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Chapter 4: Problem 66

In the following exercises, graph by plotting points. \(y=-x-2\)

Short Answer

Expert verified
Plot points \((-2, 0), (-1, -1), (0, -2), (1, -3), (2, -4)\), then draw a straight line through them.

Step by step solution

01

Create a Table of Values

To plot the graph, first choose several values for the variable \(x\), then solve for \(y\). Typically, choosing a range of values, including negative, zero, and positive values, works well. For this equation, let鈥檚 choose \(-2, -1, 0, 1, 2\) for \(x\).
02

Calculate Corresponding \(y\) Values

Substitute each chosen \(x\) value into the equation \(y = -x - 2\) to find the corresponding \(y\) values. For example: \- If \(x = -2\), then \(y = -(-2) - 2 = 2 - 2 = 0\) \- If \(x = -1\), then \(y = -(-1) - 2 = 1 - 2 = -1\) \- If \(x = 0\), then \(y = -0 - 2 = -2\) \- If \(x = 1\), then \(y = -1 - 2 = -3\) \- If \(x = 2\), then \(y = -2 - 2 = -4\)
03

Plot the Points on the Coordinate Plane

Take each pair \((x, y)\) from the table and plot them onto a coordinate plane. The points are \((-2, 0), (-1, -1), (0, -2), (1, -3), (2, -4)\).
04

Draw the Line

Once all points are plotted, draw a straight line through the points to represent the equation \(y = -x - 2\). Ensure the line extends across the graph and has arrows on both ends to show it continues infinitively.

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Key Concepts

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

Plotting Points
To plot points, you first need to know their coordinates. Points are written as \(x, y\), where \(x\) is the horizontal position and \(y\) is the vertical position. Find each pair of \(x\) and \(y\) values using the given equation. For the equation \(y = -x - 2\), you could use the pairs \((-2, 0)\), \((-1, -1)\), \( (0, -2)\), \( (1, -3)\), and \( (2, -4)\). After you have the pairs, mark these points on the coordinate plane.
Coordinate Plane
The coordinate plane is a two-dimensional space where each location is defined by two numbers, known as coordinates. The horizontal axis is called the x-axis, and the vertical axis is the y-axis. The center point where the x-axis and y-axis intersect is called the origin, labeled as \(0, 0\). To plot points effectively, start from the origin and move horizontally to the x value, then vertically to the y value.
Table of Values
A table of values is a helpful way to organize pairs of \(x\) and \(y\) coordinates. Start by choosing a range of x values, both negative and positive, then use the equation to calculate corresponding y values.

For example, if the equation is \(y = -x - 2\) and you choose x values \(-2, -1, 0, 1, 2\):
\(\begin{align*} \text{For } x = -2, & \text{ } y = -(-2) - 2 = 0 \ \text{For } x = -1, & \text{ } y = -(-1) - 2 = -1 \ \text{For } x = 0, & \text{ } y = -0 - 2 = -2 \ \text{For } x = 1, & \text{ } y = -1 - 2 = -3 \ \text{For } x = 2, & \text{ } y = -2 - 2 = -4 \end{align*}\).
This will give you a table to plot on the coordinate plane.
Linear Equations
Linear equations are equations of the first degree, which means they graph to a straight line. The general form is \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. For \(y = -x - 2\), the slope is \(-1\) and the y-intercept is \(-2\). You can see the line by plotting the points calculated earlier and then drawing a straight line through them. This line extends in both directions infinitely, which is why it's important to add arrows at both ends.

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