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Chapter 3: Problem 33

Graph each linear equation. Plot four points for each line. $$y=\frac{1}{2} x-1$$

Short Answer

Expert verified
To graph \( y = \frac{1}{2} x - 1 \), plot the points (0, -1), (2, 0), (4, 1), and (-2, -2), then draw a line through them.

Step by step solution

01

Understand the equation format

The given equation is in the slope-intercept form, which is written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Here, \( y = \frac{1}{2} x - 1 \), so the slope \( m = \frac{1}{2} \) and the y-intercept \( b = -1 \).
02

Find the y-intercept

The equation's y-intercept is the point where the line crosses the y-axis. This happens when \( x = 0 \). Substitute \( x = 0 \) into the equation to get \( y = \frac{1}{2}(0) - 1 = -1 \). Therefore, the y-intercept is the point (0, -1).
03

Calculate another point using the slope

The slope \( \frac{1}{2} \) indicates a rise of 1 unit for every 2 units of run. Starting from the y-intercept (0, -1), move 2 units to the right (along the x-axis) and 1 unit up. This gives the point (2, 0).
04

Calculate two more points on the line

Choose another value for \( x \). Using \( x = 4 \): \( y = \frac{1}{2}(4) - 1 = 2 - 1 = 1 \). This gives the point (4, 1). Now, choose \( x = -2 \): \( y = \frac{1}{2}(-2) - 1 = -1 - 1 = -2 \). This gives the point (-2, -2).
05

Plot the points and draw the line

Plot the four points found: (0, -1), (2, 0), (4, 1), (-2, -2) on graph paper. Draw a straight line through these points to represent the equation \( y = \frac{1}{2} x - 1 \).

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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 is a popular way to write the equation of a line. It is written as \( y = mx + b \), where:
  • \( m \) is the slope of the line.
  • \( b \) is the y-intercept.
This form makes it easy to identify the line's slope and where it crosses the y-axis. For instance, in the equation \( y = \frac{1}{2} x - 1 \), the slope \( m \) is \( \frac{1}{2} \) and the y-intercept \( b \) is \( -1 \). These elements are crucial for graphing the line.
plotting points
Plotting points involves finding coordinates that satisfy the linear equation and marking them on a graph. Here's a simple way to do it:
  • Find the y-intercept. Convert \( x \) to 0 and solve for \( y \).
  • Use the slope to find other points. Start from the y-intercept and move according to the slope.
  • Choose additional values for \( x \) to calculate corresponding \( y \) values.
For example, with \( y = \frac{1}{2} x - 1 \):
  • The y-intercept (0, -1)
  • Move 2 units right and 1 unit up to get (2, 0)
  • For \( x = 4 \), \( y = 1 \), giving the point (4, 1)
  • For \( x = -2 \), \( y = -2 \), giving the point (-2, -2)
Plot these points and you can draw the line through them.
y-intercept
The y-intercept is the point where the line crosses the y-axis. To find it in the equation \( y = mx + b \), set \( x \) to 0 and solve for \( y \).
For the equation \( y = \frac{1}{2} x - 1 \):
  • Set \( x = 0 \)
  • \( y = \frac{1}{2}(0) - 1 \)
  • This simplifies to \( y = -1 \)
So, the y-intercept is the point (0, -1). It's one of the key points you plot when graphing a line.
calculating slope
Slope indicates the steepness of a line and is defined as the ratio of the vertical change (rise) to the horizontal change (run). It's found by identifying two points on the line:
\ \text{Slope} = \frac{\text{rise}}{\text{run}} \( m = \frac{\text{change in } y}{\text{change in } x} \)
For instance, if we start at the y-intercept (0, -1) of the equation \( y = \frac{1}{2}x - 1 \):
  • From (0, -1), we move 2 units to the right and 1 unit up.
  • So, \( \text{rise} = 1 \) and \( \text{run} = 2 \)
  • \( \text{Slope} = \frac{1}{2} \)
By understanding how to calculate slope, you can easily graph any linear equation by moving according to its rise and run.

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Most popular questions from this chapter

Find the equation of line l in each case and then write it in standard form with integral coefficients. Line \(l\) has slope 5 and goes through \(\left(0, \frac{1}{2}\right)\).

Determine whether each pair of lines is parallel, perpendicular, or neither. $$y=3 x-8, x+3 y=7$$

Find all intercepts for each line. Some of these lines have only one intercept. $$6 x+3=0$$

Graph each pair of lines in the same coordinate system using the slope and y-intercept. $$ \begin{aligned} &y=-2 x+4\\\ &y=-2 x+2 \end{aligned} $$

Is it possible for a line to be in only one quadrant? Two quadrants? Write a rule for determining whether a line has positive, negative, zero, or undefined slope from knowing in which quadrants the line is found.
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