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Chapter 3: Problem 2

Use the slope-intercept form to graph each equation. $$ y=-4 x-1 $$

Short Answer

Expert verified
Plot \((0, -1)\), use a slope of -4 to find another point, and draw the line.

Step by step solution

01

Identify the Slope and Y-intercept

The equation given is \( y = -4x - 1 \). This is in the slope-intercept form \( y = mx + b \). Here, \( m \), the slope, is \(-4\), and \( b \), the y-intercept, is \(-1\).
02

Plot the Y-intercept

To start graphing, plot the y-intercept on the graph. This is the point where the line crosses the y-axis. For \( b = -1 \), mark the point \((0, -1)\) on the graph.
03

Use the Slope to Determine the Next Point

The slope \( m = -4 \) can be expressed as \(-4/1\), meaning for every 1 unit you move to the right on the x-axis, move 4 units down on the y-axis. From the y-intercept \((0, -1)\), move to the right 1 unit to \( x = 1 \), then 4 units down to \( y = -5 \). Plot the point \((1, -5)\) on the graph.
04

Draw the Line

Connect the points \((0, -1)\) and \((1, -5)\) with a straight line. Extend the line across the graph to represent all the solutions to the equation \( y = -4x - 1 \).

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Key Concepts

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

Graphing Linear Equations
Graphing linear equations is a fundamental skill in algebra, allowing you to visualize relationships between variables. To graph a linear equation in the slope-intercept form, start by identifying the two main components: the slope and the y-intercept. This form is given by the equation \( y = mx + b \), where \( m \) represents the slope and \( b \) the y-intercept.

Steps to Graphing a Linear Equation:
  • Identify the Slope and Y-Intercept: Recognizing these components is key. They give you the starting point and direction of the line.
  • Plot the Y-Intercept: Start by plotting the y-intercept, this is where the line will cross the y-axis.
  • Use the Slope: From the y-intercept, use the slope to find another point on the line.
  • Draw the Line: Connect the points to form a straight line extending across the graph.
Practicing these steps helps in understanding how linear equations depict a constant rate of change in real-world situations.Exploring these concepts through graphing can make these abstract numbers come alive and provide better insights into their properties.
Slope
The slope is a key concept in understanding linear equations, representing the rate at which one quantity changes relative to another. It is often denoted by the letter \( m \) in equations. The slope indicates the steepness and the direction of the line on a graph.

Understanding the Slope:
  • Positive Slope: When the line ascends from left to right, indicating a positive rate of change.
  • Negative Slope: A line that descends from left to right signals a negative rate of change, as seen with a slope of \(-4\) as in our example equation.
  • Zero Slope: A horizontal line indicative of no change in \( y \) as \( x \) changes.
  • Undefined Slope: A vertical line showing an undefined or infinite change.
Expressing a slope like \(-4\) as \(-4/1\) makes it easier to use for graphing: step to the right 1 unit in the x-direction and move down 4 units in the y-direction. This directional movement helps in plotting the path the linear equation takes on the graph.
Y-Intercept
The y-intercept is a crucial element of the slope-intercept form, marking the point where the line crosses the y-axis. It is symbolized by \( b \) in the equation \( y = mx + b \). Determining and plotting the y-intercept provides the starting point for graphing the entire line.

Importance of the Y-Intercept:
  • Starting Point: This point, usually written as \((0, b)\), is where the line meets the y-axis.
  • Easy to Plot: Even if the slope is complex, the y-intercept gives a simple coordinate to begin your graph.
  • Provides Context: It represents the value of y when x is zero, offering practical understanding like initial conditions in real-world scenarios.
In the example equation \( y = -4x - 1 \), the y-intercept is \(-1\), found easily by locating the point \((0, -1)\) on the graph. Understanding the role of the y-intercept simplifies the process of graphing linear equations and gives insight into their behavior at the origin.

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Most popular questions from this chapter

Minh, a psychology student, kept a record of how much time she spent studying for each of her 20 -point psychology quizzes and her score on each quiz. $$ \begin{array}{|l|c|c|c|c|c|c|c|c|} \hline \text { Hours Spent Studying } & 0.50 & 0.75 & 1.00 & 1.25 & 1.50 & 1.50 & 1.75 & 2.00 \\ \hline \text { Quiz Score } & 10 & 12 & 15 & 16 & 18 & 19 & 19 & 20 \\ \hline \end{array} $$ a. Write the data as ordered pairs of the form (hours spent studying, quiz score). b. In your own words, write the meaning of the ordered pair (1.25,16) C. Create a scatter diagram of the paired data. \(\mathrm{Be}\) sure to label the axes appropriately. d. What might Minh conclude from the scatter diagram?

Graph each inequality. $$ y \leq 0 $$

Solve each equation for y. See Section 2.5. $$ x+y=5 $$

Write an equation in standard form of the line that contains the point (-1,2) and is parallel to (has the same slope as) the line \(y=3 x-1\)

Find an equation of each line described. Write each equation in slope- intercept form when possible. Slope \(-2, y\) -intercept (0,-4)
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