Question

(b) what transformations have occurred?
a shifts left 3 and up 1
b shifts left 3 and down 1
c shifts right 3 and down 1
d shifts up 3 and right 1
(c) what is the asymptote of the function?
a x = -3
b y = 3
c y = -3
d x = 3
(d) what are the domain and range of the function?
a domain: (-3, ∞)
range: (-∞, ∞)

Explanation:

Part (a) - Transformations

To determine function transformations, recall the form \( y = f(x - h)+k \), where \( h \) is horizontal shift (\( h>0 \) right, \( h<0 \) left) and \( k \) is vertical shift (\( k>0 \) up, \( k<0 \) down). Assuming the parent function (e.g., exponential, logarithmic) and transformed form, if the transformation is left 3 (so \( h=-3 \)) and down 1 ( \( k = - 1 \)), option B matches.

For functions like logarithmic or exponential (common in transformation problems), vertical asymptotes for \( \log(x - h) \) is \( x=h \), horizontal asymptotes for \( a^{x - h}+k \) is \( y = k \). If the transformation involved a horizontal shift left 3 (from part a), for a logarithmic - like function (with vertical asymptote), the asymptote would be \( x=-3 \) (since \( h=-3 \) in \( x - h=x+3 \), asymptote \( x=-3 \)), matching option A.

For a function with vertical asymptote \( x = - 3 \) (from part b, e.g., logarithmic function \( y=\log(x + 3)-1 \)), the domain is all \( x>-3 \) (i.e., \( (-3,\infty) \)). For range, logarithmic functions (or transformed ones) have range \( (-\infty,\infty) \), matching option A.

Answer:

B. shifts left 3 and down 1

Part (b) - Asymptote