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The function f(x)=x2 is transformed 3 times to become f(−x+4)+3 . Place the transformations in the order in which they occurred.

Type 1 for: translate left 4 units

Type 2 for: translate up 3 units

Type 3 for: reflect over the y-axis

(2 points)
First Transformation:

Second Transformation:

Third Transformation:

1 answer

To analyze the transformation of the function \( f(x) = x^2 \) to become \( f(-x + 4) + 3 \), let's break down the steps:

  1. Start with the original function:
    \( f(x) = x^2 \)

  2. Reflect over the y-axis:
    This changes the function to \( f(-x) \). Thus, the reflection over the y-axis is the first transformation.

    • Type 3 for reflect over the y-axis.

  3. Translate left 4 units:
    Now we have \( f(-x + 4) \), which can be rewritten as \( f(-(x - 4)) \). This shows that we are translating the graph left by 4 units.

    • Type 1 for translate left 4 units.

  4. Translate up 3 units:
    Finally, we add 3 to get \( f(-x + 4) + 3 \), which translates the graph up by 3 units.

    • Type 2 for translate up 3 units.

Putting this all together, the order of transformations is:

  • First Transformation: Type 3 (reflect over the y-axis)
  • Second Transformation: Type 1 (translate left 4 units)
  • Third Transformation: Type 2 (translate up 3 units)