -3) and be consistent with a slope of \( \frac{3}{2} \).
Now, let's examine the options:
1. **First Option**: A line passing through the points (0, -3) and (2, 0):
- The point (0, -3) is the y-intercept.
- The line rises from (0, -3) to (2, 0), which means it has a positive slope.
- It shows a rise of 3 (from -3 to 0) for a run of 2 (from 0 to 2), which is a slope of \( \frac{3}{2} \).
2. **Second Option**: A line passing through (−2, 0) and (0, −3):
- The line crosses y at (0, -3), but it decreases more than it rises when calculating based on these two points.
3. **Third Option**: A line passing through (−4, −3) and (2, 1):
- The crossing point is not (0, -3). Plus, the rise is not consistent with our slope.
4. **Fourth Option**: A line passing through (−1, 1) and (1, 1):
- This is flat, with no y-intercept at (0, -3) and does not match.
From this analysis, the correct representation for the function \( y - 3 = \frac{3}{2}(x - 4) \) is the **first option**, which has the points (0, -3) and (2, 0). It has the correct slope and intercept.