To graph a line with a slope of [tex]\( 4 \)[/tex] that passes through the point [tex]\( (3, 0) \)[/tex], we will follow these steps:
### Step 1: Understand the Point-Slope Form
The point-slope form of the equation of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where:
- [tex]\( m \)[/tex] is the slope of the line,
- [tex]\( (x_1, y_1) \)[/tex] is a point on the line.
Given:
- Slope [tex]\( m = 4 \)[/tex],
- Point [tex]\( (x_1, y_1) = (3, 0) \)[/tex].
### Step 2: Plug Into the Point-Slope Form
Substitute the given slope and point into the point-slope form equation:
[tex]\[ y - 0 = 4(x - 3) \][/tex]
Simplify this to:
[tex]\[ y = 4(x - 3) \][/tex]
This is the equation of the line in point-slope form.
### Step 3: Simplify to Slope-Intercept Form
Convert the equation to the slope-intercept form [tex]\( y = mx + b \)[/tex] by distributing the slope on the right-hand side:
[tex]\[ y = 4x - 12 \][/tex]
This is the slope-intercept form of the equation where [tex]\( m = 4 \)[/tex] (the slope) and [tex]\( b = -12 \)[/tex] (the y-intercept).
### Step 4: Plot the Line
1. Identify Key Points:
- Starting Point: The point [tex]\( (3, 0) \)[/tex] is given.
- Y-Intercept: From the equation [tex]\( y = 4x - 12 \)[/tex], the y-intercept is [tex]\( -12 \)[/tex]. Therefore, another point is [tex]\( (0, -12) \)[/tex].
2. Draw Axes:
- Draw the x-axis and y-axis on a graph.
3. Plot Points:
- Plot the starting point [tex]\( (3, 0) \)[/tex].
- Plot the y-intercept [tex]\( (0, -12) \)[/tex].
4. Draw the Line:
- Draw a straight line through these points. Ensure that the line extends in both directions, indicating the correct slope.
### Step 5: Label the Graph
- Label the axes (x-axis and y-axis).
- Provide a title for the graph, such as "Graph of the Line with Slope 4 Through Point (3, 0)".
- Optionally, label the points [tex]\( (3, 0) \)[/tex] and [tex]\( (0, -12) \)[/tex] on the graph.
### Recap of Key Points
- Equation in point-slope form: [tex]\( y = 4(x - 3) \)[/tex]
- Equation in slope-intercept form: [tex]\( y = 4x - 12 \)[/tex]
- Points on the line: [tex]\( (3, 0) \)[/tex] and [tex]\( (0, -12) \)[/tex]
By following these steps, you can successfully graph the line with slope [tex]\( 4 \)[/tex] that passes through the point [tex]\( (3, 0) \)[/tex].
