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

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

1 answer

To analyze the function \( f(x) = \frac{1}{2}(x+7)^2 + 4 \) in terms of transformations from its parent function, which is \( f(x) = x^2 \), we can break it down into components.

  1. Horizontal Shift: The term \( (x + 7) \) indicates a horizontal shift. Since it is \( +7 \), the graph shifts to the left by 7 units.

  2. Vertical Stretch/Compression: The coefficient \( \frac{1}{2} \) in front of the squared term indicates a vertical compression by a factor of \( \frac{1}{2} \). This means the graph is "squished" towards the x-axis compared to the parent function.

  3. Vertical Shift: The \( +4 \) at the end indicates a vertical shift upwards by 4 units.

Now, let's summarize the transformations based on the options provided:

  • Left 7 units (due to \( +7 \) in the function inside the square)
  • Up 4 units (due to the \( +4 \) at the end)
  • Vertical compression (due to the factor of \( \frac{1}{2} \))

The other transformations you listed (going down 7 or 4 units, right shifts, reflections) do not apply to this function.

Final Summary:

The transformations that occurred are:

  • Left 7 units
  • Up 4 units
  • Vertical compression