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).