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Magazine Chapter 3: Problem 34
Graph each linear equation. Plot four points for each line. $$y=\frac{1}{3} x-2$$
Short Answer
Expert verified
Plot points (0, -2), (-6, -4), (-3, -3), (3, -1), (6, 0), then draw the line connecting them.
Step by step solution
01
- Understand the Equation
The given equation is in the slope-intercept form, which is \(y = mx + b\). Here, \(m = \frac{1}{3}\) is the slope and \(b = -2\) is the y-intercept.
02
- Determine the Y-Intercept
The y-intercept is the value of \(y\) when \(x = 0\). In this equation, \(b = -2\), so plot the point (0, -2)on the graph.
03
- Calculate Additional Points
To plot additional points, choose different values for \(x\). Use the slope \(\frac{1}{3}\), which means for each increase of 3 units in \(x\), \(y\) increases by 1 unit. Choose these values for \(x\): -6, -3, 3, and 6. Calculate the corresponding \(y\) values.
04
- Plot Points for \(x = -6\)
Substitute \(x = -6\) into the equation: \(y = \frac{1}{3}(-6) - 2 = -2 - 2 = -4\). Plot the point (-6, -4).
05
- Plot Points for \(x = -3\)
Substitute \(x = -3\) into the equation: \(y = \frac{1}{3}(-3) - 2 = -1 - 2 = -3\). Plot the point (-3, -3).
06
- Plot Points for \(x = 3\)
Substitute \(x = 3\) into the equation: \(y = \frac{1}{3}(3) - 2 = 1 - 2 = -1\). Plot the point (3, -1).
07
- Plot Points for \(x = 6\)
Substitute \(x = 6\) into the equation: \(y = \frac{1}{3}(6) - 2 = 2 - 2 = 0\). Plot the point (6, 0).
08
- Draw the Line
Using a ruler, draw a straight line through the four points: (0, -2), (-6, -4), (-3, -3), (3, -1), and (6, 0). This is the graph of the equation \(y = \frac{1}{3} x - 2\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
slope-intercept form
When graphing linear equations, understanding the slope-intercept form is crucial. The slope-intercept form of a linear equation is given by the formula: \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept. In simpler terms, this form tells us how the line rises or falls and its intersection with the y-axis. Given an equation like \( y = \frac{1}{3}x - 2 \), \( m = \frac{1}{3} \) and \( b = -2 \). This means the line climbs by 1 unit for every 3 units it moves to the right, and it crosses the y-axis at \( y = -2 \).
plotting points
To draw a linear equation, you need to plot multiple points on a graph. Start with the y-intercept. If \( b = -2 \), you plot the first point at \( (0, -2) \). To get more points, you choose different values for \( x \) and calculate the corresponding \( y \) values using the equation. For example, choosing \( x = -6 \) and substituting into \( y = \frac{1}{3}x - 2 \), we get \( y = -4 \). Continue this process with other values like \( x = -3, 3, 6 \) to get additional points \( (-3, -3), (3, -1), (6, 0) \). Plot these points on the graph.
linear functions
Linear functions are mathematical expressions that create straight lines when graphed. They have the general form \( y = mx + b \). These functions are predictable because of their constant rate of change, dictated by the slope \( m \). This means for every equal step you take along the x-axis, there's a consistent step along the y-axis. The given exercise demonstrates how you can use a linear function to produce a line. Once you understand how to derive points and plot them, connecting these points will reveal a straight line.
slope
The slope of a line, represented as \( m \) in the slope-intercept equation, indicates how steep the line is. The slope is calculated by the rise divided by the run, or the change in \( y \) over the change in \( x \). For instance, a slope of \( \frac{1}{3} \) means that the line rises 1 unit for every 3 units moved to the right. Positive slopes ascend, while negative slopes descend. In our example, the slope tells us that for each increase of 3 units in \( x \), \( y \) increases by 1 unit, resulting in a gentle upward slope.
y-intercept
The y-intercept of a linear equation is the point where the line crosses the y-axis. In the equation \( y = mx + b \), \( b \) represents this point. In our example, \( b = -2 \), meaning the line crosses the y-axis at \( y = -2 \). Plotting this point is the initial step in graphing the line. It's easy to find because it doesn't require any calculation 鈥� simply look at the value of \( b \). From there, use the slope to find additional points and complete the line. The y-intercept is essential as it gives you a concrete point to start plotting your graph.
