Answer :
To determine which graph represents the function given by the equation \(y - 3 = \frac{3}{2}(x - 4)\), we need to rewrite this equation in the slope-intercept form \(y = mx + b\).
### Step-by-step solution:
1. Start with the given equation:
[tex]\[ y - 3 = \frac{3}{2}(x - 4) \][/tex]
2. Expand the right-hand side:
[tex]\[ y - 3 = \frac{3}{2}x - \frac{3}{2} \cdot 4 \][/tex]
3. Simplify the multiplication:
[tex]\[ y - 3 = \frac{3}{2}x - 6 \][/tex]
4. Isolate \(y\) by adding 3 to both sides of the equation:
[tex]\[ y = \frac{3}{2}x - 6 + 3 \][/tex]
5. Simplify the constants on the right-hand side:
[tex]\[ y = \frac{3}{2}x - 3 \][/tex]
Now the equation is in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
### Interpretation of the terms:
- Slope (m): The slope \(m\) is \(\frac{3}{2}\), which means the line rises 3 units for every 2 units it runs to the right.
- Y-intercept (b): The y-intercept \(b\) is \(-3\), meaning the line crosses the y-axis at \(y = -3\).
### Key points to identify the graph:
1. Y-intercept: The graph should intersect the y-axis at \(-3\).
2. Slope: From the y-intercept, for every 2 units you move to the right, the graph should move up 3 units.
### Example plotting:
- Starting at the point (0, -3) on the y-axis:
- Move right 2 units to the point (2, -3)
- From there, move up 3 units to get to the point (2, 0)
- This gives another point on the line (2, 0).
- Connect these points to form a straight line with the specified slope and intercept.
By checking the characteristics of different graphs and ensuring the graph crosses the y-axis at -3 and follows the slope of [tex]\(\frac{3}{2}\)[/tex], you can determine the correct graph representing the function [tex]\(y - 3 = \frac{3}{2}(x - 4)\)[/tex].
### Step-by-step solution:
1. Start with the given equation:
[tex]\[ y - 3 = \frac{3}{2}(x - 4) \][/tex]
2. Expand the right-hand side:
[tex]\[ y - 3 = \frac{3}{2}x - \frac{3}{2} \cdot 4 \][/tex]
3. Simplify the multiplication:
[tex]\[ y - 3 = \frac{3}{2}x - 6 \][/tex]
4. Isolate \(y\) by adding 3 to both sides of the equation:
[tex]\[ y = \frac{3}{2}x - 6 + 3 \][/tex]
5. Simplify the constants on the right-hand side:
[tex]\[ y = \frac{3}{2}x - 3 \][/tex]
Now the equation is in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
### Interpretation of the terms:
- Slope (m): The slope \(m\) is \(\frac{3}{2}\), which means the line rises 3 units for every 2 units it runs to the right.
- Y-intercept (b): The y-intercept \(b\) is \(-3\), meaning the line crosses the y-axis at \(y = -3\).
### Key points to identify the graph:
1. Y-intercept: The graph should intersect the y-axis at \(-3\).
2. Slope: From the y-intercept, for every 2 units you move to the right, the graph should move up 3 units.
### Example plotting:
- Starting at the point (0, -3) on the y-axis:
- Move right 2 units to the point (2, -3)
- From there, move up 3 units to get to the point (2, 0)
- This gives another point on the line (2, 0).
- Connect these points to form a straight line with the specified slope and intercept.
By checking the characteristics of different graphs and ensuring the graph crosses the y-axis at -3 and follows the slope of [tex]\(\frac{3}{2}\)[/tex], you can determine the correct graph representing the function [tex]\(y - 3 = \frac{3}{2}(x - 4)\)[/tex].
