Given the function check all the transformations that occurred from the graph of the parent function

Question

f(x)= 1/2(x_+7)^2 + 4
Up 7 units
Up 4 units
Down 7 units
Down 4 units
Leafy 7 units
Left 4 units
Right 7 units
Right 4 units
Vertical stretch
Reflection x-axis
Reflection y-axis

Answer 1

To analyze the transformations that occurred from the graph of the parent function \( f(x) = x^2 \) to the given function \( f(x) = \frac{1}{2}(x + 7)^2 + 4 \), we can break it down into parts by looking at how it differs from the parent function.

1. Parent function: \( f(x) = x^2 \)

2. Horizontal Shift:

   • The term \( (x + 7)^2 \) indicates a shift to the left by 7 units. The standard transformation for horizontal shifts is that if \( g(x) = (x - h)^2 \), then the graph shifts right by \( h \) units. Here, because of the plus sign, it shifts left.

3. Vertical Stretch/Compression:

   • The coefficient \( \frac{1}{2} \) indicates a vertical compression (not a stretch). A factor less than 1 compresses the graph vertically.

4. Vertical Shift:

   • The \( + 4 \) at the end indicates a shift up by 4 units.

Putting this together, we can summarize the transformations:

• Left 7 units
• Vertical compression by a factor of \( \frac{1}{2} \)
• Up 4 units

Therefore, the transformations that occurred are:

• Left 7 units
• Up 4 units
• Vertical compression (not listed in your options, but important to note)

The following transformations are not applicable:

• Up 7 units
• Down 7 units
• Down 4 units
• Right 7 units
• Right 4 units
• Vertical stretch
• Reflection x-axis
• Reflection y-axis

So, the correct answers are that the graph of \( f(x) \) is shifted left 7 units and up 4 units, and it experiences a vertical compression.