Function Sketching

When prompted with a function sketching question like so...

Question: Sketch a graph of the function y = f(x) where f(x) is concave down and decreasing from (-∞, 2), has a vertical asymptote at x = 2 and horizontal asymptote at y = 5, and then is concave up and increasing from (2, ∞). f(x) has a hole at x = 3.

...there are 4 main components you need to watch out for:

  • Asymptotes
  • Rate of Change
  • Concavity
  • Holes

Asymptotes

I strongly recommend you start any function sketching problem with marking the asymptotes. It makes the rest of the problem way easier!

Just to be clear...

Asymptotes are a form of discontinuity found in functions. They are "lines" parallel to the x-axis or y-axis where the function approaches the value and gets infinitely close, but does not cross and reach that value.

The problem will explicitly declare these asymptotes like so:

Question: Sketch a graph of the function y = f(x) where f(x) is concave down and decreasing from (-∞, 2), has a vertical asymptote at x = 2 and horizontal asymptote at y = 5, and then is concave up and increasing from (2, ∞). f(x) has a hole at x = 3.

Let's start with the vertical asymptote, and then we'll move onto the horizontal one.

Vertical asymptotes

In this situation, the vertical asymptote is at x = 2.

Question: Sketch a graph of the function y = f(x) where f(x) is concave down and decreasing from (-∞, 2), has a vertical asymptote at x = 2 and horizontal asymptote at y = 5, and then is concave up and increasing from (2, ∞). f(x) has a hole at x = 3.

To visualize this, let's identify x = 2 on our graph.

Then, considering that this is a vertical asymptote, we'll draw a dotted vertical line at x = 2!

This symbolizes that our function will get infinitely close to x = 2, but never cross or reach that value.

Horizontal asymptotes

In this situation, the horizontal asymptote is at y = 5.

Question: Sketch a graph of the function y = f(x) where f(x) is concave down and decreasing from (-∞, 2), has a vertical asymptote at x = 2 and horizontal asymptote at y = 5, and then is concave up and increasing from (2, ∞). f(x) has a hole at x = 3.

To visualize this, let's identify y = 5 on our graph.

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