# Transformations of Functions

When we "transform" a function, we change it somehow. This might sound like an obvious conclusion, but functions can change in many different ways. In addition, transformations represent very specific changes. It makes sense to understand all of the various transformations that can take place. As we will see, even a subtle change can have a notable effect on a function when it is graphed on a Cartesian plane.

## The meaning of a transformation

A transformation is a term used to describe the manipulation of the shape or position of something on a Cartesian plane. We can transform individual points, lines, or entire shapes.

Our starting point is called the "pre-image." This is the thing that
we're planning to transform. The image we get *after* the
transformation is the 'image."

The most important thing to remember about transformation is that very specific rules apply. There are certain types of transformations, and these transformations occur in certain ways.

In order to illustrate changes in a more straightforward manner, we'll use the quadratic parent graph:

$y={x}^{2}$

## Horizontal shifts

One of the simplest transformations is called a "horizontal shift." This is what it looks like when we transform a function using a horizontal shift:

As we can see, the pre-image is in red, and this forms a parabola.

But when we replace ${\left(x\right)}^{2}$ with ${\left(x-b\right)}^{2}$ , where b is positive, the parabola shifts to the right. This is a horizontal shift.

If we use ${\left(x+b\right)}^{2}$ , where b is positive, instead, we get a shift to the left:

## Vertical shifts

If we wanted to shift our function vertically, we would need to swap $f\left(x\right)$ with $f\left(x\right)+c$ , where c is positive. This would result in our parabola moving upwards at a rate equal to c. Take a look:

And if we wanted to shift the parabola downward, we would simply swap $f\left(x\right)$ with $f\left(x\right)-c$ , where c is positive. See how the parabola shifts:

## Reflections

A reflection is a little more complicated. This involves "flipping" the figure over so that it faces the opposite direction. Let's take a look:

What do we need to do to make this happen? Simple: Replace x with -x. As we can see, the figure has been shifted across the y-axis. In other words, it has been flipped horizontally.

But what if we want to flip it in the vertical direction? In order to create this transformation, we would need to replace $f\left(x\right)$ with $-f\left(x\right)$ . This causes the figure to be reflected across the x-axis. In other words, it has been flipped vertically. Take a look:

## Horizontal stretch and compression

We can also compress our figure by replacing x with $n\times x$ . This results in horizontal compression by a factor of n. Notice how skinny the blue figure has become in comparison to the red figure (the pre-image):

But what if we wanted to stretch the figure instead of compressing it? In that case, we would have to replace x with $\frac{x}{n}$ . This would cause a horizontal stretch by a factor of n. Take a look at what happens when we do that:

## Vertical stretch and compression

Of course, we can also compress and stretch vertically. Let's start by creating a vertical stretch by replacing $f\left(x\right)$ with $n\times f\left(x\right)$ . This causes a vertical stretch by a factor of n. Take a look:

Now let's try the same concept, but with a compression instead of a stretch. We can do this by replacing $f\left(x\right)$ with $\frac{f\left(x\right)}{n}$ . This will cause a vertical compression by a factor of n. Take a look:

## Transformations and congruence

You might remember that if an object can be transformed into another shape using only turns (rotation), flips (reflection), or slides (shifts), then the two shapes are congruent.

If we need to resize an object in order to make it into another shape, then the two shapes are not congruent.

## Other types of transformations

Transformations illustrate the magic of the Cartesian plane. Before this concept, there was no clear link between algebra and geometry. But after Descartes' contribution, we can now see how even a slight alteration to a function can have a radical change in its displacement and shape on a plane.

There are a few other types of transformations you might need to memorize:

- Rotation: This is when an object rotates around a central point. This central point is often the origin, but not always.
- Shear: When you shear a pre-image, you change its shape. Its area remains unchanged, but its interior angles change. For example, a square becomes a rhombus when sheared.
- Dilation: Dilation changes the object's size. For example, a map is a heavily dilated version of the place that it represents.

You should also know that we can sort transformations into two main categories:

- Rigid Transformations: These types of transformations do not change the shape or the size of the pre-image. Examples include reflection, rotation, and translation.
- Non-Rigid Transformations: These types of transformations can change the size or shape of the pre-image. Examples include dilation and shear.

## Topics related to the Transformations of Functions

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