Question

question 11 (multiple choice worth 5 points) (03.02r mc) how does the graph of g(x) = (x - 1)³ + 5 compare to the parent function f(x) = x³? g(x) is shifted 1 unit to the right and 5 units up. g(x) is shifted 5 units to the right and 1 unit up. g(x) is shifted 1 unit to the left and 5 units up. g(x) is shifted 5 units to the right and 1 unit down.

Explanation:

To determine the transformation of the graph of \( g(x) = (x - 1)^3 + 5 \) from the parent function \( f(x) = x^3 \), we use the rules of function transformations:

1. Horizontal Shift: For a function of the form \( f(x - h) \), the graph is shifted \( h \) units to the right if \( h > 0 \) and \( h \) units to the left if \( h < 0 \). In \( g(x) = (x - 1)^3 + 5 \), we have \( h = 1 \), so the graph is shifted 1 unit to the right.

1. Vertical Shift: For a function of the form \( f(x) + k \), the graph is shifted \( k \) units up if \( k > 0 \) and \( k \) units down if \( k < 0 \). In \( g(x) = (x - 1)^3 + 5 \), we have \( k = 5 \), so the graph is shifted 5 units up.

Combining these transformations, the graph of \( g(x) \) is shifted 1 unit to the right and 5 units up from the graph of the parent function \( f(x) = x^3 \).

Answer:

The correct option is: "g(x) is shifted 1 unit to the right and 5 units up." (Assuming this is one of the provided options, likely the first one based on the description.)