A sketch of the graph of a continuous function is described below

MATHEMATICS · Multi-part · A Level · Mathematics · Algebra and Functions · Algebra and Functions

Problem

A sketch of the graph of a continuous function

p

is described below. • The graph intersects the

x

–axis at

(-4,0)

and

(5,0)

. • As

x\to -2^-,

the

y

–values decrease without bound. • As

x\to -2^+,

the

y

–values increase without bound. • For large positive

x

, the points on the graph lie close to the line

y=\tfrac12x+3

. • For large negative

x

, the points on the graph lie close to the line

y=\tfrac12x+3

as well. • Between

x=-2

and

x=5

the graph is always above the

x

–axis. Use this verbal description to analyze asymptotic behavior. (1) Identify every vertical asymptote of

y=p(x)

and support your answer with limit notation expressed in terms of

p

. Your explanation must reference the one–sided behavior stated in the description. (2) Derive the equation of the oblique (slant) asymptote and explain, using an appropriate limit, how the graph’s long–run behavior verifies your result. (3) Explain why the oblique asymptote you found cannot be a horizontal asymptote, citing relevant properties of limits at infinity. (4) Sketch a possible graph of

y=p(x)

that satisfies all given conditions. Label the vertical and oblique asymptotes clearly and indicate the intercepts.

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