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Chapter 1: Problem 16

Graph each equation .Let $x=-3,-2,-1,0,1,2, and 3. $$ y=x+2 $$

Short Answer

Expert verified
The graph of the equation \(y = x + 2\) is a straight line slanting upwards and passing through the points for which the x-values are -3, -2, -1, 0, 1, 2, and 3.

Step by step solution

01

Find the corresponding y for each x value

Substitute the given x-values into the equation to find the corresponding y-values. For example, when \(x = -3\), \(y = -3 + 2 = -1\). Similarly, find y for all other x-values - which are -3, -2, -1, 0, 1, 2, and 3.
02

Plot the points

Each pair (x,y) represents a point in the 2D plane. Plot these points on the graph. For instance, for \(x = -3\) and \(y = -1\), mark this point on the graph.
03

Draw the line

After plotting all the points, you can see that they align on a straight line. Draw this line, which is 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.

Coordinate Plane
The coordinate plane is like a map used to locate points. It has two axes: the horizontal x-axis and the vertical y-axis. These axes intersect at the origin, marked as (0,0). Each point on this plane is identified by a pair of numbers (x, y), called coordinates.

  • The x-coordinate tells you how far to move left or right from the origin.
  • The y-coordinate tells you how far to move up or down.
Moving along these axes helps us plot and visualize mathematical equations, making sense of how different values interact.
Plotting Points
Plotting points involves placing dots at specific locations on the coordinate plane. Each dot represents a coordinate pair (x, y) that tells where to place the point on the graph.

For instance, if you have the point (-3, -1), start at the origin. Move 3 units to the left on the x-axis, and then 1 unit down on the y-axis. Place a dot there.

  • Repeat this process for each point given in the equation.
  • Ensure each point is accurately placed to reflect the relationship described by the equation.
Linear Functions
Linear functions are mathematical expressions that create straight lines when graphed. They have the form \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.

  • The slope \(m\) describes how steep the line is.
  • The y-intercept \(b\) tells where the line crosses the y-axis.
These functions are simple yet powerful tools to describe constant rates of change or proportional relationships between variables.
Equation of a Line
The equation of a line in the form \(y = x + 2\) is a linear function that represents a straight line. Each x-value you substitute results in a corresponding y-value, like a formula that transforms x into y.

  • For \(x = -3\), \(y = -1\).
  • Continue this process for other values to get a set of (x, y) pairs.
Once you have these points, you can draw a straight line through them, visually representing the equation's solution. This straight line provides a clear visual interpretation of the relationship between x and y.

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Most popular questions from this chapter

Solve inequality using a graphing utility. Graph side separately. Then determine the values of \(x\) for which the graph for the left side lies above the graph for the right side. \(-3(x-6)>2 x-2\)

Describe how to solve an absolute value inequality involving the symbol \(>\). Give an example.

Solve absolute value inequality. \(\left|\frac{3 x-3}{9}\right| \geq 1\)

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