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Chapter 4: Problem 45

Graph the line containing the given point and with the given slope. $$(6,2) ; m=-4$$

Short Answer

Expert verified
The line containing the point (6, 2) and with a slope of -4 has the equation \(y = -4x + 26\). Plot the given point (6, 2) and the y-intercept (0, 26), then connect them with a straight line to graph the line.

Step by step solution

01

Use the point-slope form

Write the point-slope form using the given point (6, 2) and slope -4: \[y - 2 = -4(x - 6)\]
02

Expand and simplify the equation

Expand the equation to get the equation of the line containing the given point and with the given slope: \[y - 2 = -4x + 24\] Add 2 to both sides of the equation to rewrite it in slope-intercept form: \[y = -4x + 26\]
03

Identify the y-intercept

The slope-intercept form of the equation of a line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In the equation \(y = -4x + 26\), the y-intercept is 26. This means that the line crosses the y-axis at the point (0, 26).
04

Plot the given point and the y-intercept

To start graphing the line, plot the given point (6, 2) and the y-intercept (0, 26) on the coordinate plane.
05

Draw the line

Connect the points (6, 2) and (0, 26) with a straight line. This line represents the equation \(y = -4x + 26\), and it contains the given point (6, 2) and has a slope of -4.

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Key Concepts

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

Point-Slope Form
Point-slope form is a way to write the equation of a straight line. It's especially useful when you have a point on the line and the slope.
The formula for point-slope form is:\[ y - y_1 = m(x - x_1) \]where:
  • \((x_1, y_1)\) is a point on the line
  • \(m\) is the slope
For example, if you know the line passes through the point \((6, 2)\) with a slope of \(-4\), you can plug these values into the formula:\[ y - 2 = -4(x - 6) \]This equation represents the line with the specific slope and point given.
To graph from here, you might want to convert it into another form that's easy to understand, like the slope-intercept form.
Slope-Intercept Form
The slope-intercept form is a popular way to express the equation of a line. It's very straightforward because you can see the slope and where the line crosses the y-axis right away.
The formula is:\[ y = mx + b \]where:
  • \(m\) is the slope
  • \(b\) is the y-intercept
To convert the point-slope form to the slope-intercept form, you simply expand and solve for \(y\). For instance:
Starting with:\[ y - 2 = -4(x - 6) \]Expanding gives:\[ y = -4x + 24 + 2 \]Simplifying, we find:\[ y = -4x + 26 \]Now the equation is in slope-intercept form, showing a slope \(-4\) and a y-intercept at \(26\). This is an easy form for graphing too.
Y-Intercept
The y-intercept is a crucial part of understanding and graphing a line. It's the point where the line crosses the y-axis, which happens when \(x = 0\).
In our slope-intercept form equation:\[ y = -4x + 26 \]the y-intercept \(b\) is \(26\). This tells us that the line crosses the y-axis at the point \((0, 26)\).
When you're graphing, starting with the y-intercept gives you an anchor point on the graph. From \((0, 26)\), you can use the slope to identify other points on the line.
  • For a slope of \(-4\), it means you go down 4 units and right 1 unit from the y-intercept, each time.
This makes it simple to construct the full line on a graph.

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