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Chapter 5: Problem 34

Graph the line containing the given point and with the given slope. $$(-3,-2) ; m=4$$

Short Answer

Expert verified
The equation of the line in slope-intercept form is \(y = 4x + 10\). The y-intercept is \(b = 10\), or the point \((0, 10)\). To graph the line, plot the given point \((-3, -2)\), use the slope \(m = 4\) to plot another point by moving 4 units up and 1 unit to the right, and plot the y-intercept \((0, 10)\). Draw a straight line through these points.

Step by step solution

01

Use the point-slope form to find the equation of the line

We are given the point \((-3, -2)\) and the slope \(m = 4\). Using these values, we can plug them into the point-slope form: \(y - (-2) = 4(x - (-3))\)
02

Simplify the equation

Now, we'll simplify the equation: \(y + 2 = 4(x + 3)\)
03

Rewrite the equation in slope-intercept form

Expand and isolate y to rewrite the equation in slope-intercept form: \(y + 2 = 4x + 12\) \(y = 4x + 10\)
04

Identify the y-intercept

From the slope-intercept form, we can see that the y-intercept is \(b = 10\), which means the line intersects the y-axis at the point \((0, 10)\).
05

Graph the line

To graph the line: 1. Plot the given point \((-3, -2)\). 2. Use the slope of \(m = 4\), which means that the line rises by 4 units for each unit it moves to the right. Starting from the given point, move 4 units up and 1 unit to the right, and plot another point there. 3. Plot the y-intercept \((0, 10)\). 4. Draw a straight line through the points. Now, you have successfully graphed the line containing the given point and with the given slope.

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Key Concepts

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

Point-Slope Form
The point-slope form is a useful equation for graphing lines when you know a point on the line and the slope. It's written as \( y - y_1 = m(x - x_1) \), where \( m \) is the slope, and \((x_1, y_1)\) is a point on the line. In the original exercise, we have the point \((-3, -2)\) and the slope \(m = 4\). By substituting these values into the point-slope form, we get:
  • \( y - (-2) = 4(x - (-3)) \)
Simplifying this, we have:
  • \( y + 2 = 4(x + 3) \)
This equation represents the line using information from both the point and the slope.

Remember, the point-slope form is great for quickly constructing the equation of a line once you have a point and a slope.
Slope-Intercept Form
The slope-intercept form is another way to write the equation of a line. This form is particularly helpful for graphing, as it directly reveals the slope and y-intercept. It's written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. From the exercise, we started with the point-slope form:
  • \( y + 2 = 4(x + 3) \)
We expand and solve for \( y \) to convert it to slope-intercept form:
  • \( y + 2 = 4x + 12 \)
  • \( y = 4x + 10 \)
Now, the equation \( y = 4x + 10 \) shows us that the slope of the line is \( 4 \) and the y-intercept is \( 10 \).

This form is excellent for graphing since it provides immediate understanding of how the line behaves and where it crosses the y-axis.
Y-Intercept
The y-intercept is a key feature of a line and represents the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), the y-intercept is \( b \). From our derived equation \( y = 4x + 10 \), we see that the y-intercept is \( 10 \).
  • This means the line crosses the y-axis at \((0, 10)\).
To graph the line, knowing the y-intercept helps in quickly locating one point on the graph.

Along with the slope, the y-intercept allows us to draw the line accurately. It's a simple yet powerful piece of information that simplifies graphing and understanding linear equations.

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Most popular questions from this chapter

Write a system of equations and solve. A movie theater charges \(\$ 9.00\) for an adult's ticket. S7. OO for a ticket for scriors 60 and over, and \(\$ 6,00\) for a child's ticket. For a particular movie, the theater sold a total of 290 tickets. which brought in \(\$ 2400\). The number of semors' 'tickets sold was twice the number of children's tickets sold. Determine the number of ndults", scniors', and children's tickets sold.

The best-selling paper towel brands in 2002 were Bounty, Brawny, and Scott. Together they accounted for \(59 \%\) of the market. Bounty's market share was \(25 \%\) more than Brawny's, and Scott's market share was \(2 \%\) Iess than Brawny's. What percentage of the market did each brand hold in \(2002 ?\) (Source: \(U S A\) Today. Oct. 23 \(2003, p .3 B\) from Information 91影视, lnc.)

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Determine the domain of each relation, and determine whether each relation describes \(y\) as a function of \(x .\) $$y=4 x-7$$
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