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Chapter 5: Problem 9

Find the slope and the \(y\) -intercepts of the lines with the given equations. Sketch the graphs. $$y=-2 x+4$$

Short Answer

Expert verified
Slope: -2, Y-intercept: 4 at (0, 4).

Step by step solution

01

Identify the Equation Form

The given equation is in the form of the slope-intercept form, which is written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept.
02

Determine the Slope

In the equation \( y = -2x + 4 \), the coefficient of \( x \) is \(-2\). Thus, the slope \( m \) of the line is \(-2\).
03

Determine the Y-intercept

The constant term in the equation \( y = -2x + 4 \) is \( 4 \). Thus, the \( y \)-intercept \( b \) is \( 4 \). This represents the point \( (0, 4) \) on the graph.
04

Sketch the Graph

Begin by plotting the \( y \)-intercept, \( (0, 4) \), on the graph. From this point, use the slope to find another point on the graph. Since the slope is \(-2\), go down 2 units and right 1 unit to plot the next point \( (1, 2) \). Draw a line through these points to complete the graph.

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Key Concepts

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

Understanding the Slope-Intercept Form
The slope-intercept form of a linear equation is a popular and widely used format that makes it easy to graph linear equations. It's expressed as \( y = mx + b \). Here, \( m \) represents the slope of the line and \( b \) denotes the \( y \)-intercept. Knowing this form allows you to quickly determine two essential characteristics of the linear equation:
  • Slope \( m \): This tells you how steep the line is and the direction in which it tilts. A positive slope means the line rises as it moves from left to right, while a negative slope indicates it falls.
  • Y-intercept \( b \): This is the point where the line crosses the \( y \)-axis.
Seeing an equation in this form immediately gives you enough information to sketch the graph without needing further calculations.
Breaking Down: Graphing Linear Equations
Graphing linear equations when they are in slope-intercept form can be quick and straightforward. Follow these steps to successfully graph any line in this form:
  • Locating the \( y \)-intercept: Start by plotting the \( y \)-intercept \( b \) on the \( y \)-axis. For \( y = -2x + 4 \), the intercept is \( 4 \), making the point \( (0, 4) \).
  • Using the slope: The slope \( m \) provides the rise over run. With a slope of \(-2\), you move down 2 units (a negative rise) and right 1 unit. This brings you to a new point \( (1, 2) \).
  • Connecting the points: Draw a line through these points. This continuous line represents all solutions to the equation \( y = -2x + 4 \).
Graphing can be a visual exploration of the relationship between \( x \) and \( y \), helping strengthen understanding as you plot the line.
Steps to Finding Slope
The slope of a line is a measure of its steepness or inclination. It shows how much \( y \) changes for a unit change in \( x \). When you have an equation in the form \( y = mx + b \), identifying the slope is straightforward:
  • Find the coefficient of \( x \): In our equation \( y = -2x + 4 \), the coefficient of \( x \) is \(-2\), indicating that each step right in the \( x \) direction results in a 2 unit decrease in \( y \).
  • Interpret the slope: A negative slope like \(-2\) implies a downward slant from left to right. Conversely, a positive slope would indicate an upward slant.
Understanding the slope not only aids in graphing but also gives insight into the behavior and relationships within real-world scenarios where linear equations model trends and changes.

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