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Function transformation involves changing the position or shape of the graph of a function. This can be achieved through various operations, such as shifting, stretching, compressing, or reflecting. In the context of a high school algebra or precalculus course, students typically learn about basic transformations including horizontal and vertical shifts, reflections over the x and y axes, and stretching or compressing by changing the scale factor.

When working with transformations, it's important to understand the order of operations: first the inside of the function is affected (horizontal shifts or stretches), then the outside (vertical shifts or stretches). For example, when adding or subtracting a number from the variable before applying the function (like replacing x with x+5), you're performing a horizontal transformation. If you add or subtract after applying the function (like adding 4 to the entire function), you’re executing a vertical transformation.

A horizontal shift moves a graph left or right in relation to the y-axis. To implement a horizontal shift of a function, you add or subtract a constant from the input variable, x. If you subtract a positive number (like -5), the graph will shift to the right; if you add (like +5), the graph shifts to the left.

For instance, consider the function f(x) = x^2. To shift this parabola 5 units to the left, you replace every x with (x + 5), yielding the new function g_1(x) = (x + 5)^2. It's crucial to adjust the input variable x, not the entire function, to avoid changing the graph's shape or direction.

Conversely, a vertical shift moves a graph up or down along the y-axis. This is done by adding or subtracting a constant to the entire function. If you add a positive constant, the graph shifts upward; subtracting moves it downward.

After horizontally shifting our example function to g_1(x) = (x + 5)^2, a vertical shift upwards by 4 units means you add 4 to g_1(x), resulting in the final transformed function g(x) = g_1(x) + 4 = (x + 5)^2 + 4.

A quadratic function is a type of polynomial that has a degree of 2, which forms a U-shaped curve known as a parabola when graphed. The standard form of a quadratic function is f(x) = ax^2 + bx + c.

The graph's vertex represents the maximum or minimum point, and the a, b, and c coefficients control the opening direction, vertical position, width, and horizontal position of the parabola. Function transformations apply to quadratic functions just as they do to linear functions, but the resulting graph is a parabola rather than a straight line. For example, the original function f(x) = x^2 + 2 is a simple quadratic function with its vertex at (0, 2). Applying the described horizontal and vertical shifts will move this vertex to the point (-5, 6), as can be seen in the transformed function g(x) = (x+5)^2 + 6.