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the function f(x) = x^2 is transformed 3 times to become f(-x+4) + 3. place the transformations in the order in which they occured

type 1 for: translate left 4 units
type 2 for: translate up 3 units
type 3 for: reflect over the x-axis

1 answer

To transform the function \( f(x) = x^2 \) into \( f(-x+4) + 3 \), we can break down the transformations step by step:

  1. Reflect over the x-axis: The transformation \( f(-x) \) reflects the graph of \( f(x) \) over the x-axis. This changes \( x^2 \) to \( (-x)^2 \), which is still \( x^2 \), thus this step can be considered as reflecting the function.

  2. Translate left 4 units: The part \( -x + 4 \) can be rewritten as \( -(x - 4) \), indicating a left shift. Therefore, \( f(-x + 4) \) translates the graph 4 units to the left.

  3. Translate up 3 units: Finally, the term \( +3 \) at the end means that the entire graph is shifted up by 3 units.

Now, placing these transformations in the order in which they occurred, we get:

  1. Reflect over the x-axis (3)
  2. Translate left 4 units (1)
  3. Translate up 3 units (2)

So the order is:

  • 3 (reflect over the x-axis)
  • 1 (translate left 4 units)
  • 2 (translate up 3 units)