To describe the transformation of the parent function $f(x) = x^2$ to the transformed function $f(x) = -(x-2)^2 + 3$, let's carefully analyze the transformations step by step:

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Parent Function

The parent function is $f(x) = x^2$, which is a standard upward-opening parabola centered at $(0, 0)$.

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Transformed Function

The transformed function is given as:

$f(x) = -(x-2)^2 + 3$

To describe the transformations, compare the given transformed function to the general transformation form of a quadratic function:

$f(x) = a(x-h)^2 + k$

Here:

• $a$ controls the vertical stretch/compression and reflection.
• $(h, k)$ is the vertex of the parabola.
• $h$ indicates a horizontal shift.
• $k$ indicates a vertical shift.

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Step-by-Step Transformations

1. Horizontal Shift:
   The term $(x-2)$ inside the square indicates a shift 2 units to the right.

   • From $x$ to $x-2$, the parabola moves horizontally to the right.

2. Vertical Reflection:
   The negative sign $-$ in front of $(x-2)^2$ reflects the parabola over the $x$-axis.

   • The parabola now opens downward instead of upward.

3. Vertical Shift:
   The constant $+3$ at the end shifts the entire graph 3 units upward.

   • The new vertex is located at $(2, 3)$.

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Summary of Transformations

1. Shifted 2 units to the right (due to $(x-2)$).
2. Reflected over the $x$-axis (due to the negative sign).
3. Shifted 3 units upward (due to $+3$).

The new vertex of the transformed parabola is $(2, 3)$, and it opens downward.

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Final Transformation Description:

• Rightward shift of $2$ units.
• Reflection across the $x$-axis.
• Upward shift of $3$ units.

Would you like a graph or a step-by-step illustration of how these shifts look?

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Related Questions:

1. What are the key features (vertex, axis of symmetry, direction) of the parent and transformed functions?
2. How would the function change if the reflection were removed (no negative sign)?
3. What happens if the vertical shift is increased further, say to $+5$ instead of $+3$?
4. How would a vertical stretch or compression factor $a$ affect the graph of $f(x)$?
5. Can you determine the range and domain of the transformed function?

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Tip: To identify transformations, compare the equation to its general form step by step, noting shifts, reflections, and changes to the vertex.