Graphing a function by plugging in points

I love showing this engaging example to my students to emphasize the importance of the various curve-sketching techniques that are taught in Precalculus and Calculus.

Problem. Sketch the graph of f(x) = x^5 - 5x^3 + 4x + 6.

“Solution”. Let’s plug in some convenient points, graph the points, and then connect the dots to produce the graph.

  • f(-2) = (-32) - 5(-8) + 4(-2) + 6 = 6
  • f(-1) = (-1) - 5(-1) + 4(-1) + 6 = 6
  • f(0) = (0) - 5(0) + 4(0) + 6 = 6
  • f(1) = (1) - 5(1) + 4(1) + 6 = 6
  • f(2) = (32) - 5(8) + 4(2) + 6 = 6

That’s five points (shown in red), and surely that’s good enough for drawing the picture. Therefore, we can obtain the graph by connecting the dots (shown in blue). So we conclude the graph is as follows.

linearquinticgreen line

Of course, the above picture is not the graph of f(x) = x^5 - 5x^3 + 4x + 6, even though the five points in red are correct. I love using this example to illustrate to students that there’s a lot more to sketching a curve accurately than finding a few points and then connecting the dots.

Previous Post
Next Post
Leave a comment

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.

%d bloggers like this: