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Chapter 11: Problem 298

In the following exercises, graph by plotting points. $$ y=-3 x $$

Short Answer

Expert verified
Plot points (-2, 6), (-1, 3), (0, 0), (1, -3), and (2, -6); then connect them with a straight line.

Step by step solution

01

Create a Table of Values

Select a few values for the variable x and calculate the corresponding y values using the equation \( y = -3x \).
02

Calculate y-values

For instance, choose x = -2, -1, 0, 1, 2. Then compute y:\( y = -3(-2) = 6 \)\( y = -3(-1) = 3 \)\( y = -3(0) = 0 \)\( y = -3(1) = -3 \)\( y = -3(2) = -6 \)
03

Create a Table of Values

Complete the table with the pairs of x and y values:\[\begin{array}{c|c}x & y \hline-2 & 6 \-1 & 3 \0 & 0 \1 & -3 \2 & -6 \end{array}\]
04

Plot the Points on the Graph

On a coordinate plane, plot the points (-2, 6), (-1, 3), (0, 0), (1, -3), and (2, -6).
05

Draw the Line

Connect the points with a straight line. This straight line represents the graph of the equation \( y = -3x \).

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Key Concepts

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

plotting points
When plotting points on a graph, you place individual coordinate pairs \( (x, y) \) on a grid. Each point corresponds to a unique pair of x (horizontal) and y (vertical) values from your table of values.
Here's how you plot points:
  • Start with the x-value. Move horizontally along the x-axis to this value.
  • Next, move vertically to the y-value from this x position. Mark this spot.
  • Repeat for each pair in your table.
By plotting these points, you can visually interpret the relationship between x and y values.
coordinate plane
A coordinate plane, also known as a Cartesian plane, is a two-dimensional surface defined by two intersecting lines: the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants.
Each point on the plane can be defined by an ordered pair \( (x, y) \) where:
  • \( x \) (first number) shows the position along the horizontal axis.
  • \( y \) (second number) shows the position along the vertical axis.
Using a coordinate plane helps you see how x and y values interact visually by plotting points and drawing lines or curves between them.
linear relationships
Linear relationships occur when there is a constant rate of change between two variables, resulting in a straight line when graphed. In the equation \( y = -3x \), it's clear that as \( x \) increases, \( y \) decreases at a consistent rate of -3.
To identify a linear relationship, pay attention to these properties:
  • The graph is a straight line.
  • There is a constant slope (rate of change).
  • The relationship can be described with a linear equation of the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
This equation's slope, -3, signifies that for each unit increase in \( x \), \( y \) decreases by 3 units. Hence, it represents a negative linear relationship.
table of values
Creating a table of values is an essential step in graphing equations. It helps you organize x and y values systematically, making it easier to plot points on a graph.
Here's how to create and use a table of values:
  • Select a range of x values. For example, \( x = -2, -1, 0, 1, 2 \).
  • Use your equation to solve for the corresponding y values. For \( y = -3x \), calculate y for each chosen x value.
  • Compile these values into a table with columns for x and y.

    Example:
    \( \begin{array}{c|c} x & y \hline -2 & 6 \ -1 & 3 \ 0 & 0 \ 1 & -3 \ 2 & -6 \end{array} \)
This table allows a clear view of how x and y pairs correspond, making it straightforward to plot and draw the graph accurately.

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In the following exercises, graph the line given a point and the slope. $$ (-3,3) ; m=2 $$

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