Transformations are super important for understanding how a quadratic function looks when you graph it. A basic quadratic function can be written like this:

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

In this equation, $a$, $b$, and $c$ are just numbers that help define the function. When you graph this function, it creates a shape called a parabola. Here are some key transformations that change how the graph looks:

1. Vertical Shifts

• What It Is: Moving the graph up or down.
• Effect: If you add or subtract a number $k$ from the function, the graph goes up or down.
• Example: In the function $g(x) = ax^2 + bx + (c+k)$, if $k$ is positive, the graph goes up by $k$ units. If $k$ is negative, it goes down by $k$ units.

2. Horizontal Shifts

• What It Is: Moving the graph left or right.
• Effect: If you change $x$ to $(x-h)$ in the function, the graph shifts to the side.
• Example: In the function $h(x) = a(x-h)^2 + b(x-h) + c$, if $h$ is positive, the graph moves to the right by $h$ units. If $h$ is negative, it moves to the left by $h$ units.

3. Vertical Stretch and Compression

• What It Is: Changing how steep the parabola is.
• Effect: Altering the number $a$ causes the graph to stretch or compress vertically.
• Example: If $|a| > 1$, the graph stretches and looks taller and narrower. If $0 < |a| < 1$, the graph compresses and looks shorter and wider.

4. Reflection

• What It Is: Flipping the graph over a line.
• Effect: If $a$ is a negative number, the graph opens down instead of up.
• Example: In the function $f(x) = -ax^2 + bx + c$, the graph flips over the x-axis, changing from an upward shape to a downward shape.

5. Horizontal Stretch and Compression

• What It Is: Changing the width of the parabola from side to side.
• Effect: If you replace $x$ with $(kx)$ and $|k| > 1$, the graph gets narrower. If $0 < |k| < 1$, it gets wider.
• Example: In the function $j(x) = a(kx)^2 + b(kx) + c$, as $k$ gets bigger, the parabola becomes narrower. As $k$ gets smaller, it becomes wider.

Summary of Transformations:

• Vertical Shift: Add or subtract a number $k$.
• Horizontal Shift: Change $x$ to $(x-h)$.
• Vertical Stretch/Compression: Change the $a$ value (increase it to stretch, decrease it to compress).
• Reflection: Use a negative $a$ to flip the graph over the x-axis.
• Horizontal Stretch/Compression: Change $k$ in $(kx)$ to adjust the width.

By understanding these transformations, students can better predict what a graph will look like based on a quadratic function. This skill can help them accurately graph quadratics and explore different quadratic equations and their features more easily.