Transformations of Parent Graphs |
Notes
Generic Transformations of Functions
Again, the “parent functions” assume that we have the simplest form of the function; in other words, the function either goes through the origin (0, 0), or if it doesn’t go through the origin, it isn’t shifted in any way.
When a function is shifted, stretched (or compressed), or flipped in any way from its “parent function“, it is said to be transformed, and is a transformation of a function. Functions are typically transformed either vertically or horizontally.
T-charts are extremely useful tools when dealing with transformations of functions. For example, if you know that the quadratic parent function is being transformed 2 units to the right, and 1 unit down, we can create the original t-chart, following by the transformation points on the outside of the original points. Then we can plot the “outside” points to get the newly transformed functions.
When looking at the equation of the function, however, we have to be careful.
When functions are transformed on the outside of the part, you move the function up and down and do the “regular” math, as we’ll see in the examples below. These are vertical transformations or translations.
When transformations are made on the inside of the part, you move the function back and forth (but do the “opposite” math – basically since if you were to isolate the x, you’d move everything to the other side). These are horizontal transformations or translations.
When functions are transformed on the outside of the part, you move the function up and down and do the “regular” math, as we’ll see in the examples below. These are vertical transformations or translations.
When transformations are made on the inside of the part, you move the function back and forth (but do the “opposite” math – basically since if you were to isolate the x, you’d move everything to the other side). These are horizontal transformations or translations.
Vertical Transformations
Here are the rules and examples of when functions are transformed on the “outside” (notice that the y values are affected). The t-charts include the points (ordered pairs) of the original parent functions, and also the transformed or shifted points.
Notice that the first two transformations are translations, the third is a dilation, and the last is a reflection.