Problem 64 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

To graph an equation in a rectangular coordinate system, understand the equation, prepare a table of values, plot the points, draw the resulting curve, and validate the graph.

Step 1: Understand the Equation

An equation gives a relationship between variables, often x and y in a two-dimensional coordinate system. A common form of such equations is \(y = f(x)\), where y changes in response to changes in the x-variable following the rule defined by function \(f\).

Step 2: Prepare a Table of Values

Choose several values for \(x\) and find the corresponding \(y\)-values using the equation. Prepare a table with columns for \(x\) and \(y\), and write down each pair of corresponding values.

Step 3: Plot Points onto the Coordinate System

On the rectangular coordinate system, plot the (\(x,y\)) pairs identified in step 2. The x-coordinate corresponds to the horizontal position, with positive values to the right and negative values to the left of the origin. The y-coordinate corresponds to the vertical position, with positive values above and negative values below the origin.

Step 4: Draw the Curve

Once a sufficient number of points are plotted, they can be used to draw a line or curve. The exact number of points needed often depends on the complexity of the function. In most cases, the curve will be a straight line for a linear equation and a parabola for a quadratic equation.

Step 5: Check the Graph

Finally, validate the plot by picking a point not used in step 2, substitute the \(x\)-coordinate into the function to get a \(y\)-value, and check that the (\(x,y\)) pair lies on the graph.