To graph the line given the slope [tex]\( m = 6 \)[/tex] and the y-intercept [tex]\( b = -3 \)[/tex], 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:
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Given:
- [tex]\( m = 6 \)[/tex]
- [tex]\( b = -3 \)[/tex]
So, the equation of the line is:
[tex]\[ y = 6x - 3 \][/tex]
### 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 [tex]\( x = 0 \)[/tex]. Substituting [tex]\( x = 0 \)[/tex] into the equation:
[tex]\[
y = 6(0) - 3 = -3
\][/tex]
So, the y-intercept is at [tex]\( (0, -3) \)[/tex].
- Another Point Using the Slope (m): The slope [tex]\( m = 6 \)[/tex] indicates that for every unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 6. From the y-intercept [tex]\( (0, -3) \)[/tex], we can use the slope to find another point. Starting from [tex]\( (0, -3) \)[/tex], if we increase [tex]\( x \)[/tex] by 1, [tex]\( y \)[/tex] increases by 6:
[tex]\[
\text{New } x = 0 + 1 = 1
\][/tex]
[tex]\[
\text{New } y = -3 + 6 = 3
\][/tex]
So, another point on the line is [tex]\( (1, 3) \)[/tex].
### Step 3: Plot the Points and Draw the Line
1. Plot the y-intercept [tex]\( (0, -3) \)[/tex] on the graph.
2. Plot the point [tex]\( (1, 3) \)[/tex], 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 [tex]\( x \)[/tex]-values and substituting them into the equation [tex]\( y = 6x - 3 \)[/tex].
For [tex]\( x = -1 \)[/tex]:
[tex]\[
y = 6(-1) - 3 = -6 - 3 = -9 \quad \text{(point(-1, -9))}
\][/tex]
For [tex]\( x = 2 \)[/tex]:
[tex]\[
y = 6(2) - 3 = 12 - 3 = 9 \quad \text{(point (2, 9))}
\][/tex]
### Step 5: Draw the Line
1. Draw the line through the points [tex]\( (0, -3) \)[/tex], [tex]\( (1, 3) \)[/tex], [tex]\( (-1, -9) \)[/tex] and [tex]\( (2, 9) \)[/tex].
2. Extend the line in both directions to cover all necessary values of [tex]\( x \)[/tex] on your graph.
### Summary:
You should see a straight line with a steep positive slope of 6, crossing the y-axis at [tex]\( -3 \)[/tex]. 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 [tex]\( y = 6x - 3 \)[/tex].
