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Chapter 3: Problem 39

Graph each linear equation. Plot four points for each line. $$y=\frac{1}{2} x$$

Short Answer

Expert verified
Plot points (-2,-1), (0,0), (2,1), (4,2) and connect with a straight line.

Step by step solution

01

- Identify the linear equation

The equation given is in the slope-intercept form: \(y=\frac{1}{2} x\)
02

- Create a table of values

Choose four values for \(x\). Common choices are -2, 0, 2, and 4. Substitute these values into the equation to find corresponding \(y\) values.
03

- Calculate corresponding values of \(y\)

For \(x = -2\): \[y=\frac{1}{2}(-2) = -1\]For \(x = 0\): \[y=\frac{1}{2}(0) = 0\]For \(x = 2\): \[y=\frac{1}{2}(2)=1\]For \(x = 4\): \[y=\frac{1}{2}(4)=2\]
04

- Plot the points

Plot the points \((-2,-1)\), \((0,0)\), \((2,1)\), and \((4,2)\) on the coordinate plane.
05

- Draw the line

Connect the plotted points with a straight line. This line is the graph of the equation \(y=\frac{1}{2}x\).

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form of a linear equation is one of the most common and useful ways to represent a linear equation. This form is written as \(y = mx + b\), where
  • \(m\) is the slope of the line
  • \(b\) is the y-intercept, which is where the line crosses the y-axis
In the given equation, \(y = \frac{1}{2}x\), the term \(\frac{1}{2}\) represents the slope (\(m\)). Since there is no constant term, it means \(b = 0\), indicating that the line passes through the origin (0,0). The slope, \(m\), indicates the steepness of the line. In this case, for every unit increase in \(x\), \(y\) increases by half a unit.
Coordinate Plane
A coordinate plane is a two-dimensional surface where we can plot points, lines, and curves. It is defined by a horizontal axis (x-axis) and a vertical axis (y-axis).
  • The point where the x-axis and y-axis intersect is called the origin, which has coordinates (0,0)
  • The plane itself is divided into four quadrants
When graphing a linear equation, we plot points on this plane and then draw a line through these points. Each point on the plane has coordinates (x,y), representing its position relative to the axes. Understanding the coordinate plane is crucial for visualizing and solving problems involving linear equations.
Plotting Points
Plotting points on the coordinate plane involves finding the specific locations that satisfy the given equation.
  • First, choose values for \(x\) and substitute them into the equation to find the corresponding \(y\) values
  • The resulting pairs (x,y) are your points
For our equation \(y = \frac{1}{2}x\), let's choose the points given in the solution: -2, 0, 2, and 4 for \(x\).
  • For \(x = -2\), \(y = \frac{1}{2}(-2) = -1\). So, the point is (-2,-1)
  • For \(x = 0\), \(y = \frac{1}{2}(0) = 0\). So, the point is (0,0)
  • For \(x = 2\), \(y = \frac{1}{2}(2) = 1\). So, the point is (2,1)
  • For \(x = 4\), \(y = \frac{1}{2}(4) = 2\). So, the point is (4,2)
Once these points are plotted, they show the path of the linear equation.
Linear Function
A linear function is a function that produces a straight line when graphed on the coordinate plane. It can be expressed in various forms, but the slope-intercept form \(y = mx + b\) is very common.
  • The slope (\(m\)) tells us how steep the line is
  • The y-intercept (\(b\)) tells us where the line crosses the y-axis
For the line \(y = \frac{1}{2}x\), since \(b = 0\), the line crosses the origin. The slope \(\frac{1}{2}\) shows that for every increase of 1 in \(x\), \(y\) increases by half. By plotting multiple points and connecting them, we visualize the function as a straight line. This visual representation helps us understand relationships between variables and solve equations more easily.

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