Vocabulary words: **4**

This section is all about **transformations**, which are ways of taking the points (or solutions) of a graph and mapping them to new ones. For now, we’re just going to focus on transforming a function by adding or multiplying by a number, and also by negating it.

**Translation** is just a fancy way of saying ‘move’. In order to move a function, you need to add or subtract a value to it. Say you have a function $f(x)$ that you want to translate. Adding (or subtracting) a value to the whole function would move it vertically, so $f(x)+k$. Keep in mind that $f(x) = y$, so you are modifying the y value when you do that.

So, if I wanted to move the function $f(x) = x^2$ up two, I would just add 2, so it becomes $f(x) = x^2 + 2$.

Horizontal movement requires that we modify the $x$ value. If we want to move our parabola to the right 2 units, that means instead of $(0,0)$ being the vertex, it would now be at $(2,0)$. So, we need our 2 to behave like a 0, and to do that we’re going to subtract.

The short version of all this is that to move a function horizontally, you need $f(x-h)$, where n is the number of units. Keep in mind that it’s subtraction. Adding would move the function left.

**Reflecting** a function is when it’s moved across a line. We’ll just be looking at flipping over the $x$- and $y$-axes, but it can be done with other lines.

Reflecting over the $x$-axis, or a vertical reflection, means you are again modifying the $y$ values of a function. A reflection over the $x$-axis means $f(x)$ becomes $-f(x)$. Notice how we are modifying the whole function. $y = x^3$ becomes $y = -x^3$.

With horizontal reflections, meaning over the $y$-axis, we want to only modify the $x$ values. So, $f(x)$ becomes $f(-x)$. $y = x^3$ becomes $y= (-x)^3$.

Sometimes called stretches and compressions, **dilation** is when the distance a point is from either the $x$- or $y$ axis is scaled. This involves multiplying your functions, or just the $x$ component. So, $2f(x)$ would double the $y$ values, stretching it vertically. Though, $f(2x)$ would actually compress the function horizontally. This is similar to the horizontal translation earlier. The effect is opposite when shooting for horizontal transformations.

Transformation combination on Desmos

- $a$ is a vertical stretch/shrink*
- $b$ is for horizontal stretch/shrink*
- $h$ is horizontal shift (note the negative)
- $k$ is vertical shift

* For vertical stretch/shrinks, if $a>1$ then it is a stretch. If $0<a<1$ then it is a shrink. The opposite is true for horizontal. If it’s a negative, then it is also a reflection in the same direction.