To determine which graph represents the equation [tex]\( y = \frac{3}{2} x - 2 \)[/tex], we can use several steps. We'll start by identifying key points on the line using some selected [tex]\( x \)[/tex]-values.
### Step-by-Step Solution:
1. Identify the Slope and Y-Intercept:
- The slope ([tex]\( m \)[/tex]) is [tex]\( \frac{3}{2} \)[/tex].
- The y-intercept ([tex]\( b \)[/tex]) is [tex]\(-2\)[/tex].
2. Calculate Points on the Line:
We'll use a few distinct [tex]\( x \)[/tex]-values to calculate corresponding [tex]\( y \)[/tex]-values using the equation [tex]\( y = \frac{3}{2} x - 2 \)[/tex].
- For [tex]\( x = -2 \)[/tex]:
[tex]\[
y = \frac{3}{2}(-2) - 2 = -3 - 2 = -5
\][/tex]
Point: (-2, -5)
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
y = \frac{3}{2}(0) - 2 = 0 - 2 = -2
\][/tex]
Point: (0, -2)
- For [tex]\( x = 2 \)[/tex]:
[tex]\[
y = \frac{3}{2}(2) - 2 = 3 - 2 = 1
\][/tex]
Point: (2, 1)
- For [tex]\( x = 4 \)[/tex]:
[tex]\[
y = \frac{3}{2}(4) - 2 = 6 - 2 = 4
\][/tex]
Point: (4, 4)
- For [tex]\( x = 6 \)[/tex]:
[tex]\[
y = \frac{3}{2}(6) - 2 = 9 - 2 = 7
\][/tex]
Point: (6, 7)
3. List the Points:
The points calculated are:
- (-2, -5)
- (0, -2)
- (2, 1)
- (4, 4)
- (6, 7)
4. Plot the Points:
- Start by plotting the y-intercept (0, -2).
- Plot the remaining points: (-2, -5), (2, 1), (4, 4), and (6, 7).
5. Draw the Line:
- Connect the points with a straight line, ensuring that the line extends in both directions.
### Conclusion:
The correct graph will show a line passing through the points (-2, -5), (0, -2), (2, 1), (4, 4), and (6, 7), with a slope that rises [tex]\(\frac{3}{2}\)[/tex] for every unit it moves to the right. This results in a line that starts at an intercept of -2 on the y-axis and ascends upwards.
