Question

which set of transformations can be used to obtain the graph of $g(x) = 2(x + 3)^3$ from the graph of $f(x) = x^3$?
a. a vertical stretch by a factor of 2, followed by a shift of 3 units right
b. a vertical stretch by a factor of 2, followed by a shift of 3 units left
c. a horizontal stretch by a factor of 2, followed by a shift of 3 units right
d. a horizontal stretch by a factor of 2, followed by a shift of 3 units left

Explanation:

Step1: Analyze Vertical Stretch

For a function \( y = f(x) \), a vertical stretch by a factor of \( a \) (where \( a>1 \)) transforms it to \( y = a\cdot f(x) \). Here, \( f(x)=x^3 \), and \( g(x) \) has a factor of 2 multiplying \( (x + 3)^3 \), so first, we have a vertical stretch by factor 2: \( y = 2x^3 \).

Step2: Analyze Horizontal Shift

For a function \( y = f(x) \), a horizontal shift left by \( h \) units ( \( h>0 \)) transforms it to \( y = f(x + h) \). From \( y = 2x^3 \), we get \( y = 2(x + 3)^3 \) by replacing \( x \) with \( x+3 \), which is a shift 3 units left.

Step3: Evaluate Options

• Option A: Shift right is wrong, eliminate.
• Option B: Vertical stretch (factor 2) then shift left (3 units) matches, keep.
• Option C: Horizontal stretch is wrong (vertical stretch here), eliminate.
• Option D: Horizontal stretch is wrong, eliminate.

Answer:

B. a vertical stretch by a factor of 2, followed by a shift of 3 units left