What are the transformations from the parent​ function?

When referring to "transformations from the parent function," this usually involves changes to a basic or "parent" function that result in a new graph. The parent function represents the simplest form of a particular type of function, and transformations change its appearance, position, or size.

Here are the common types of transformations:

1. **Translation (Shift)**:
   - **Vertical Shift**: Moves the graph up or down.
     - \( f(x) + c \): Shifts the graph upward by \(c\) units.
     - \( f(x) - c \): Shifts the graph downward by \(c\) units.
   - **Horizontal Shift**: Moves the graph left or right.
     - \( f(x - c) \): Shifts the graph right by \(c\) units.
     - \( f(x + c) \): Shifts the graph left by \(c\) units.

2. **Reflection**:
   - **Reflection over the x-axis**:
     - \( -f(x) \): Flips the graph upside down (over the x-axis).
   - **Reflection over the y-axis**:
     - \( f(-x) \): Flips the graph horizontally (over the y-axis).

3. **Stretching and Compressing**:
   - **Vertical Stretch/Compression**:
     - \( a \cdot f(x) \): Stretches the graph vertically by a factor of \(a\) if \(a > 1\), or compresses it if \(0 < a < 1\).
   - **Horizontal Stretch/Compression**:
     - \( f(bx) \): Compresses the graph horizontally by a factor of \(b\) if \(b > 1\), or stretches it if \(0 < b < 1\).

4. **Rotation**:
   - Although less common in basic function transformations, certain combinations of reflections and stretches/compressions can appear to rotate graphs about a point.

### Example:
For a parent function \( f(x) = x^2 \) (a parabola), applying the transformation \( g(x) = -2(x - 3)^2 + 4 \) involves:
- A reflection over the x-axis (the negative sign).
- A vertical stretch by a factor of 2.
- A horizontal shift right by 3 units.
- A vertical shift upward by 4 units.

### Relative questions:
1. What are the transformations for the parent function \( f(x) = \sin(x) \) with \( g(x) = 2\sin(x - \pi) + 1 \)?
2. How does a horizontal stretch differ from a horizontal shift?
3. What effect does a reflection over the y-axis have on the graph of \( f(x) = x^3 \)?
4. Can multiple transformations occur simultaneously, and how would you represent them?
5. How would the graph of \( f(x) = \sqrt{x} \) change if it were reflected and stretched vertically?

**Tip**: Always pay attention to the order of transformations. Horizontal changes (shifts and stretches/compressions) are applied before vertical ones when interpreting function transformations.

What are the transformations from the parent function?

Explore the transformations of parent functions with detailed explanations on shifts, reflections, stretches, and compressions. Learn how these transformations affect the appearance and position of function graphs, suitable for Grades 9-12.

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