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Chapter 4: Problem 63

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

Short Answer

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

Step by step solution

01

- Choose values for x

Select a few values for the variable x. It is helpful to choose both negative and positive values. For instance, let's choose x = -2, -1, 0, 1, 2.
02

- Calculate corresponding y values

For each selected value of x, substitute it into the equation to calculate the corresponding y value. For x = -2: y = -2 + 2 = 0For x = -1: y = -1 + 2 = 1For x = 0: y = 0 + 2 = 2For x = 1: y = 1 + 2 = 3For x = 2: y = 2 + 2 = 4
03

- Plot the points on the graph

Plot the points on the coordinate plane: (-2, 0), (-1, 1), (0, 2), (1, 3), (2, 4).
04

- Draw the line through the points

Draw a straight line that passes through all the plotted points. This line represents 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.

Coordinate Plane
The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface on which we can plot points, lines, and curves.
It is formed by two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical).
These lines intersect at the origin, which has coordinates (0,0).

Each point on the coordinate plane is represented by an ordered pair \(x, y\).
The first number, x, tells you how far to move horizontally from the origin.
The second number, y, tells you how far to move vertically.

Dividing the plane into four quadrants helps in understanding the locations of all points:
  • The first quadrant (top-right) is where both x and y are positive.
  • The second quadrant (top-left) is where x is negative, and y is positive.
  • The third quadrant (bottom-left) is where both x and y are negative.
  • The fourth quadrant (bottom-right) is where x is positive, and y is negative.
Plotting Points
Plotting points is about locating points on the coordinate plane.
You start with an equation or a list of points, like (-2, 0), (-1, 1), (0, 2), (1, 3), and (2, 4).
Let's break down the steps to plot these points:

  • First, identify the x-value (the first number in the pair). Move that many units left or right from the origin.
  • Next, identify the y-value (the second number in the pair). Move that many units up or down from the x position.
  • Mark the spot where you end up. That鈥檚 one of your points!

Repeat these steps for each point you need to plot, and you will have a set of points that show the relationship described by the equation.
Linear Equation
A linear equation is an equation that allows us to represent a straight line on the coordinate plane.
The general form of a linear equation is \( y = mx + b \), where:
  • \(m\) is the slope of the line, telling us how steep the line is.
  • \(b\) is the y-intercept, showing us where the line crosses the y-axis.
For instance, in the equation \( y = x + 2 \), the slope \(m\) is 1 (since x is equivalent to 1x), and the y-intercept \(b\) is 2.
This means the line will rise one unit for every unit it runs to the right and will cross the y-axis at y = 2.

To graph this equation, select several values of x, compute the corresponding y values using the equation, and then plot the points \( (x, y) \).
Lastly, draw a straight line through these points to get the graphical representation of the linear equation.

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Most popular questions from this chapter

Find an equation of a line parallel to the given line and contains the given point. Write the equation in slope-intercept form. line \(y+7=0,\) point(1,-1)

Find an equation of a line parallel to the given line and contains the given point. Write the equation in slope-intercept form. line \(y=-2 x-3\), point (-1,3)

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Graph the linear inequality \(y \geq-\frac{1}{3} x-2\)

Graph the linear inequality \(x \leq 4\)
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