PRE CALCULUS Polynomials and Rational Functions 3

Question 1

Which transformation maps f(x) = x³ to g(x) = −(x + 2)³ + 5?

Accepted Answer

Reflect over x-axis, shift left 2, up 5

Answer

The negative reflects over the x-axis, (x + 2) shifts left 2, and + 5 shifts up 5.

Question 2

What is the oblique (slant) asymptote of f(x) = (x² + 3x − 4)/(x − 1)?

Accepted Answer

y = x + 4

Answer

Dividing x² + 3x − 4 by x − 1 gives quotient x + 4 with remainder 0, so the slant asymptote is y = x + 4.

Question 3

How many positive real zeros does f(x) = x⁵ − 3x⁴ + x² − 2 have at most, by Descartes' Rule of Signs?

Accepted Answer

3

Answer

The sign changes in coefficients (+, −, 0, +, 0, −) are: + to −, − to +, + to − = 3 sign changes, so at most 3 positive real zeros.

Question 4

Which end behavior describes f(x) = −2x⁴ + 7x − 1?

Accepted Answer

Down left, down right

Answer

Even degree with negative leading coefficient means both ends point downward.

Question 5

Find all vertical asymptotes of f(x) = (x − 3)/[(x + 1)(x − 3)].

Accepted Answer

x = −1 only

Answer

The factor (x − 3) cancels, creating a hole at x = 3; only x = −1 remains as a vertical asymptote.

PRE CALCULUS Practice Test