when describing transformations how do you write the equation of a graph

To write the equation of a transformed graph, you need to consider how each transformation affects the parent function. Transformations include translations, reflections, stretches, and compressions. Here is a general way to write the equation of a transformed graph based on different types of transformations:

### General form of a transformed function:
For a given function \( f(x) \), the transformation can be written as:

\[
g(x) = a \cdot f(b(x - h)) + k
\]

Where:
- \( a \): **Vertical stretch/compression** or **reflection** factor. If \( |a| > 1 \), it stretches the graph vertically. If \( 0 < |a| < 1 \), it compresses vertically. If \( a < 0 \), it reflects over the x-axis.
- \( b \): **Horizontal stretch/compression** or **reflection** factor. If \( |b| > 1 \), it compresses the graph horizontally. If \( 0 < |b| < 1 \), it stretches horizontally. If \( b < 0 \), it reflects over the y-axis.
- \( h \): **Horizontal translation** (shift). Positive \( h \) shifts the graph to the right, negative \( h \) shifts it to the left.
- \( k \): **Vertical translation** (shift). Positive \( k \) shifts the graph upward, negative \( k \) shifts it downward.

### Examples:
1. **Vertical shift**: 
   If you have \( f(x) = x^2 \), shifting it up by 3 units would be written as:
   \[
   g(x) = f(x) + 3 = x^2 + 3
   \]

2. **Horizontal shift**:
   If you shift \( f(x) = \sqrt{x} \) to the right by 2 units:
   \[
   g(x) = f(x - 2) = \sqrt{x - 2}
   \]

3. **Reflection across the x-axis**:
   Reflecting \( f(x) = \sin(x) \) across the x-axis:
   \[
   g(x) = -\sin(x)
   \]

4. **Vertical stretch by a factor of 2**:
   Stretching \( f(x) = \ln(x) \) vertically by a factor of 2:
   \[
   g(x) = 2\ln(x)
   \]

5. **Combining multiple transformations**:
   Reflecting \( f(x) = x^3 \), shifting it 3 units left and 4 units up:
   \[
   g(x) = -f(x + 3) + 4 = -(x + 3)^3 + 4
   \]

### Summary:
The function \( g(x) = a \cdot f(b(x - h)) + k \) encapsulates the transformation of the parent function \( f(x) \) based on shifts, stretches, compressions, and reflections.

Would you like further details or have any questions?

### Five questions to expand:
1. How do horizontal and vertical stretches affect the graph differently?
2. What happens if you apply both horizontal and vertical translations together?
3. How do you write the equation for a reflection over the y-axis?
4. What effect does a negative value for \( b \) have on the graph?
5. How can we combine multiple transformations in a single equation?

### Tip:
To visualize transformations, start with the basic parent function and apply each transformation step-by-step (stretch, shift, reflect).

Learn how to write the equation of a transformed graph based on translations, stretches, compressions, and reflections. This problem includes key concepts in algebra and graph transformations. Suitable for students in Grades 9-12.

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