Question

Consider the function below. f(x) = x2 + 5 − x (a) Find the vertical asymptote(s). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) x = Find the horizontal asymptote(s). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) y = (b) Find the interval where the function is decreasing. (Enter your answer in interval notation.) (c) Find the interval where the function is concave up. (Enter your answer in interval notation.) (d) Use this information to sketch the graph of f. (Do this on paper. Your instructor may ask you to turn in this graph.)

Answer 1

a) DNE

b) Decreasing(-∞, 0.5) and Increasing (0.5, ∞)

c) (-∞, ∞)

d) See attached picture.

Explanation:

The function x² + 5 - x is a quadratic function. Its graph is a parabola or a U-shaped function. Quadratic functions do not have asymptotic behavior. There are no vertical or horizontal asymptotes. According to its graph attached below, the function starts by decreasing from negative infinity to 0.5. At the vertex it changes to increasing from 0.5 to infinity. This is written as (-∞, 0.5) and (0.5, ∞). It is concave up over its entire graph.

a) DNE

b) Decreasing(-∞, 0.5) and Increasing (0.5, ∞)

c) (-∞, ∞)

d) See attached picture.