To graph the line given the slope \(\( m = 6 \)\) and the y-intercept \(\( b = -3 \)\), we follow these steps:
### Step 1: Understand the Equation of the Line
The standard form of the equation of a line in slope-intercept form is:
\(\[ y = mx + b \]\)
where \(\( m \)\) is the slope and \(\( b \)\) is the y-intercept.
Given:
- \(\( m = 6 \)\)
- \(\( b = -3 \)\)
So, the equation of the line is:
\(\[ y = 6x - 3 \]\)
### Step 2: Identify Key Points for Plotting
- Y-intercept (b): The y-intercept is the point where the line crosses the y-axis. This occurs when \(\( x = 0 \)\). Substituting \(\( x = 0 \)\) into the equation:
\(\[
y = 6(0) - 3 = -3
\]\)
So, the y-intercept is at \(\( (0, -3) \)\).
- Another Point Using the Slope (m): The slope \(\( m = 6 \)\) indicates that for every unit increase in \(\( x \)\), \(\( y \)\) increases by 6. From the y-intercept \(\( (0, -3) \)\), we can use the slope to find another point. Starting from \(\( (0, -3) \)\), if we increase \(\( x \)\) by 1, \(\( y \)\) increases by 6:
\(\[
\text{New } x = 0 + 1 = 1
\]\)
\(\[
\text{New } y = -3 + 6 = 3
\]\)
So, another point on the line is \(\( (1, 3) \)\).
### Step 3: Plot the Points and Draw the Line
1. Plot the y-intercept \(\( (0, -3) \)\) on the graph.
2. Plot the point \(\( (1, 3) \)\), derived using the slope.
3. Draw the line through these points, extending it across the graph.
### Step 4: Additional Points for Verification (Optional)
To ensure accuracy, you can calculate a few more points by choosing different \(\( x \)\)-values and substituting them into the equation \(\( y = 6x - 3 \)\).
For \(\( x = -1 \)\):
\(\[
y = 6(-1) - 3 = -6 - 3 = -9 \quad \text{(point(-1, -9))}
\]\)
For \(\( x = 2 \)\):
\(\[
y = 6(2) - 3 = 12 - 3 = 9 \quad \text{(point (2, 9))}
\]\)
### Step 5: Draw the Line
1. Draw the line through the points \(\( (0, -3) \)\), \(\( (1, 3) \)\), \(\( (-1, -9) \)\) and \(\( (2, 9) \)\).
2. Extend the line in both directions to cover all necessary values of \(\( x \)\) on your graph.
### Summary:
You should see a straight line with a steep positive slope of 6, crossing the y-axis at \(\( -3 \)\). The more points you plot, the more accurate your graph will be, but two points are sufficient to define the line. This line represents the equation \(\( y = 6x - 3 \)\).
