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Chapter 3: Problem 14

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

Short Answer

Expert verified
Choose x-values, calculate corresponding y-values, plot the points, and draw the line.

Step by step solution

01

- Choose x-values

Start by selecting a few values for x. It's often helpful to choose a range of values that includes both negative and positive numbers. For example, choose x-values: -3, -2, -1, 0, 1, 2, 3.
02

- Calculate corresponding y-values

Substitute each chosen x-value into the equation to find the corresponding y-values. For example: - When x = -3: y = -(-3) - 2 = 3 - 2 = 1 - When x = -2: y = -(-2) - 2 = 2 - 2 = 0 - When x = -1: y = -(-1) - 2 = 1 - 2 = -1 - When x = 0: y = -(0) - 2 = -2 - When x = 1: y = -(1) - 2 = -1 - 2 = -3 - When x = 2: y = -(2) - 2 = -4 - When x = 3: y = -(3) - 2 = -5
03

- Plot the points

Create a Cartesian plane with the x-axis and y-axis. Plot each pair of (x, y) values obtained from the previous step: - Plot (-3, 1) - Plot (-2, 0) - Plot (-1, -1) - Plot (0, -2) - Plot (1, -3) - Plot (2, -4) - Plot (3, -5)
04

- Draw the line

After plotting all points, draw a straight line through them. This line represents the graph of the equation y = -x - 2.

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Key Concepts

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

plotting points
Plotting points is a fundamental skill in graphing linear equations.
To start, you need pairs of coordinates \(x,y\). In our example, we use the equation \( y = -x - 2 \) to find points.
We can choose any values for x. Let鈥檚 pick seven values: -3, -2, -1, 0, 1, 2, and 3.
Next, substitute these x-values into the equation to find corresponding y-values.
For instance, if \( x = -3 \), substitute it into the equation: \( y = -(-3) - 2 = 1 \).
Continue this process for all chosen x-values.
The pairs (-3, 1), (-2, 0), (-1, -1), (0, -2), (1, -3), (2, -4), and (3, -5) are obtained.
Lastly, plot these points on the Cartesian plane.
Cartesian plane
The Cartesian plane, named after Ren茅 Descartes, is a grid that enables us to plot points and graph functions.
The horizontal axis is called the x-axis, and the vertical axis is the y-axis.
Where they intersect is the origin, denoted as \( (0,0) \).
The plane is divided into four quadrants:
  • The first quadrant contains positive x and y values.
  • The second quadrant contains negative x and positive y values.
  • The third quadrant contains negative x and y values.
  • The fourth quadrant contains positive x and negative y values.

Using our equation \( y = -x - 2 \), the points (-3, 1), (-2, 0), etc., can be plotted.
Each point \( (x, y) \) shows a specific location on the plane.
When all points are accurately plotted, you can see the linear relationship more clearly.
linear functions
Linear functions are a type of function where the graph forms a straight line.
The general form of a linear equation is \( y = mx + b \).
This is known as the slope-intercept form.
In this form, 'm' represents the slope of the line, and 'b' represents the y-intercept.
In our equation \( y = -x - 2 \), it is already in slope-intercept form. Here, the slope (m) is -1, and the y-intercept (b) is -2.
Linear functions exhibit a constant rate of change and are foundational in algebra.
Graphing linear functions helps visualize the relationship between x and y.
slope-intercept form
The slope-intercept form of a linear equation makes it easy to graph.
This form \( y = mx + b \) highlights two key features: the slope and the y-intercept.
Slope (m) indicates the steepness and direction of the line.
A positive slope means the line rises as it moves from left to right; a negative slope means it falls.
The y-intercept (b) is where the line crosses the y-axis.
For \( y = -x - 2 \), the negative slope of -1 tells us that the line descends one unit vertically for every one unit it moves horizontally.
The y-intercept of -2 shows the line crosses the y-axis at -2.
Knowing the slope and y-intercept allows for quick and accurate graphing of the line.

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