Question

find the value of k for the function g. then describe the graph of g as a transformation of the graph of f.
select the correct choice below, and fill in the answer box(es) to complete your choice.
a. k is equal to □ the graph of g is compressed horizontally compared to the graph of f.
b. k is equal to □ the graph of g(x) is translated □ unit(s) down compared to the graph of f(x).
c. k is equal to □ the graph of g is stretched horizontally compared to the graph of f.
d. k is equal to 2 the graph of g(x) is translated 2 unit(s) up compared to graph of f(x).
e. k is equal to □ the graph of g(x) is translated □ unit(s) to the left compared to the graph of f(x).
f. k is equal to □ the graph of g is stretched vertically compared to the graph of f.
g. k is equal to □ the graph of g is compressed vertically compared to the graph of f

Explanation:

Step1: Analyze Vertical Translation

To find the vertical shift, compare the y - intercepts of \( f(x) \) and \( g(x) \). The y - intercept of a linear function \( y = mx + b \) is \( b \). If \( g(x) \) is a vertical translation of \( f(x) \), the value of \( k \) (the vertical shift) is the difference in their y - intercepts. From the graph, if \( g(x) \) is translated up by 2 units, then \( k = 2 \).

Step2: Identify the Transformation

A vertical translation of a function \( y = f(x) \) to \( y=f(x)+k \) means if \( k>0 \), it is a translation up by \( k \) units. So if \( k = 2 \), the graph of \( g(x) \) is \( f(x)+2 \), which is a translation 2 units up.

Answer:

D. \( k \) is equal to 2. The graph of \( g(x) \) is translated 2 unit(s) up compared to the graph of \( f(x) \)