Question

how does the graph of $g(x) = 3x^3 + 6$ differ from the graph of its parent function $f(x) = x^3$? select two that apply.
a the graph is translated 3 units up.
b the graph is stretched vertically.
c the graph is compressed vertically.
d the graph is translated 6 units up.
e the graph is translated 6 units right.

Explanation:

Step1: Analyze Vertical Stretch/Compression

For a function \( y = a f(x) \), if \( |a|>1 \), the graph is vertically stretched by a factor of \( |a| \). Here, \( g(x)=3x^{3}+6 \) and \( f(x)=x^{3} \), so \( a = 3>1 \), so the graph is stretched vertically (matches option B).

Step2: Analyze Vertical Translation

For a function \( y = f(x)+k \), if \( k>0 \), the graph is translated \( k \) units up. Here, \( g(x)=3x^{3}+6 \), so \( k = 6>0 \), so the graph is translated 6 units up (matches option D).

Step3: Eliminate Other Options

• Option A: The vertical translation is 6 units, not 3, so A is wrong.
• Option C: \( |a| = 3>1 \) means stretch, not compression, so C is wrong.
• Option E: There is no horizontal shift (the form for horizontal shift is \( f(x - h) \)), so E is wrong.

Answer:

B. The graph is stretched vertically.
D. The graph is translated 6 units up.