Graph Transformations involve modifying the appearance of a function's graph.
Common transformations include shifting the graph horizontally or vertically, stretching or compressing it, and reflecting it across the x-axis or y-axis.
These transformations help visualize how changes in the function's equation affect its graph.

Translating a graph doesn't affect its size, shape, or orientation; it simply moves the graph to a different position on the coordinate plane.
For instance, translating the graph of a parabola doesn't change its curvature; it merely shifts the location of the vertex and the path of the curve.
It is a straightforward process that involves adding or subtracting constants from the input or output of the function, resulting in a shift of the graph on the coordinate plane.

Reflections change the graph's orientation by flipping it across the specified axis or line.
This transformation can significantly alter the graph's appearance, especially in terms of which direction it opens or faces.
Unlike translations, which maintain the graph's orientation while shifting its position, reflections can create a mirror image or an upside-down version of the original graph.

When stretching or compressing graphs, the key visual changes involve the graph's dimensions along the x or y-axis.
These transformations allow for the graph to maintain its original shape and orientation but change its scale, making it possible to closely examine the behavior of the function over different intervals or to fit the graph into a specific viewing window.

Combining transformations allows for precise control over the positioning, orientation, and size of a function's graph.
By understanding how each transformation works and carefully considering the order of operations, you can achieve any desired modification to the graph.
This capability is especially valuable in graphically solving equations, modeling real-world situations, and analyzing functions in advanced mathematics and engineering contexts.

Understanding inverse functions is crucial in various mathematical contexts, from solving equations to modeling real-world scenarios where two variables are inversely related.
Remember, not all functions have inverses that are also functions, primarily due to the requirement for a function to be bijective.
The process of finding the inverse involves algebraic manipulation and a clear understanding of the relationship between a function and its inverse.