Sketch the Graph on the Cartesian Plane

Problem Solving > Sketch the Graph on the Cartesian Plane

Let’s say you were given a function like f(x) = x² – 2 (or, equivalently, y = x² – 2) and asked to sketch the graph on the Cartesian Plane. In order to solve the problem you need to:

1. Choose your x-values to input into the formula. Usually you’ll want to choose integers (whole numbers plus zero and negatives like -10, 3, 1, 7…) because they are easier to work with.
2. Create a few (x, y) coordinates.
3. Plot the points on a Cartesian plane (also called an x-y graph).
4. Draw the curve or line of the graph.

Sketch the Graph on the Cartesian Plane: Example

Example question #1: Sketch the graph of y = x² – 2 on the Cartesian plane.

Step 1: Choose your x-values. It’s up to you what values to choose for your x-values, but pick numbers that are easy to calculate. A good starting point is a few values around zero: -2, -1, 0, -1, 2. Or you could try -10, -5, 0, 5, 10.

Step 2: Input your x-values from Step 1 into the function. To do this, you replace the “x” in the formula with the number you choose for x. Let’s use -2, -1, 0, -1, 2.

• y = x^-2 – 2 = 2
• y = x^-1 – 2 = -1
• y = x⁰ – 2 = -2
• y = x¹ – 2 = -1
• y = x² – 2 = 2

Step 3: Create a table of values from Steps 1 and 2. This step is optional, but it can help you see the (x, y) points a little more clearly:

Step 4: Plot the points.

Step 5: Draw the curve or line of the graph. This is really just connecting the dots.


If you aren’t sure at this point about the shape of the graph, go back to Step 1 and choose a couple more x-values to complete the picture.

Step 6: Check your work. You could use a graphing calculator: I used the one at Desmos.com to create the graphs here.
If you can’t use a graphing calculator, throw in a random x-value as an input, like x = 9.2 or x = 7.5. If you have drawn your graph correctly, your new point will fall on the line you drew.

Tip: Unless you have a linear function, which always gives a straight line graph, draw a curve from each point to the next instead of a series of straight lines. For example, don’t do this:

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