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Chapter 3: Problem 16

In the following exercises, graph by plotting points. $$ y=-2 x $$

Short Answer

Expert verified
Plot the points (-2, 4), (-1, 2), (0, 0), (1, -2), and (2, -4), then connect them with a straight line.

Step by step solution

01

Understand the Equation

First, understand that the equation given is a linear equation of the form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. Here, \(y = -2x\), making the slope \(-2\) and the y-intercept \(0\).
02

Choose Values for x

To plot points, choose a set of values for \(x\). Common choices are \(-2, -1, 0, 1, \text{and } 2\).
03

Calculate Corresponding y Values

For each value of \(x\) chosen, substitute it into the equation to find the corresponding \(y\) values.When \(x = -2\), \(y = -2(-2) = 4\)When \(x = -1\), \(y = -2(-1) = 2\)When \(x = 0\), \(y = -2(0) = 0\)When \(x = 1\), \(y = -2(1) = -2\)When \(x = 2\), \(y = -2(2) = -4\)
04

Plot the Points

Using the pairs \((x, y)\) calculated, plot these points on the graph: \((-2, 4)\), \((-1, 2)\), \(0, 0)\), \(1, -2)\), and \(2, -4)\).
05

Draw the Line

Connect the points with a straight line. This line represents the graph of the equation \(y = -2x\).

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Key Concepts

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

linear equations
A linear equation is an equation that, when graphed, forms a straight line. The general formula for a linear equation is typically written as \( y = mx + c \), where:
  • \( y \) represents the dependent variable (or output).
  • \( x \) represents the independent variable (or input).
  • \( m \) is the slope of the line.
  • \( c \) is the y-intercept, or the point where the line crosses the y-axis.
In the original exercise, the linear equation is \( y = -2x \), which means that the slope \( m \) is -2 and the y-intercept \( c \) is 0. This is a foundational concept for understanding how to graph linear equations.
slope
The slope of a line tells us how steep the line is and the direction in which it tilts. It's calculated as the change in \( y \) over the change in \( x \) (often referred to as 'rise over run'). In our equation \( y = -2x \), the slope \( m \) is -2. This slope indicates:
  • A downward tilt from left to right.
  • A steeper decline since the value is negative and greater than 1 in absolute value.
This means that for every 1 unit increase in \( x \), \( y \) decreases by 2 units. Understanding the slope helps you figure out how to connect points when drawing the linear graph.
plotting points
Plotting points is the step where you choose values for \( x \) and calculate corresponding \( y \) values based on the linear equation. Here鈥檚 how it was done in the exercise:
  • For \( x = -2 \), \( y = -2(-2) = 4 \)
  • For \( x = -1 \), \( y = -2(-1) = 2 \)
  • For \( x = 0 \), \( y = -2(0) = 0 \)
  • For \( x = 1 \), \( y = -2(1) = -2 \)
  • For \( x = 2 \), \( y = -2(2) = -4 \)
You then plot these (\( x \), \( y \)) pairs on the graph:
  • (-2, 4)
  • (-1, 2)
  • (0, 0)
  • (1, -2)
  • (2, -4)
This gives you points through which you draw a straight line to represent the equation.
y-intercept
The y-intercept is where the line crosses the y-axis. For the equation \( y = -2x \), the y-intercept \( c \) is 0. This means the line passes through the origin, or the point (0, 0).
Knowing the y-intercept is useful because it gives you a starting point when graphing the line. After plotting the y-intercept, you can use the slope to find other points on the line and draw the graph accurately.
For instance, starting from (0, 0) and using the slope of -2, you can determine another point by moving 1 unit to the right (positive direction on the x-axis) and 2 units down (negative direction on the y-axis).

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Most popular questions from this chapter

In the following exercises, (a) graph each function (b) state its domain and range. Write the domain and range in interval notation. $$ f(x)=-2 x^{2} $$

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