Let be the curve with equation

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

Problem

Let

C

be the curve with equation

y=\log_3 x

. A series of transformations is applied to obtain a new curve

C_2

as follows: • Step 1: Reflect

C

in the

y

-axis to obtain curve

C_1

. • Step 2: Translate

C_1

two units to the left. • Step 3: Translate the resulting curve three units upward. (i) Obtain the equation of

C_1

. (ii) Obtain, in simplest form, the equation of

C_2

after all three steps. (iii) Find the coordinates of the image of the point

(3,1)

C

under the combined transformations. (iv) State the domain, range, and equation of the vertical asymptote of

C_2

. (v) Sketch

C_2

, clearly indicating the image of

(3,1)

and the asymptote.