Which of the following tables shows the correct order of combinations to transform f(x)=x2

Question

to f(2x+2)+9
?(1 point)
Responses

x Step 1: x+2
Step 2: 2(x+2)
Step 3: 2(x+2)2
Step 4: 2(x+2)2+9
−1
1 2 4 13
2 4 8 64 73x Step 1: x plus 2 Step 2: 2 left parenthesis x plus 2 right parenthesis Step 3: 2 left parenthesis x plus 2 right parenthesis squared Step 4: 2 left parenthesis x plus 2 right parenthesis squared plus 9 negative 1 1 2 4 13 2 4 8 64 73

x Step 1: x+9
Step 2: 2(x+9)
Step 3: 2(x+9)+2
Step 4: (2(x+9)+2)2
−1
8 16 18 324
2 11 22 24 576x Step 1: x plus 9 Step 2: 2 left parenthesis x plus 9 right parenthesis Step 3: 2 left parenthesis x plus 9 right parenthesis plus 2 Step 4: left parenthesis 2 left parenthesis x plus 9 right parenthesis plus 2 right parenthesis squared negative 1 8 16 18 324 2 11 22 24 576

x Step 1: 2x
Step 2: 2x+2
Step 3: (2x+2)2
Step 4: (2x+2)2+9
−1
−2
0 0 9
2 4 6 36 45x Step 1: 2 x Step 2: 2 x plus 2 Step 3: left parenthesis 2 x plus 2 right parenthesis squared Step 4: left parenthesis 2 x plus 2 right parenthesis squared plus 9 negative 1 negative 2 0 0 9 2 4 6 36 45

x Step 1: x2
Step 2: 2x2
Step 3: 2x2+2
Step 4: 2x2+2+9
−1
1 2 4 13
2 4 8 10 19

Answer 1

To find the correct order of transformations that transform \( f(x) = x^2 \) into \( f(2x + 2) + 9 \), we can break this down step by step using the given transformations:

1. Start with \( f(x) = x^2 \).
2. Step 1: Replace \( x \) with \( (x + 2) \): This means we shift the graph 2 units to the left, giving us \( f(x + 2) = (x + 2)^2 \).
3. Step 2: Now multiply by 2: This means we stretch the graph vertically by a factor of 2, yielding \( 2f(x + 2) = 2(x + 2)^2 \).
4. Step 3: Finally, we add 9: So we perform \( 2(x + 2)^2 + 9 \).

Putting these steps together, we can see that the correct transformation sequence is as follows:

• Step 1: \( x + 2 \)
• Step 2: \( 2(x + 2) \)
• Step 3: \( 2(x + 2)^2 \)
• Step 4: \( 2(x + 2)^2 + 9 \)

Now we can check the responses you gave:

1. The first response seems to contain the correct sequence:
   • \( -1, 1, 2, 4, 13 \) (for \( f(2x + 2) + 9 \))

This matches the operations we've outlined.

Continuing with the verification of values based on our transformation, we can calculate:

1. Start with values of \( x = -1, 0, 1, 2 \).
2. Calculate \( f(2x + 2) + 9 \):
   • For \( x = -1 \): \( f(2(-1) + 2) + 9 = f(0) + 9 = 0 + 9 = 9 \)
   • For \( x = 0 \): \( f(2(0) + 2) + 9 = f(2) + 9 = 4 + 9 = 13 \)
   • For \( x = 1 \): \( f(2(1) + 2) + 9 = f(4) + 9 = 16 + 9 = 25 \)
   • For \( x = 2 \): \( f(2(2) + 2) + 9 = f(6) + 9 = 36 + 9 = 45 \)

Thus, the final values should be \( 9, 13, 25, 45 \).

Among the given options, it looks like the closest match based on transformation steps is:

Response 1:

• \( \text{Step 1}: x+2\)
• \( \text{Step 2}: 2(x+2) \)
• \( \text{Step 3}: 2(x+2)^2 \)
• \( \text{Step 4}: 2(x+2)^2 + 9 \)

This aligns with your transformation sequence.

The final answer is: Response 1.

Answer 2

thats not it

Answer 3

I apologize for the confusion. Let's go through the transformation from \( f(x) = x^2 \) to \( f(2x + 2) + 9 \) more closely to find the correct sequence.

Given Transformation

We want to start with \( f(x) = x^2 \) and transform it into \( f(2x + 2) + 9 \).

Breaking Down the Function

1. Start with the inputs to the function:

   • We want to move from \( x \) to \( 2x + 2 \). This means we first scale \( x \) by 2 and then translate it by adding 2.

2. Substituting into the function:

   • \( f(x) = x^2 \) transforms into \( f(2x + 2) = (2x + 2)^2 \).

3. Final transformation involving the constant:

   • Finally, we need to add 9 to this: \( f(2x + 2) + 9 = (2x + 2)^2 + 9 \).

Step-by-Step Transformation

• Step 1: Start with \( x \).

• Step 2: Apply the transformation \( 2x + 2 \) (scaling and translating):

  • Here we see the comparison to the inputs with \( x + 2 \) is not correct, instead we are performing transformations to get to \( 2x + 2 \).

• Step 3: Apply the squaring function:

  • This gives us \( (2x + 2)^2 \).

• Step 4: Add the constant \( 9 \):

  • Hence we finalize with \( (2x + 2)^2 + 9 \).

Options Available

Now let's re-evaluate the provided options, ensuring we match the operations correctly:

1. Option 1:

   • Step 1: \( x + 2 \)
   • Step 2: \( 2(x + 2) \):
   • Step 3: \( 2(x + 2)^2 \)
   • Step 4: \( 2(x + 2)^2 + 9 \)

2. Option 2:

   • Step 1: \( x + 9 \) (incorrect, it should start with \( x \))

3. Option 3:

   • Step 1: \( 2x \)
   • Step 2: \( 2x + 2 \)
   • Step 3: \( (2x + 2)^2 \)
   • Step 4: \( (2x + 2)^2 + 9 \)
   • This looks valid, as it correctly shows the transformations needed to reach the desired output.

4. Option 4:

   • Step 1: \( x^2 \) (starts incorrectly)

Conclusion

Based on the breakdown and analysis of each step, Option 3 correctly represents the sequence of transformations from \( f(x) = x^2 \) to \( f(2x + 2) + 9 \).

Thank you for your patience, and the corrected response is Option 3.