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Chapter 11: Problem 102

In the following exercises, graph each equation. $$ y=x $$

Short Answer

Expert verified
Plot the points (-2,-2), (-1,-1), (0,0), (1,1), and (2,2) and draw a straight line through them.

Step by step solution

01

Understand the Equation

The given equation is a linear equation in the form of y = mx + b, where m is the slope and b is the y-intercept. In this equation, y = x, the slope m is 1 and the y-intercept b is 0.
02

Identify Key Points

Choose key points to plot for the equation y = x. You can select points where x is -2, -1, 0, 1, and 2.
03

Calculate Corresponding y-values

Calculate the y-values using the equation y = x for the chosen x-values: When x = -2, y = -2 When x = -1, y = -1 When x = 0, y = 0 When x = 1, y = 1 When x = 2, y = 2.
04

Plot the Points

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

Draw the Line

Connect the points with a straight line. This line represents the equation y = x.

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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 makes a straight line when graphed on a coordinate plane. An example of a linear equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Every point on the line is a solution to the equation. For instance, the equation \( y = x \) has a slope of 1 and a y-intercept of 0, which means the line passes through the origin. To graph a linear equation, you need to find multiple points that lie on the line and connect them.
Coordinate Plane
A coordinate plane is a two-dimensional plane formed by two perpendicular lines called axes. The horizontal line is the x-axis, and the vertical line is the y-axis.
The point where these axes intersect is called the origin, and has coordinates (0, 0).
Each point on the plane is represented by an ordered pair (x, y), where x is the distance from the y-axis, and y is the distance from the x-axis.
Having a basic understanding of the coordinate plane is essential for graphing equations.
Slope-Intercept Form
The slope-intercept form is a way to express linear equations. An equation in slope-intercept form looks like \( y = mx + b \). Here:
  • \( m \) is the slope which shows the steepness of the line. It tells how much y changes for a change in x.
  • \( b \) is the y-intercept, the point where the line crosses the y-axis.
For example, in \( y = x \), the slope \( m = 1 \) (which means 1 unit up for every 1 unit right), and \( b = 0 \) (the line crosses the y-axis at 0).
This form makes it easy to graph because you can start at the y-intercept and use the slope to find other points on the line.
Plotting Points
Plotting points is the first step in graphing any equation. To plot points, you need a set of coordinates (x, y). For example, to graph \( y = x \):
  • Choose several values for x like -2, -1, 0, 1, 2.
  • Calculate the corresponding y-values (which are the same as x-values in this case).
So, you get points (-2, -2), (-1, -1), (0, 0), (1, 1), and (2, 2).
Next, mark these points on the coordinate plane and connect them with a straight line. This step-by-step process helps visualize the function鈥檚 behavior.

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Most popular questions from this chapter

Find the slope of the line \(y=2 .\)

In the following exercises, for each ordered pair, decide a. if the ordered pair is a solution to the equation. b. if the point is on the line. \(y=\frac{2}{3} x-1\) (a) \((0,-1)\) (b) \((3,1)\) (c) \((-3,-3)\) (d) \((6,4)\)

Find the \(x\) -intercept and \(y\) -intercept of the equation \(3 x-y=6\)

In the following exercises, graph by plotting points. $$ -x+y=4 $$

Use the slope formula to find the slope of the line between (0, -4) and (5, 2).
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