Graph transformations involve altering the appearance of a function's graph to create a new version. These transformations can be categorized into three main types: translations, reflections, and stretches.
Firstly, let's examine translations. This transformation shifts the function to a different position on the graph. To describe a translation, a vector in the form (a, b) is used, with a representing the horizontal translation and b representing the vertical translation. For example, a translation of (-5, 3) would indicate a movement of 5 units to the left and 3 units upward on the graph.
To better understand translations, think of coordinates on a graph. Just like a negative x coordinate would be on the left side, a negative a value in the vector shifts the function to the left. Similarly, a positive y coordinate would be on the top part, and a positive b value in the vector moves the function upwards.
Next, let's explore stretches. This transformation can be horizontal or vertical, causing the graph to either enlarge or shrink in that direction. For instance, a stretch of 4h means each x-coordinate is multiplied by 4, while the y-coordinate remains the same. This can also be written as f(x) = 4hf(x), where f(x) is the original function and hf(x) is the stretched function.
Reflections are the third type of transformation, where the entire function is inverted in a line of reflection. This means the graph is flipped over a designated line. Reflections can be either horizontal or vertical and can also be expressed as a function.
Now, let's see how these transformations can be combined. When using a combination of graphical transformations, it is crucial to follow the order of operations for graph transformations, which is stretch -> reflection -> translation. This means the function is first stretched, then reflected, and finally translated.
The key takeaways from graph transformations are the three main types: stretches, reflections, and translations. Translations are described using a vector in the form (a, b), where a represents the horizontal translation and b represents the vertical translation. Stretches involve changing the size of the graph, while reflections involve flipping the graph over a line of reflection. The correct order for combining transformations is stretch -> reflection -> translation.
Let's now address some common questions about graph transformations:
