Algebra Transformations

The Joy of a Teacher is the Success of his Students. - Samuel Dominic Chukwuemeka


Welcome to Algebra Transformations

Samdom For Peace I greet you this day,
First: read the notes. Second: view the videos. Third: solve the questions/solved examples. Fourth: check your solutions with my thoroughly-explained solutions. Fifth: check your answers with the calculators as applicable.
Comments, ideas, areas of improvement, questions, and constructive criticisms are welcome. You may contact me.
If you are my student, please do not contact me here. Contact me via the school's system. Thank you for visiting!!!

Samuel Dominic Chukwuemeka (Samdom For Peace) B.Eng., A.A.T, M.Ed., M.S

Algebra Transformations


Objectives

Students will:

(1.) List the toolbox functions.

(2.) Describe the concept of the transformation of functions.

(3.) Describe the transformations done to a parent function to give the child function.

(4.) Calculate the transformed coordinate of a parent function on the child function.

(5.) Discuss some applications of the transformations of functions.


The Function Family or Parent Functions or Toolbox Functions

Biology: Parents give birth to children

just as in

Mathematics: Parent functions "give birth" to child functions.
Well, actually, parent functions are transformed to give child functions.
In other words, the transformation of parent functions lead to child functions.

Teacher: What do you usually do to graph any function?
Student: It depends on the function.
We can graph Linear Functions using Table of Values.
We can also graph Linear Functions using the Intercepts - the $x-intercept$ and the $y-intercept$
Teacher: You answered well.
What about Quadratic Functions?
Student: We can graph Quadratic Functions using Table of Values.
We can also graph Quadratic Functions using the Vertex and the Intercepts (the $x-intercept$ and the $y-intercept$)
Teacher: Very good answer.
What about Cubic Functions? Absolute Value Functions?
Student: We can graph the functions using the Table of Values.
Teacher: Very good!
Why are we learning this topic?



Why Are We Learning The Transformation of Functions?
What Does Transformation of Functions Mean?

Rather than using the Table of Values to graph each child function, we can graph only the parent function using the Table of Values.
Then, we just transform the parent function to give the child function.

Biology: The husband and wife do several positions before the "man scores a goal/goals into the woman". ☺☺☺
In other words, the husband and wife do several transformations for the wife to be pregnant and give birth to child/children.

Mathematics: There are several transformations done to the parent function in order to give birth to the child function.
The parent function can move up and down - Vertical Shift
The parent function can move left and right - Horizontal Shift
The parent function can turn over vertically - Vertical Reflection - Reflection across the $x-axis$
The parent function can turn over horizontally - Horizontal Reflection - Reflection across the $y-axis$
The parent function can be stretched vertically - Vertical Stretch
The parent function can be stretched horizontally - Horizontal Stretch
The parent function can be compressed vertically - Vertical Compression
The parent function can be compressed horizontally - Horizontal Compression

So, we can just transform the parent functions to give the child functions.
We can use Table of Values to graph the parent function.
Then, we can use any of those transformations on the parent function to give us the child function.
We can also use a combination of transformations (Transformation Combo) to give child functions.

Food and Nutrition: (Burger King, McDonalds, Jacks, Wendy's): Combo of cheeseburger, fries, and drink

just as in

Mathematics: Combination(Combo) of transformations to give child functions.
Combination of transformations is when we use more than one transformation to get the child function.


Teacher: What happens when we have several operations in an arithmetic or algebraic operations?
Student: We use the Order of Operations
Teacher: In that sense, what happens when we have a child function that was got from several (more than one) transformation?
Student: I guess we should use the Order of Transformations
Teacher: That is correct!!!
Student: So, what is the order of transformations when you have more than one transformation?
Teacher: We shall get to that.
However, just know this: the horizontal transformations have preeminence over the vertical transformations.
Student: Why is that?
Teacher: What do you think?
Horizontal is inside
Vertical is outside
Do you start your journey from "inside" and work your way "outside" OR do you start from "outside" and work your way "inside"?
Student: You begin from "inside" to "outside".
Teacher: Correct!
This reminds me of a popular African Proverb
Student: What is it?
Teacher: It states that Charity begins at home
Do you want another reason?
Student: Sure...
When the husband and wife are alone at night in the room, which position is preeminent - horizonal or vertical?
Student: I do not know
Teacher: That's okay. Just know that when you have any child function that is formed as a result of a combination of transformations, the horizontal transformations should be done before the vertical transformations.


The Parent Functions

For now, we shall focus on these parent functions.
The parent functions are:

(1.) Identity Function or Linear Function: $y = x$

(2.) Quadratic Function or Squaring Function: $y = x^2$

(3.) Cubic Function of Cubing Function: $y = x^3$

(4.) Positive Square root Function: $y = \sqrt{x}$

(5.) Cube Root Function: $y = \sqrt[3]{x}$

(6.) Absolute Value Function: $y = |x|$

(7.) Reciprocal Function: $y = \dfrac{1}{x}$


Later, we shall discuss these parent functions:

(9.) Exponential Function: $y = a^x$ and $y = e^x$

(10.) Logarithmic Function: $y = \log_a{x}$ and $y = \log_e{x}$

(11.) Trigonometric Function: $y = \sin x$ and $y = \cos x$

SAMDOM FOR PEACE Pneumonic for the Transformation of Functions

HOSH - Horizontal Shift

HORE - Horizontal Reflection

HOST - Horizontal Stretch

HOCO - Horizontal Compression

VECO - Vertical Compression

VEST - Vertical Stretch

VERE - Vertical Reflection

VESH - Vertical Shift


Notable Notes for the Transformation of Functions

Solved Examples


(1.) ACT A point at $(-5, 7)$ in the standard $(x, y)$ coordinate plane is translated right $7$ coordinate units and down $5$ coordinate units.
What are the coordinates of the point after the translation?


$ Point: (-5, 7) \\[3ex] x = -5, y = 7 \\[3ex] HOSH\:\: 7 \:\:units\:\: right: x \:\:changes\:\: \\[3ex] -5 + 7 = 2 \\[3ex] (-5, 7) \rightarrow (2, 7) \\[3ex] x = 2, y = 7 \\[3ex] VESH\:\: 5 \:\:units\:\: down: y \:\:changes\:\: \\[3ex] 7 - 5 = 2 \\[3ex] (2, 7) \rightarrow (2, 2) $

(2.) ACT In the standard $(x, y)$ coordinate plane, $A'$ is the image resulting from the reflection of the point $A(2, -3)$ across the $y-axis$.
What are the coordinates of $A'$?


Reflection across the y-axis is Horizontal Reflection (HORE)
Only $x-value$ change
$y-value$ does not change

$ Point: (2, -3) \\[3ex] x = 2, y = -3 \\[3ex] HORE \\[3ex] (2, -3) \rightarrow (-2, -3) $

(3.) ACT In the standard $(x, y)$ coordinate plane, the coordinates of the $y-intercept$ of the graph of the function $y = f(x)$ are $(0, -2)$.
What are the coordinates of the $y-intercept$ of the graph of the function $y = f(x) - 3$?


$ Point: (0, -2) \\[3ex] x = 0, y = -2 \\[3ex] f(x) - 3 = VESH\:\: 3\:\: units\:\: down;\:\: y\:\: changes \\[3ex] -2 - 3 = -5 \\[3ex] (0, -2) \rightarrow (0, -5) $