Problem 40 - Algebra 1

Question

Use a table of values to graph the equation. \(y=3 x+2\)

Step-by-Step Solution

Answer

After plotting the points (-2, -4), (-1, -1), (0, 2), (1, 5), and (2, 8) on the graph and then connecting them, the line we obtain is the graph of the equation \(y = 3x + 2\).

Step 1: Determine the x values

Choose any arbitrary values for x. Typically, it is good to select a few negative, zero, and positive values. Let's go with -2, -1, 0, 1, and 2.

Step 2: Substituting the x-values into the equation

After the x-values are selected, substitute each of the x-values in the equation \(y = 3x + 2\) to find the corresponding y-values. That will give us the following pairs: (-2, -4), (-1, -1), (0, 2), (1, 5) and (2, 8).

Step 3: Plot the points on a graph

Plot the pairs of (x, y) you have obtained from the previous step onto a graph. The points will be (-2, -4), (-1, -1), (0, 2), (1, 5), and (2, 8). When you connect these points, it forms a straight line, which is the graph of the equation \(y = 3x + 2\).

Key Concepts

Table of Values

To begin graphing a linear equation like \(y = 3x + 2\), a table of values is an invaluable tool. This table helps you systematically organize chosen input values, known as \(x\)-values, and their corresponding output values or \(y\)-values. When creating this table:

• Choose a range of \(x\)-values. Picking a combination of negative, zero, and positive numbers can offer a balanced view of the line.
• Substitute each \(x\)-value into the equation to solve for \(y\). For \(-2, -1, 0, 1,\) and \(2\), substituting into the equation \(y = 3x + 2\) gives \(y\)-values of \(-4, -1, 2, 5,\) and \(8\), respectively.
• List these pairs \((-2, -4), (-1, -1), (0, 2), (1, 5), (2, 8)\) in the table. These coordinate pairs are crucial for plotting on the graph.

Using a table of values efficiently showcases the relationship between the variables in the equation, making the graphing process visually and conceptually smoother.

Coordinate System

Graphing within a coordinate system is essential for visualizing relationships in equations. This system, also known as the Cartesian Plane, is divided into four quadrants by the \(x\)-axis (horizontal line) and \(y\)-axis (vertical line). Understanding how to use this system involves:

• Recognizing the position of each quadrant - the upper right is the first, upper left is the second, lower left is the third, and lower right is the fourth.
• Identifying coordinates. A point, such as \((2,8)\), indicates a position on the plane, where \(2\) is the distance from the origin along the \(x\)-axis, and \(8\) is the distance along the \(y\)-axis.
• Each quadrant provides unique information about the sign of coordinates. For example, the first quadrant contains positive \(x\) and \(y\) values.

The coordinate system allows you to systematically plot points derived from equations and visualize the graphical representation of mathematical relationships.

Plotting Points

When graphing a linear equation, plotting points on the coordinate plane is the next critical step. This involves the careful placement of the calculated coordinate pairs on the graph:

• Start with your prepared table of values. Each pair, such as \((-2, -4)\), reflects a precise point.
• Locate each \(x\) value on the horizontal axis and move vertically until you reach the corresponding \(y\) value. Place a point at this intersection.
• Repeat for all points: \((-2, -4), (-1, -1), (0, 2), (1, 5),\) and \((2, 8)\).
• Once all points are plotted, connect them with a straight line. This illustrates the linear relationship described by the equation \(y = 3x + 2\).

Precision in plotting is vital, as each point contributes to the overall structure of the graph. Ensuring points align correctly will help demonstrate the equation's consistent rate of change or slope.