Understanding Quadratic Transformations

Quadratic transformations help us understand how parabolas look in coordinate geometry. They involve moving the graph up or down, shifting it left or right, flipping it, or changing its shape.

The basic form of a quadratic function is:

$$f(x) = ax^2 + bx + c$$

Types of Transformations:

1. Vertical Shifts:

   • You can move the graph up or down by adding or subtracting a number (let's call it $k$):
   $$f(x) = ax^2 + bx + (c + k)$$
   • What happens: If $k$ is positive, the graph goes up. If $k$ is negative, it goes down.

2. Horizontal Shifts:

   • To move the graph left or right, we change $x$ like this:
   $$f(x) = a(x - h)^2 + k$$
   • What happens: If $h$ is positive, the graph shifts left. If $h$ is negative, it shifts right.

3. Reflections:

   • A reflection happens if $a$ is negative:
   $$f(x) = -ax^2 + bx + c$$
   • What happens: The graph flips over the x-axis.

4. Stretches and Compressions:

   • We can make the graph taller (stretched) or shorter (compressed) by changing the value of $a$:
   $$f(x) = ax^2 \text{ (changing } a\text{)}$$
   • What happens: If $|a|$ is greater than 1, the graph stretches. If $|a|$ is less than 1, it compresses.

Important Points:

• The vertex form of a quadratic function is:

$$f(x) = a(x-h)^2 + k$$

• The point called the vertex $(h, k)$ is very important. It helps us find the highest or lowest point on the graph.
• The line called the axis of symmetry is at $x = h$. This line cuts the parabola into two identical parts, which helps us understand how the graph reflects and shifts.