If there is one prayer that you should

- Samuel Dominic Chukwuemeka
**pray/sing** every day and every hour, it is the
LORD's prayer (Our FATHER in Heaven prayer)

It is the **most powerful prayer**.
A **pure heart**, a **clean mind**, and a **clear conscience** is necessary for it.

For in GOD we live, and move, and have our being.

- Acts 17:28

The

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

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

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.

**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?
*

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 theTable of Values.

Then, we just transform the parent function to give the child function.

In other words, the husband and wife do several transformations for the wife to be pregnant and give birth to child/children.

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

The parent function can turn over horizontally: Horizontal Reflection: Reflection across the

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

just as in

Combination of transformations is when we use more than one transformation to get the child function.

However, just know this: the horizontal transformations have preeminence over the vertical transformations.

Do you start your journey from "inside" and work your way "outside" OR do you start from "outside" and work your way "inside"?

This reminds me of a popular

Do you want another reason?

When the husband and wife are alone at night in the room, which position is preeminent - horizonal or vertical?

**HOSH**: Horizontal Shift

**HORE**: Horizontal Reflection

**HOST**: Horizontal Stretch

**HOCO**: Horizontal Compression

**VECO**: Vertical Compression

**VEST**: Vertical Stretch

**VERE**: Vertical Reflection

**VESH**: Vertical Shift

For now, we shall focus on these parent functions.

The parent functions are:

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

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

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

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

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

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

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

(8.) Natural Exponential Function: $y = e^x$

(9.) Natural Logarithmic Function: $y = \log_e{x}$

(10.) Exponential Function: $y = a^x$ where $a \gt 1$

(11.) Exponential Function: $y = a^x$ where $0 \lt a \lt 1$

(12.) Logarithmic Function: $y = \log_a{x}$ where $a \gt 1$

(13.) Logarithmic Function: $y = \log_a{x}$ where $0 \lt a \lt 1$

(14.) Trigonometric Functions (Several of them. Please review Trigonometry)

Properties | Linear Function: $y = x$ | Quadratic Function: $y = x^2$ |
---|---|---|

Graph | ||

Domain | D = (−∞, ∞) | D = (−∞, ∞) |

Range | R = (−∞, ∞) | R = [0, ∞) |

Increasing Interval(s) | f(x) ↑ for x ∈ (−∞, ∞) |
f(x) ↑ for x ∈ (0, ∞) |

Decreasing Interval(s) | Not Applicable | f(x) ↓ for x ∈ (−∞, 0) |

Symmetry |
f(−x) = −f(x) Odd |
f(−x) = f(x) Even |

End Behavior |
As x → −∞, y → −∞ As x → ∞, y → ∞ Down/Up |
As x → −∞, y → ∞ As x → ∞, y → ∞ Up/Up |

Chukwuemeka, S.D (2016, April 30). *Samuel Chukwuemeka Tutorials - Math, Science, and Technology.*
Retrieved from https://www.samdomforpeace.com

Coburn, J., & Coffelt, J. (2014). *College Algebra Essentials* (3rd ed.).
New York: McGraw-Hill

Sullivan, M. (2020). *Precalculus.* (11th ed.). Pearson.

CrackACT. (n.d.). Retrieved from http://www.crackact.com/act-downloads/

CSEC Math Tutor. (n.d). Retrieved from https://www.csecmathtutor.com/past-papers.html

Desmos. (n.d.). Desmos Graphing Calculator. https://www.desmos.com/calculator

DLAP Website. (n.d.). Curriculum.gov.mt.
https://curriculum.gov.mt/en/Examination-Papers/Pages/list_secondary_papers.aspx

Geogebra. (2019). Graphing Calculator - GeoGebra. Geogebra.org.
https://www.geogebra.org/graphing?lang=en

GCSE Exam Past Papers: Revision World. Retrieved April 6, 2020, from
https://revisionworld.com/gcse-revision/gcse-exam-past-papers

51 Real SAT PDFs and List of 89 Real ACTs (Free) : McElroy Tutoring. (n.d.).
Mcelroytutoring.com. Retrieved December 12, 2022,
from https://mcelroytutoring.com/lower.php?url=44-official-sat-pdfs-and-82-official-act-pdf-practice-tests-free