Problem 35 - College Algebra

Question

For each of the following exercises, construct a table and graph the equation by plotting at least three points. $$y=\frac{1}{3} x+2$$

Step-by-Step Solution

Answer

Plot points: (-3, 1), (0, 2), and (3, 3) on a graph, and draw a line through them.

Step 1: Choose Values for x

Select three values for \( x \) to use to find corresponding \( y \) values. It's often easiest to choose simple numbers. Let's choose \( x = -3, 0, 3 \).

Step 2: Calculate Values of y

For each chosen \( x \) value, calculate \( y \) using the equation \( y = \frac{1}{3} x + 2 \):- For \( x = -3 \), \( y = \frac{1}{3}(-3) + 2 = -1 + 2 = 1 \).- For \( x = 0 \), \( y = \frac{1}{3}(0) + 2 = 0 + 2 = 2 \).- For \( x = 3 \), \( y = \frac{1}{3}(3) + 2 = 1 + 2 = 3 \).

Step 3: Construct a Table

Create a table listing the \( x \) and \( y \) values you calculated:\[\begin{array}{c|c} x & y \\hline -3 & 1 \ 0 & 2 \ 3 & 3 \\end{array}\]

Step 4: Plot the Points on a Graph

On a graph, plot the points \((-3, 1)\), \((0, 2)\), and \((3, 3)\). Use the table from the previous step as reference for these coordinates.

Step 5: Draw the Line

Once the points are plotted, draw a straight line through them, as the equation \( y = \frac{1}{3} x + 2 \) is a linear equation. The line should extend in both directions, passing through the plotted points.

Key Concepts

Coordinate System

To successfully graph linear equations, understanding the coordinate system is foundational. The coordinate system is composed of two perpendicular number lines: the horizontal line is known as the x-axis and the vertical line is the y-axis. These axes intersect at a point called the origin, which has the coordinates (0, 0).
When you want to plot a point on this grid, you use an ordered pair, or coordinates, written as \((x, y)\). The first number represents how far along the x-axis you move, and the second number tells how far you move along the y-axis.

• The x-axis increases as you move to the right and decreases to the left.
• The y-axis increases as you move up and decreases as you move down.

Each point you plot gives a position on the graph where the x-value and y-value meet. Understanding the coordinate system lays the groundwork for identifying and drawing the graph of a linear equation.

Slope-Intercept Form

The equation of a line is often given in a form called the slope-intercept form. This form is written as \(y = mx + b\), where:

• \(m\) is the slope of the line.
• \(b\) is the y-intercept, which is the point where the line crosses the y-axis.

In our exercise, the equation \(y = \frac{1}{3}x + 2\) is already in slope-intercept form:

• The slope \(m\) is \(\frac{1}{3}\), meaning the line rises one unit for every three units it runs along the x-axis.
• The y-intercept \(b\) is 2, indicating the line crosses the y-axis at the point \((0, 2)\).

The slope gives the direction and steepness of the line, while the y-intercept provides a starting point for graphing. By understanding these components, you can easily graph any linear equation in slope-intercept form.

Table of Values

Constructing a table of values is a helpful step to graph a linear equation. It involves choosing specific x-values and calculating the corresponding y-values. This gives you several points to plot on the graph.
Follow these straightforward steps to build a table of values:

• Select a few values for \(x\) (commonly nice, simple numbers) to work with.
• Substitute each \(x\) into the equation to find the matching \(y\).
• Record these \((x, y)\) pairs into a table.

For the equation \(y = \frac{1}{3}x + 2\), we chose \(x = -3, 0,\) and \(3\). This resulted in \(y\) values of 1, 2, and 3 respectively, forming a table of values with points:

• \((-3, 1)\)
• \((0, 2)\)
• \((3, 3)\)

Once these points are plotted, they clearly show the line's path, allowing you to draw the graph confidently. This approach ensures accuracy when graphing on the coordinate system.