Problem 64 - College Algebra

Question

Explain how to graph an equation in the rectangular coordinate system.

Step-by-Step Solution

Answer

Graphing an equation in the rectangular coordinate system involves understanding the coordinate system, identifying the equation to be plotted, finding the slope and y-intercept from the equation, and drawing a straight line through the points determined by the slope and y-intercept. For our given example, the line graph of the equation \(y = 3x - 2\) intercepts the y-axis at -2 and rises 3 units for each unit moved to the right.

Step 1: Understand the rectangular coordinate system

Before starting to graph anything, we first need to understand the rectangular coordinate system. It is composed of two perpendicular number lines: the horizontal line called \(x\)-axis and the vertical line called \(y\)-axis. The intersection of the two axes is the point called the origin, where both \(x\) and \(y\) coordinates equal to 0. The location of every point in this system can be described by an ordered pair of numbers, \((x, y)\), where the first number represents the \(x\)-coordinate (horizontal position relative to the origin) and the second number represents the \(y\)-coordinate (vertical position relative to the origin).

Step 2: Identify the equation

Identify the equation you are tasked with graphing. For our example, let's use a linear equation in its slope-intercept form \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. The slope tells how steep the line is, and the y-intercept tells where the line crosses the y-axis.

Step 3: Determine the slope and y-intercept

From the chosen equation, determine the values of the slope and the y-intercept. The slope is the number in front of the \(x\) and the y-intercept is the number added or subtracted from \(mx\). For example, in \(y = 3x - 2\), the slope is 3, and the y-intercept is -2.

Step 4: Plot the y-intercept

On the graph, find the value of the y-intercept on the y-axis and put a dot. In our example, put a dot on -2 on the y-axis.

Step 5: Use the slope to find the next point

Starting from the y-intercept, use the slope to find the next point. In our example, the slope is 3, which means for every 1 unit increase in \(x\), \(y\) increases by 3 units. So, from the y-intercept (-2), go right 1 unit (increase in \(x\)) and up 3 units (increase in \(y\)) and mark another dot.

Step 6: Draw the line

Once you have two or more points, you can draw a straight line through them. This line represents the graph of the equation.