Problem 92 - Algebra 1

Question

Use a table of values to graph the equation. $$ y=x+2 $$

Step-by-Step Solution

Answer

The graphed line passes through the points (-1,1), (0,2), and (1,3). This represents the equation \(y=x+2\).

Step 1: Setup table of values

Start by setting up a table with two columns, one for \(x\) and one for \(y\). Pick several values for \(x\) as shown: \n For \(x = -1, 0, 1\).

Step 2: Substitute \(x\) into \(y = x + 2\) to obtain corresponding \(y\)

Substitute the selected values of \(x\) into the equation to get corresponding \(y\) values. The results are as follows: \n When \(x = -1\), \(y = -1+2 = 1\), \n When \(x = 0\), \(y = 0+2 = 2\), \n When \(x = 1\), \(y = 1+2 = 3\).

Step 3: Plotting the points and graph the line

With the pairs (-1, 1), (0, 2), and (1, 3), plot these points on the graph and draw a line that passes through them. This line is the graph of the equation \(y=x+2\).

Key Concepts

Table of Values

To begin graphing the linear equation \(y = x + 2\), we start with a table of values. This is essentially a handy tool to organize the information needed to plot the graph. Imagine it as a simple chart with two columns: one marked for \(x\) values and the other for the resulting \(y\) values based on our equation.

Begin by choosing values for \(x\). They can be any numbers you like, but it's often easiest to start with small whole numbers, such as -1, 0, and 1. This makes calculations straightforward and easy to handle on a graph.

Next, substitute each chosen \(x\) value into the equation \(y = x + 2\) to find the corresponding \(y\) value:

• For \(x = -1\), calculate \(y = -1 + 2 = 1\).
• For \(x = 0\), calculate \(y = 0 + 2 = 2\).
• For \(x = 1\), calculate \(y = 1 + 2 = 3\).

Writing these in the table gives us pairs of points: (-1, 1), (0, 2), and (1, 3). These pairs represent coordinates that will be crucial for our graph.

Plotting Points

Once you have your table of values and respective point pairs, it's time to plot these points on the graph. Picture a coordinate plane, much like a map, where each point is a precise location described by an \(x\) and \(y\) value.

Start by locating each point on your coordinate plane:

• Locate the point (-1, 1). Here, move left to \(x = -1\) and then up to \(y = 1\).
• Plot (0, 2), moving not left or right from the origin, then up to \(y = 2\).
• Plot (1, 3) by moving right to \(x = 1\) and then up to \(y = 3\).

Mark these spots clearly. These coordinates will serve as guides in visualizing your graphing line. It's essential to ensure accuracy, as even a small shift can lead to errors in the graph.

Coordinate Plane

Understanding the coordinate plane is fundamental when graphing linear equations. Think of it as a two-dimensional space that lets us represent algebraic equations visually.

The coordinate plane consists of two perpendicular lines:

• The horizontal line, known as the \(x\)-axis.
• The vertical line, known as the \(y\)-axis.

Where these two lines intersect is called the origin, represented as \((0, 0)\).

Every point on this plane is defined by a pair of numbers, \((x, y)\), where \(x\) tells you how far to move horizontally and \(y\) indicates vertical movement. When plotting linear equations, these coordinates provide the exact spots to mark your points.

In our example with the equation \(y = x + 2\), spreading our points across the plane reveals the equation as a line, simple yet precise. The ability to translate algebra into geometry via the coordinate plane gives us a deeper understanding and visual evidence of mathematical relationships.