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Chapter 3: Problem 6

Graph the given functions. $$y=-2 x$$

Short Answer

Expert verified
The equation \(y=-2x\) is a line through the origin with a slope of -2.

Step by step solution

01

Understanding the Slope-Intercept Form

The given function is in slope-intercept form, which is generally written as \(y = mx + b\). In this form, \(m\) represents the slope of the line, and \(b\) is the y-intercept. For the function \(y = -2x\), \(m = -2\) and \(b = 0\). This means that the line will cross the origin \((0, 0)\) and have a slope of -2.
02

Plotting the Y-intercept

The y-intercept \(b = 0\) tells us that the line crosses the y-axis at the origin. So, the first point to plot on the graph is \((0, 0)\).
03

Using the Slope to Find Another Point

The slope \(m = -2\) can be interpreted as 'rise over run.' A slope of -2 means that for every unit you move to the right (along the x-axis), you move down 2 units. Start from the y-intercept \((0,0)\), move one unit to the right to \(x = 1\), and then move two units down to \(y = -2\). Plot this point at \((1, -2)\).
04

Drawing the Line

Now that you have two points, \((0,0)\) and \((1,-2)\), you can draw a straight line through these points. This line represents the graph of the function \(y = -2x\).
05

Verifying with Additional Points

To ensure accuracy, try finding a third point. Start again from \((0,0)\), move another unit right to \(x = 2\), and 2 more units down to \(y = -4\). Plot \((2, -4)\) and check consistency with your line.

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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 straightforward way to represent a line on a graph. It's expressed as \( y = mx + b \), where:
  • \( m \) is the slope of the line, indicating its steepness.
  • \( b \) is the y-intercept, the point where the line crosses the y-axis.
In this form, you can quickly identify both the slope and the y-intercept, simplifying the process of graphing. By examining the given function \( y = -2x \), you see that it perfectly fits the slope-intercept form with a slope \( m = -2 \) and a y-intercept \( b = 0 \). This means the line starts at the origin,
since there is no vertical shift, then proceeds downwards to the right as determined by the slope.
Plotting Points Using the Equation
Graphing a linear function involves plotting key points on the coordinate plane that lies on the line. To begin, use the y-intercept from the function, which is the first point:
  • For \( y = -2x \), the y-intercept \( b = 0 \) means the line crosses the y-axis at \( (0, 0) \).
With this starting point plotted, we can use the slope to locate additional points. Each plotted point helps ensure the accuracy of the graph and provides a clearer picture of the line's direction and angle.
Begin plotting with the y-intercept because it simplifies aligning subsequent points using the slope.
Determining the Slope of a Line
The slope is crucial in defining how steep or flat a line is. Represented by \( m \) in the slope-intercept form, it acts as a measure of the line's "rise over run." Simply put:
  • The "rise" refers to the change in y-values (vertical movement).
  • The "run" refers to the change in x-values (horizontal movement).
A slope of \( -2 \) tells us that for every unit increase in \( x \) (moving right), the line moves down 2 units in \( y \).
Starting from the y-intercept \( (0, 0) \), if you move 1 unit to the right (\( x = 1 \)), you simultaneously move 2 units down, putting you at \( (1, -2) \). Plotting a few more points using this pattern ensures accuracy in the visualization.
The Role of the Y-Intercept
The y-intercept is a significant starting point for any linear graph. Defined as \( b \) in the equation \( y = mx + b \), it is the point where the line crosses the y-axis:
  • In our example, \( b = 0 \) indicates the line passes through the origin \( (0, 0) \).
The y-intercept provides a baseline for plotting the line using its slope. Starting at \( (0, 0) \) allows us to systematically apply the slope's pattern and identify further points along the line.
This consistent approach makes it easier to verify the accuracy of the plotted slope visually.

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