The image you've uploaded is a worksheet asking to write equations for transformations of a function $y = f(x)$. Let me explain what I observe:

The table includes transformations such as horizontal/vertical stretches, reflections across the x-axis or y-axis, and corresponding mapping notations.

The task is to write the transformed function and include the mapping notation. Below are the transformations mentioned:

First Column:

1. Horizontal stretch by a factor of $\frac{3}{7}$, vertical stretch by $\frac{5}{9}$, and reflection across the x-axis:
   • Equation: The function after these transformations would be written as: $y = -\frac{5}{9} f\left(\frac{7}{3} x\right)$
   • Mapping Notation: $(x, y) \mapsto \left( \frac{7}{3}x, -\frac{5}{9}y \right)$

Second Column:

2. Horizontal stretch by a factor of $\frac{1}{4}$, vertical stretch by $\frac{2}{5}$, and reflection over the y-axis:
   • Equation: The transformed function will be: $y = \frac{2}{5} f\left(-4x\right)$
   • Mapping Notation: $(x, y) \mapsto (-4x, \frac{2}{5}y)$

Third Column:

3. Horizontal stretch by a factor of $\frac{3}{4}$, vertical stretch by $1$, and reflection over the y-axis:
   • Equation: Since there is no vertical stretch (it's $1$), but a horizontal stretch and a reflection over the y-axis: $y = f\left(-\frac{4}{3} x\right)$
   • Mapping Notation: $(x, y) \mapsto \left(-\frac{4}{3}x, y\right)$

Fourth Column:

4. Horizontal stretch by a factor of $2$, vertical stretch by $3$, and reflection across the y-axis:
   • Equation: This transforms the function as: $y = 3 f\left(-\frac{x}{2}\right)$
   • Mapping Notation: $(x, y) \mapsto \left(-2x, 3y\right)$

Would you like to go into more details on how these transformations affect the graph?

Related Questions:

1. What is the general formula for horizontal and vertical stretches of a function?
2. How do reflections across the x-axis and y-axis affect the function?
3. How do you combine multiple transformations in a function?
4. What is the difference between horizontal and vertical stretching in terms of the input and output of the function?
5. How do transformations apply to more complex functions, like quadratic or trigonometric functions?

Tip: Always apply horizontal transformations before vertical ones to avoid confusion when handling multiple transformations.