Pre-Calculus 12 Notes

By: Gen L.

In partnership with Hyperion University, 2024

Lesson 1: Translations

Vertical Translations

A Vertical Translation is basically sliding a graph up and down.

Example:

  • Consider the function . It has this table of values:
x -2 -1 0 1 2
y 4 1 0 1 4
  • Desmos: Graph

Example: Vertical Translation I

  • Consider the function . How does its table and graph change?
  • Well, we can rewrite it as , and that gives us a hint.
x -2 -1 0 1 2
y 1 -2 -3 -2 1
  • Congrats, we have transformed a graph!

Vertical Translation II

  • Consider the function . How does its table and graph change?
  • We know that shifts a graph down, must shift it up.
x -2 -1 0 1 2
y 5 2 1 2 5
  • Congrats, you can shift a graph vertically.

Review:

  • A vertical translation shifts a graph up and down.
  • How we write a vertical translation:

or

Horizontal Translations

A Horizontal Translation is basically sliding a graph left and right.

Example:

  • Consider the function . It has this table of values:
x -2 -1 0 1 2
y 2 1 0 1 2
  • Desmos: Graph

Example: Horizontal Translation I

  • Consider the function . How does its table and graph change?
x -2 -1 0 1 2
y 4 3 2 1 0
  • Hmm... It shifts it right 2.

Example: Horizontal Translation II

  • Consider the function . How does its table and graph change?
x -2 -1 0 1 2
y 1 0 1 2 3
  • Oh! It's opposite if its next to the .

Review:

  • A Horizontal Translation shifts a graph left and right.
  • How we write a horizontal translation:

Translations:

  • Now, we have a way to write any translation as a function.

or

  • Translations DO NOT change the shape of your graph, just the location of the points.

Practice Question:

  • Graph
  • For , pick any function.

Image Points and Mapping Notation

  • Image Points:
    • The new points created after a transformation.
  • Mapping Notation:
    • A way to write out how tranformations map a function to its image.

Example

What would the following functions look like if they were translated left 4 and up 6?

Going Backwards

If , what is our function if the vertex is translated 2 right and 3 up?

  • Starting with , we can remember our transformation formula: or
  • Now, we know that , so we just plug them in.
  • , rewrite in terms of the function:

Note:

  • You may think this is very easy (it is).
  • But trust me, it gets harder from here.

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