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Chapter 3: Problem 99

In the following exercises, graph each line with the given point and slope. $$ (-4,2) ; m=4 $$

Short Answer

Expert verified
Plot the point (-4, 2), use the slope to find another point, and draw a line through these points.

Step by step solution

01

- Identify the slope and point

The given point is (-4, 2) and the slope is m = 4. Keep these values in mind for the following steps.
02

- Write the equation in point-slope form

Use the point-slope form of the equation of a line y - y_1 = m(x - x_1) , where (x_1, y_1) = (-4, 2) and m = 4. Plug these values into the equation to get y - 2 = 4(x + 4).
03

- Simplify to obtain the slope-intercept form

Expand and simplify the equation: y - 2 = 4(x + 4) becomes y - 2 = 4x + 16 . Add 2 to both sides to get: y = 4x + 18 . This is the slope-intercept form of the equation.
04

- Plot the given point

On a graph, plot the point (-4, 2).
05

- Use the slope to plot another point

Starting from the point (-4, 2), use the slope m = 4, which means a rise of 4 units and a run of 1 unit. Move up 4 units and 1 unit to the right to plot a second point at (-3, 6).
06

- Draw the line

Draw a straight line through the points (-4, 2) and (-3, 6) . This line represents the equation y = 4x + 18.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Slope
The slope of a line (\text{slope is italicized} as \( m \)) is a measure of how steep the line is.
It is calculated by the ratio of the 'rise' (the change in the \( y \) values) to the 'run' (the change in the \( x \) values).
The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here are some key things to keep in mind:
  • A positive slope means the line rises as it moves from left to right.
  • A negative slope means the line falls as it moves from left to right.
  • A slope of zero means the line is horizontal.
  • An undefined slope means the line is vertical.
In our example: the slope \( m \) is 4, indicating the line rises steeply as it moves from left to right.
Point-Slope Form
The point-slope form of the equation of a line is particularly useful when you know a point on the line and the slope.
It is given by the formula:
\[ y - y_1 = m(x - x_1) \]
Here:
  • \( (x_1, y_1) \) is a point on the line.
  • \( m \) is the slope of the line.
In our example: the point given is \((-4, 2)\). Substituting these values into the point-slope form gives:
\[ y - 2 = 4(x + 4) \] This equation can be simplified to give the line's equation in slope-intercept form.
Slope-Intercept Form
The slope-intercept form is widely used for its simplicity in graphing linear equations.
The format of the slope-intercept form is:
\[ y = mx + b \]
Where:
  • \( m \) is the slope of the line.
  • \( b \) is the y-intercept (the point where the line crosses the y-axis).
In our equation from the previous step:
we simplify \( y - 2 = 4(x + 4) \) to get \( y = 4x + 18 \).
From this new form:
  • The slope \( m \) is still 4.
  • The y-intercept \( b \) is 18, meaning the line crosses the y-axis at (0, 18).
Plotting Points and Drawing the Line
To graph a line, we need to plot points and connect them. Let鈥檚 start with the given point and use the slope to find another.
The point provided in the example is \((-4, 2)\).
  • We start by plotting this point on the graph.
Using the slope \( m = 4 \):
  • Starting from (-4, 2), we move 4 units up (rise) and 1 unit to the right (run) to reach a new point (-3, 6).
Plot this new point (-3, 6).
Next: draw a straight line passing through both points (-4, 2) and (-3, 6).
This completes the graph of the line described by the equation \( y = 4x + 18 \).

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