A function is defined for all real numbers

MATHEMATICS · Multi-part · AP · AP Precalculus · Polynomial and Rational Functions · Polynomial and Rational Functions

Problem

A function

r

is defined for all real numbers, and selected values of

r(x)

are listed.

$x$	$50$	$100$	$200$	$400$	$800$
$r(x)$	$4.97$	$9.93$	$19.87$	$39.77$	$79.57$

Assume that, for large positive

x

, the graph of

r

approaches an oblique (slant) asymptote of the form

y=mx+b

. (a) For each

x

in the table, compute the ratio

\dfrac{r(x)}{x}

. Record your results. (b) Use the values from part (a) to estimate the slope

m

of the oblique asymptote. Justify your estimate with numerical evidence. (c) Using the slope

m

found in part (b), compute

r(x)-mx

for each

x

in the original table. Record these new values and describe their trend as

x

becomes large. (d) Use your work in parts (b) and (c) to write the equation of the oblique asymptote

y=mx+b

. Explain how your numerical findings support both the slope and the intercept of the line.