# 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.

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.

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