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Chapter 3: Problem 15

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

Short Answer

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

Step by step solution

01

- Choose Values for x

Select a few values for the variable x to plot. For simplicity, let's choose -2, -1, 0, 1, and 2.
02

- Calculate Corresponding y-values

Substitute the chosen x-values into the equation to find the corresponding y-values. For x = -2: y = 2(-2) = -4For x = -1: y = 2(-1) = -2For x = 0: y = 2(0) = 0For x = 1: y = 2(1) = 2For x = 2: y = 2(2) = 4
03

- Plot the Points

Plot each pair ewline (-2, -4), (-1, -2), (0, 0), (1, 2), and (2, 4) on the coordinate plane.
04

- Draw the Line

Connect the plotted points with a straight line. This line represents the graph of the equation y = 2x.

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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 when graphing linear equations. Each point on a graph corresponds to a pair of values, one for x and one for y. To plot points:
  • First, choose values for x. These can be any numbers, but simpler values like -2, -1, 0, 1, and 2 make calculations easier.
  • Next, substitute these x-values into the equation to find the corresponding y-values. For example, if you have the equation \( y=2x \), and you choose x = -1, then y = 2(-1) = -2, so the point is (-1, -2).
  • Finally, place these points on a graph. On a coordinate plane, x-values are shown on the horizontal axis and y-values on the vertical axis.
After plotting each point correctly, you provide a visual representation of the equation.
Coordinate Plane
The coordinate plane is a two-dimensional surface where we can plot points, lines, and curves. It consists of a horizontal axis (x-axis) and a vertical axis (y-axis). The point where these two axes intersect is called the origin, marked as (0,0). Each point on the plane is defined by an ordered pair \(x,y\). To graph linear equations:
  • First, start by marking the origin (0,0).
  • Use positive and negative values on the x and y-axes to place your points accurately.
Drawing lines or curves through well-chosen points shows how one variable relates to another. This relationship is crucial in understanding the underlying equation. For example, the equation \( y=2x \) shows a straight line passing through all points plotted by this rule on the coordinate plane.
Linear Functions
Linear functions are equations that form a straight line when graphed. They can be written in the form \( y=mx+b \), where:
  • m is the slope, which indicates how steep the line is.
  • b is the y-intercept, which shows where the line crosses the y-axis.
In the given equation \( y=2x \), the slope is 2, and there is no y-intercept (or b=0). This means for every unit increase in x, y increases by twice as much. Steps to graphing a linear function include:
  • Choose several x-values.
  • Calculate their corresponding y-values.
  • Plot the resulting points on the coordinate plane.
  • Draw a straight line through these points.
Understanding linear functions helps in visualizing relationships and predicting outcomes based on the equation's formula.

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Most popular questions from this chapter

In the following exercises, (a) graph each function (b) state its domain and range. Write the domain and range in interval notation. $$ f(x)=|x|+1 $$

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