Friday's lesson was all about the moving of graphs; translating, stretching, and reflecting. We first reviewed the parent functions of the graphs by going over what they look like. Here are the 6 he showed us:

y=x

y=x^2

y=x^3

y=|x|

y=

y=c (assuming that c=3)

You can manipulate any type of graph using the concept of f as a function of x (or any variables other than f and x). This is represented as f(x), and for the most part, replaces y. So a function such as y=x could also be written as f(x)=x because it is the function of whatever value x is. This is called function notation. We'll first look at translating graphs using the parent function f(x)=x^2.

Translation

Vertical Translation:

If you want to move the graph f(x)=x^2 up 1 unit, you would +1 on the end of the equation,

and because f(x)=x^2, you can replace x^2 with f(x) when using function notation, the process

would look something like this

Parent function Translated Equation

f(x)=x^2 y=f(x)+1

Translated Graph

To translate to the down simply subtract instead of adding (ex. y=f(x)-1). This means that the equation for vertical translation is y=f(x)+c

To move the graph to the right or left, you would use the same principal of function notation, only instead of changing the y value (f(x)) you are going to change the x value. This is easily done by inserting the value you want to move the graph by, right next to the x. There is only one change in the format of the two equations, and that is when manipulating a graph horizontally (in this case to the left or right), the sign is the opposite of what you would think. So if you're trying to move f(x)=x^2 to the right 2 units, the equation would be y=f(x-2) NOT y=f(x+2). The resulting graph would look like this:

To move the parent function to the left 2, the equation would be y=f(x+2), meaning that the equation for moving a graph horizontally is y=f(x-c)

Stretching

To make the graph narrower, you would multiply its parent function (f(x)) by however much narrower you would want to make it. For example, if you want to make it 2 times narrower, the equation would be y=2f(x). This is because you are making the parent function larger, resulting in a graph with the same x values, but with y values that move higher up on the graph. For example, to make an equation with the parent function of f(x)=x^2 2 times narrower, the equation would look like this: y=2f(x), and would yield this graph:

(the red being the parent function of f(x)=x^2 and the blue being y=2f(x))

To make a graph wider, you would multiply its x value by the amount you want enlarge it. Same basic concept as the making it narrower, so the equation if you want to make it 2 times wider would be y=f(2x). Taking all of this information into account, the equation for making graphs narrower by increasing its parent function is y=cf(x), and the equation to make a graph wider by increasing its x value is y=f(cx).

Reflecting

Reflecting a graph is the simplest one of the three. To reflect an image on the graph is basically like you are putting a mirror up to the image and it shows up on the other side. To reflect a graph vertically (meaning flip it upside down), you simply multiply the parent function (the f(x)) by negative 1. An equation with the parent function f(x)=x^2 upside down would be y=-f(x) and its graph would look like this

To reflect it to the right or left, you would multiply its x value by negative 1, making the equation look like this: y=f(-x). This is called horizontal reflection, and the equation for vertical reflection is y=-f(x). You can manipulate graphs by translating (shifting), stretching, or reflecting them. You may see one or all in a given equation, and can all be broken down as long as you remember the basic equations. A couple reminders; vertical changes effect the f(x), or y-coordinates, and horizontal changes effect the x, or x-coordinates. And finally, when horizontally changing, the things that happen inside the parenthesis (such as f(x-c)) will have an opposite effect than what you would expect. Goodnight and good luck.