Transformation rules for functions are mathematical rules that describe how the graph of a function can be transformed or modified. Here are some common transformation rules:

1. Vertical Translation: Adding or subtracting a constant value to the function's output (y-values) shifts the graph up or down.

   • Example: f(x) + c shifts the graph of f(x) c units upward, while f(x) - c shifts it c units downward.

2. Horizontal Translation: Adding or subtracting a constant value to the function's input (x-values) shifts the graph left or right.

   • Example: f(x + c) shifts the graph of f(x) c units to the left, while f(x - c) shifts it c units to the right.

3. Vertical Scaling: Multiplying the function's output by a constant value stretches or compresses the graph vertically.

   • Example: a * f(x) stretches the graph of f(x) vertically by a factor of |a|, while -a * f(x) reflects it across the x-axis and stretches it vertically.

4. Horizontal Scaling: Multiplying the function's input by a constant value stretches or compresses the graph horizontally.

   • Example: f(a * x) stretches the graph of f(x) horizontally by a factor of 1/|a|, while f(-a * x) reflects it across the y-axis and stretches it horizontally.

5. Reflection: Negating the function's output or input reflects the graph across the x-axis or y-axis, respectively.

   • Example: -f(x) reflects the graph of f(x) across the x-axis, while f(-x) reflects it across the y-axis.

These transformation rules allow us to manipulate functions and create new functions with different characteristics.

To change the coordinate point for a vertical transformation, you would add a constant value 'd' to the y-coordinate. The updated table would look like this:

| Transformation Rule | Description |

|-------------------------|--------------------------------------------------------------------------------------------------|

| Vertical Translation | Adding or subtracting a constant value to the function's output shifts the graph up or down. |

| Horizontal Translation | Adding or subtracting a constant value to the function's input shifts the graph left or right. |

| Vertical Scaling | Multiplying the function's output by a constant value stretches or compresses the graph vertically.|

| Horizontal Scaling | Multiplying the function's input by a constant value stretches or compresses the graph horizontally.|

| Reflection | Negating the function's output or input reflects the graph across the x-axis or y-axis, respectively.|print("hello world")

Parameters

In simple terms, parameters in functions allow us to modify or transform the graph of a function. In the equation g(x) = f(x) + k, the parameter k shifts the graph of f(x) vertically up or down by k units. If k is positive, the graph moves up, and if k is negative, it moves down.

In the equation g(x) = f(x-h), the parameter h shifts the graph of f(x) horizontally. If h is positive, the graph moves to the right, and if h is negative, it moves to the left.

So, by adjusting the parameters in these equations, we can change the position of the graph of f(x) to create new functions g(x).