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Chapter 3: Problem 19

Exer. 19-20: Sketch the graphs of the lines on the same coordinate plane. $$ y=x+3, \quad y=x+1, \quad y=-x+1 $$

Short Answer

Expert verified
Plot the lines by their y-intercepts and slopes on a coordinate plane to see the graph.

Step by step solution

01

Identify the Lines

First, let's identify the equations of the lines that will be graphed:1. Line 1: \(y = x + 3\).2. Line 2: \(y = x + 1\).3. Line 3: \(y = -x + 1\).These are all linear equations in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
02

Determine Key Features of Each Line

For each line, identify the slope and y-intercept:1. For \(y = x + 3\): - Slope \(m = 1\) - Y-intercept \(b = 3\)2. For \(y = x + 1\): - Slope \(m = 1\) - Y-intercept \(b = 1\)3. For \(y = -x + 1\): - Slope \(m = -1\) - Y-intercept \(b = 1\)
03

Set Up the Coordinate Plane

Prepare a coordinate plane: - Draw a vertical axis for y and a horizontal axis for x. - Use an appropriate scale for the axes, considering the y-intercepts and the slopes.
04

Plot the Y-Intercepts

Plot the y-intercepts of each line on the vertical axis:1. \(y = x + 3\): Point (0, 3).2. \(y = x + 1\): Point (0, 1).3. \(y = -x + 1\): Point (0, 1).
05

Use the Slope to Plot a Second Point and Draw the Lines

For each line, start from the y-intercept and use the slope to find another point:1. For \(y = x + 3\): - From (0, 3), slope 1 means rise 1 and run 1 to reach (1, 4). - Draw a line through points (0, 3) and (1, 4).2. For \(y = x + 1\): - From (0, 1), slope 1 means rise 1 and run 1 to reach (1, 2). - Draw a line through points (0, 1) and (1, 2).3. For \(y = -x + 1\): - From (0, 1), slope -1 means rise -1 and run 1 to reach (1, 0). - Draw a line through points (0, 1) and (1, 0).
06

Label Your Lines

Finally, label each line on your graph to distinguish them:- Label the line through (0, 3) and (1, 4) as \(y = x + 3\).- Label the line through (0, 1) and (1, 2) as \(y = x + 1\).- Label the line through (0, 1) and (1, 0) as \(y = -x + 1\).

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form of a linear equation is a way of expressing the equation of a line. This form is given by the central formula: \[ y = mx + b \]where:
  • \(m\) is the slope of the line.
  • \(b\) is the y-intercept, which is where the line crosses the y-axis.
This format is particularly useful because it allows you to quickly identify two important characteristics of the line: its slope and its y-intercept.\(m\), the slope, shows the steepness and the direction of the line. If \(m\) is positive, the line slopes upwards; if negative, it slopes downwards. The y-intercept \(b\) provides a starting point for graphing by showing where the line meets the y-axis.The slope-intercept form simplifies graphing and understanding the behavior of linear equations on the coordinate plane.
Coordinate Plane
The coordinate plane is a two-dimensional surface where we can graph equations and visualize the relationships between variables. It consists of:
  • A horizontal number line, called the x-axis.
  • A vertical number line, called the y-axis.
These axes intersect at the origin (0,0), partitioning the plane into four quadrants. Each point on the plane is described by an ordered pair \((x, y)\), where \(x\) is the horizontal position and \(y\) is the vertical position. When graphing lines:
  • The x-axis represents the input or independent variable.
  • The y-axis represents the output or dependent variable.
Understanding the coordinate plane helps in visualizing mathematical concepts and provides a foundation for plotting points, lines, and shapes effectively.
Slope and Y-Intercept
The slope and y-intercept are key features of a line, defining its direction and position on the plane.
  • Slope \(m\): This is a measure of the line's steepness and direction. It is calculated by \(\text{slope} = \frac{\text{rise}}{\text{run}}\). This ratio shows how much the line goes up or down (rise) for each step it moves horizontally (run).Positive slope: The line rises as you move right.
    Negative slope: The line falls as you move right.
  • Y-Intercept \(b\): This is the value of \(y\) when \(x=0\). It is the point where the line crosses the y-axis. It offers a clear starting point for drawing the line.
Knowing the slope and y-intercept helps in quickly sketching the line without the need for extensive calculation. They allow you to determine a second point on the line, ensuring an accurate graphing process.
Plotting Points
Plotting points on the coordinate plane is the fundamental step in graphing linear equations. Here's how to plot points easily:To graph a line, such as those given by an equation in slope-intercept form:
  • Start by identifying your y-intercept \((0, b)\) and place a point there on the y-axis.
  • Use the slope \(m\), often as a fraction \(\frac{rise}{run}\), to find another point. From the y-intercept, move up or down (rise) and then right (run). Example: For a slope of \(2\), move up 2 and right 1, reaching the new point.
After determining at least two points, draw a straight line through them, extending it across the grid. By repeating this process for each equation, you can easily visualize and see the differences or similarities in the lines, such as parallelism or intersection points.

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