To graph the line given by the equation \(\( y = 2x - 6 \)\), follow these steps:
1. Identify the Equation and Components:
- The given equation is \(\( y = 2x - 6 \)\).
- The equation is in the slope-intercept form \(\( y = mx + b \)\), where:
- \(\( m \)\) is the slope (rate of change) of the line.
- \(\( b \)\) is the y-intercept (the value of \(\( y \)\) when \(\( x = 0 \)\)).
2. Determine the Slope and Y-Intercept:
- \(\( m = 2 \)\) (slope), which means that for every unit increase in \(\( x \)\), \(\( y \)\) increases by 2 units.
- \(\( b = -6 \)\) (y-intercept), which means the line crosses the y-axis at \(\( y = -6 \)\).
3. Plot the Y-Intercept:
- Locate the point \(\( (0, -6) \)\) on the graph and put a point there. This is the y-intercept.
4. Use the Slope to Find Another Point:
- Starting from the y-intercept \(\( (0, -6) \)\), use the slope \(\( m = 2 \)\) to find another point.
- Since the slope is \(\( 2 \)\), move up 2 units in the y-direction and 1 unit in the x-direction.
- Doing this, from \(\( (0, -6) \)\), we move to the point \(\( (1, -4) \)\).
5. Plot the Second Point:
- Locate the point \(\( (1, -4) \)\) on the graph and put a point there.
6. Draw the Line:
- Draw a straight line passing through the points \(\( (0, -6) \)\) and \(\( (1, -4) \)\).
- Extend this line in both directions and ensure it covers the entire graph.
7. Verify with More Points (Optional):
- You can verify the accuracy by choosing another \(\( x \)\)-value, say \(\( x = 2 \)\):
- Plug \(\( x = 2 \)\) into the equation \(\( y = 2(2) - 6 \)\), which gives \(\( y = 4 - 6 = -2 \)\).
- Plot the point \(\( (2, -2) \)\) to ensure it lies on the line.
- Similarly, verify with a negative \(\( x \)\)-value, say \(\( x = -1 \)\):
- Plug \(\( x = -1 \)\) into the equation \(\( y = 2(-1) - 6 \)\), which gives \(\( y = -2 - 6 = -8 \)\).
- Plot the point \(\( (-1, -8) \)\) to ensure it lies on the line.
Your graph should look like this:
``
y
|
10 | .
9 | .
8 | .
7 | .
6 |
5 | .
4 | .
3 | .
2 | .
1 | .
0 o--------------------------------------| x
-1 | .
-2 | .
-3 | .
-4 o (1, -4)
-5 |
-6 o (0, -6)
-7 |
-8 o (-1, -8)
-9 |
-10 |
This shows the line \(\( y = 2x - 6 \)\) graphed on the coordinate plane. The line crosses the y-axis at \(\( y = -6 \)\) and rises with a slope of \(\( 2 \)\).
