Question

question 10 of 10
how does the graph of $f(x) = \frac{1}{2}(3)^{x + 2} - 1$ relate to its parent function?

a. the parent function has been translated down.
b. the parent function has been stretched.
c. the parent function has been translated to the left.
d. the parent function has been compressed.

Explanation:

Step1: Identify Parent Function

The parent function of \( f(x) = \frac{1}{2}(3)^{x + 2}-1 \) is \( g(x)=3^x \) (exponential function).

Step2: Analyze Transformations

• Vertical Stretch/Compression: The coefficient \( \frac{1}{2} \) (between 0 and 1) means a vertical compression of the parent function \( g(x) = 3^x \) (since \( 0 < |a| < 1 \) for \( y = a\cdot g(x) \) indicates compression).
• Horizontal Translation: For \( y = g(x + h) \), \( h = 2>0 \) means a translation 2 units to the left.
• Vertical Translation: For \( y = g(x)-k \), \( k = 1>0 \) means a translation 1 unit down.

Now check options:

• Option A: The \(-1\) at the end translates the parent function down. Correct.
• Option B: The coefficient \( \frac{1}{2} \) is a compression, not a stretch. Incorrect.
• Option C: The \( x + 2 \) (i.e., \( h = 2 \)) translates the parent function 2 units to the left. Correct.
• Option D: The coefficient \( \frac{1}{2} \) (since \( 0<\frac{1}{2}<1 \)) compresses the parent function vertically. Correct. Wait, but let's re - evaluate:

Wait, the question is about how \( f(x) \) relates to its parent function. Let's re - check each option:

• Option A: The \(-1\) term: \( f(x)= \text{[transformed } g(x)\text{]} - 1 \), so the parent function \( g(x) \) is translated down by 1 unit. Correct.
• Option B: The coefficient \( \frac{1}{2} \): For \( y = a\cdot g(x) \), if \( 0 < a<1 \), it's a vertical compression, not a stretch. So B is incorrect.
• Option C: The \( x + 2 \) in the exponent: For \( y = g(x + h) \), \( h = 2>0 \), so the parent function \( g(x) \) is translated 2 units to the left. Correct.
• Option D: The coefficient \( \frac{1}{2} \): Since \( 0<\frac{1}{2}<1 \), the parent function \( g(x) \) is vertically compressed (because multiplying by a number between 0 and 1 compresses the graph vertically). Correct.

But maybe the question allows multiple correct options? Wait, the original problem's options:

Wait, let's re - check the function \( f(x)=\frac{1}{2}(3)^{x + 2}-1 \)

• Vertical Compression: The factor \( \frac{1}{2} \) (since \( 0 < \frac{1}{2}<1 \)) compresses the parent function \( 3^x \) vertically. So D is correct.
• Horizontal Translation Left: \( x+2 \) means shift left 2 units. So C is correct.
• Vertical Translation Down: \(-1\) means shift down 1 unit. So A is correct.

But maybe there was a mis - understanding. Wait, the options are check - boxes, so multiple options can be correct. But let's confirm:

• Option A: Translated down (because of \(-1\)): Correct.
• Option B: Stretched? No, \( \frac{1}{2} \) is a compression. Incorrect.
• Option C: Translated left (because of \( x + 2 \)): Correct.
• Option D: Compressed (because of \( \frac{1}{2} \)): Correct.

Answer:

A. The parent function has been translated down.
C. The parent function has been translated to the left.
D. The parent function has been compressed.