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.
PAID CONTENT This is the end of the preview. To unlock the rest, get the Lifetime Access, MTH 141 Exam 3 Cram Kit or MTH 141 Cram Kit Bundle.
Already purchased? Click here to log in.