Calculus How To

Sketch the Graph on the Cartesian Plane

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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:
table of values

Step 4: Plot the points.
sketch the graph on the cartesian plane

Step 5: Draw the curve or line of the graph. This is really just connecting the dots.
sketch the graph 4

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 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:

Stephanie Glen. "Sketch the Graph on the Cartesian Plane" From Calculus for the rest of us!

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