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Chapter 1: Problem 52

Use the y-intercept and slope to sketch the graph of each equation. $$y=-\frac{3}{2} x$$

Short Answer

Expert verified
The y-intercept is 0 and the slope is -\(\frac{3}{2}\). Plot (0,0) and (2,-3), then draw the line.

Step by step solution

01

Identify the slope

The given equation is \(y = -\frac{3}{2}x\). The slope, which is the coefficient of \(x\), is \(-\frac{3}{2}\).
02

Identify the y-intercept

The equation \(y = -\frac{3}{2}x\) can be written as \(y = -\frac{3}{2}x + 0\). Thus, the y-intercept is 0.
03

Plot the y-intercept

Plot the y-intercept (0, 0) on the graph.
04

Use the slope to find another point

The slope \(-\frac{3}{2}\) means that for every 2 units you move to the right along the x-axis, you move down 3 units along the y-axis. Starting from the y-intercept (0, 0), move 2 units to the right and drop 3 units to plot the point (2, -3).
05

Draw the line

Draw a straight line passing through the points (0, 0) and (2, -3). This is the graph of the equation \(y = -\frac{3}{2}x\).

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Key Concepts

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

Understanding the y-intercept
In the given equation, we have \(y = -\frac{3}{2}x\). To find the y-intercept, we need to consider where the line crosses the y-axis. The y-axis is where \(x = 0\). When we rewrite the equation as \(y = -\frac{3}{2}x + 0\), we can see that the y-intercept is \(0\). This tells us that the line intersects the y-axis at the point \((0, 0)\).
This point is very important when starting to sketch the graph since it gives us a fixed point to draw from.
Grasping the slope
In the equation \(y = -\frac{3}{2}x\), the slope of the line is the coefficient of \(x\), which is \(-\frac{3}{2}\). The slope indicates how steep the line is and in which direction it moves. A slope of \(-\frac{3}{2}\) tells us that for every 2 units we move to the right along the x-axis, we move down 3 units along the y-axis.
To break it down:
  • A negative sign means the line goes downwards.
  • The number \(\frac{3}{2}\) tells us the exact rate of descent relative to horizontal movement.
Understanding the slope helps us determine other points on the line.
Plotting points on the graph
Now that we know the y-intercept and the slope, plotting points becomes straightforward. Start with the y-intercept, \((0, 0)\).
From this point, look at the slope \(-\frac{3}{2}\):
  • Move 2 units right along the x-axis.
  • Move 3 units down along the y-axis.
This brings us to the point \((2, -3)\).
Plot this point as well.
Finally, draw a straight line through \((0, 0)\) and \((2, -3)\). This line represents the equation \(y = - \frac{3}{2} x\). Knowing how to plot points and draw a line using the slope is key in graphing linear equations.

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