Which of the following tables correctly shows the order of operations that transforms f(x)=x2

Question

to f(−2x)+3
?

Option #1:

x Step 1: −2x
Step 2: (−2x)2
Step 3: (−2x)2+3

−1
2 4 7
0 0 0 3
2 −4
16 19
Option #2:

x Step 1: 2x
Step 2: (2x)2
Step 3: −(2x)2
Step 4: −(2x)2+3

−1
−2
4 −4
−1

0 0 0 0 3
2 4 16 −16
−13

Option #3:

x Step 1: x2
Step 2: −2x2
Step 3: −2x2+3

−1
1 −2
1
0 0 0 3
2 4 −8
−5

Answer 1

To determine which option correctly shows the order of operations that transforms \( f(x) = x^2 \) to \( f(-2x) + 3 \), we can break down the transformation step-by-step.

1. Start with \( f(x) = x^2 \).

2. Substitute \(-2x\) into the function \( f \): \[ f(-2x) = (-2x)^2 = 4x^2 \]

3. Add 3 to the result: \[ f(-2x) + 3 = 4x^2 + 3 \]

Now, let's analyze each of the given options to see if they reflect this transformation correctly.

Option #1:

• Step 1: \(-2x\)
• Step 2: \((−2x)^2 = 4x^2\)
• Step 3: \((−2x)^2 + 3 = 4x^2 + 3\)

The computations made in this option are consistent with our transformation steps. The transformations appear to be correct.

Option #2:

• Step 1: \(2x\)
• Step 2: \((2x)^2 = 4x^2\)
• Step 3: \(-(2x)^2 = -4x^2\)
• Step 4: \(-4x^2 + 3\)

This option also computes \((2x)^2\) correctly, but it adds an unnecessary negative sign which is not part of the transformation we intended. Thus, the result is incorrect compared to \( f(-2x) + 3 \).

Option #3:

• Step 1: \(x^2\)
• Step 2: \(-2x^2\)
• Step 3: \(-2x^2 + 3\)

This option seems to miss the transformation step of substituting \(-2x\) into the function correctly, so it does not match our intended transformation.

Summary:

Only Option #1 correctly represents the order of operations to transform \( f(x) = x^2 \) into \( f(-2x) + 3 \), yielding \( 4x^2 + 3 \).

Thus, the answer is Option #1.