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Magazine Chapter 1: Problem 8
Graph each of the following lines. \(y=-2 x\)
Short Answer
Expert verified
Graph the line passing through (0,0) and (1,-2) with a slope of -2.
Step by step solution
01
Identify the format of the equation
Notice that the equation given is in the slope-intercept form, which is typically written as \(y = mx + b\). In this form, \(m\) represents the slope and \(b\) represents the y-intercept of the line. Here, the given equation is \(y = -2x\), meaning \(m = -2\) and \(b = 0\).
02
Interpret the slope and y-intercept
The slope \(m = -2\) indicates that for every step of 1 unit increase in \(x\), \(y\) decreases by 2 units. The y-intercept \(b = 0\) means the line passes through the origin, or the point (0,0).
03
Plot the y-intercept
Start by plotting the point where the line crosses the y-axis. Since the y-intercept is 0, plot the point \((0,0)\).
04
Use the slope to find a second point
From the point (0,0), apply the slope \(-2\) to determine the next point on the line. Move right 1 unit along the x-axis (since the run is +1), and move down 2 units along the y-axis (since the rise is -2). Plot this new point at \((1, -2)\).
05
Draw the line through the points
Connect the points (0,0) and (1,-2) with a straight line. This line represents the graph of the equation \(y = -2x\). Extend the line in both directions, continuing the pattern defined by the slope.
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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 way of expressing linear equations using the format:
Understanding how to use the slope-intercept form is crucial for graphing lines easily. In this format, once you know \(m\) and \(b\), you can quickly sketch a line on the coordinate plane.
Let's break it down further:
- \(y = mx + b\)
Understanding how to use the slope-intercept form is crucial for graphing lines easily. In this format, once you know \(m\) and \(b\), you can quickly sketch a line on the coordinate plane.
Let's break it down further:
- If \(m = 0\), the line is horizontal since there is no rise and only runs along the x-axis.
- Positive \(m\) indicates the line rises, while a negative \(m\) means it descends.
- \(b\) being zero means the line passes through the origin (0,0).
Slope of a Line
The slope of a line, denoted as \(m\) in the slope-intercept form \(y = mx + b\), measures the line's steepness.
Imagine you are climbing a hill; the steeper the hill, the higher the slope. In graphs, positive slopes like uphill climbs indicate a line going upwards as you move right. Conversely, negative slopes represent downhill paths.
The slope \(m\) is defined as:
Imagine you are climbing a hill; the steeper the hill, the higher the slope. In graphs, positive slopes like uphill climbs indicate a line going upwards as you move right. Conversely, negative slopes represent downhill paths.
The slope \(m\) is defined as:
- \(m = \frac{\text{rise}}{\text{run}}\)
- If \(m = -2\), every time you move right by 1 unit (positive direction along x-axis), you move down 2 units (a negative change in y-axis).
Y-Intercept
The y-intercept is a special point where the line crosses the y-axis. It's the value of \(y\) when \(x = 0\). In the slope-intercept equation \(y = mx + b\), the \(b\) stands for the y-intercept.
- For the equation \(y = -2x\), \(b = 0\), indicating that the line crosses the y-axis at the origin.
- The y-intercept allows you to "anchor" or start the line on the graph. It's the first point you plot when graphing a line.
- A y-intercept of 0 means the line goes through the origin, simplifying the plotting process since everything starts from (0,0).
- Different y-intercepts lead to parallel lines that are equidistant but never meet, assuming the same slope.
