Question

\(\sf \Large \hspace{50 pt} \text{Transformations of Functions}\\ \sf \hspace{100 pt} \text{by Data_LG2}\)

In a function, y=f(x), the graph can be adjusted with different transformations that can produce different new equations (new graphs). The comments below summarize all the information you need to know when learning different transformation of functions.

Source: My previous notes that I'm about to throw after I finished this post and I think I get the basic guidelines from MathisFun.com

Answer 1

Thank you for sharing with us (-:

Answer 2

\(\sf \large \text{The General Term:}\\ \hspace{85 pt} \LARGE y= \color{blue}{a}f(\color{red}{k}(x-\color{green}{d}))+\color{orange}{c}\)

\(\sf \large where\\ \color{blue}{a}\)is vertical stretch/compression

|\(\sf \color{blue}{a}\)| > 1 stretches
|\(\sf \color{blue}{a}\)| < 1 compresses
\(\sf \color{blue}{a}\) < 0 vertical reflection (down or up)

\(\sf \color{red}{k}\) is horizontal stretch/compression

|\(\sf \color{red}{k}\) | > 1 compresses
|\(\sf \color{red}{k}\) | < 1 stretches
\(\sf \color{red}{k}\) < 0 horizontal reflection (left or right)

\(\sf \color{green}{d}\) is horizontal shift

\(\sf \color{green}{d}\) < 0 shifts to the right
\(\sf \color{green}{d}\) > 0 shifts to the left

\(\sf \color{orange}{c}\) is vertical shift

\(\sf \color{orange}{c}\) > 0 shifts upward
\(\sf \color{orange}{c}\) < 0 shifts downward



\(\sf \text{If Equation is Given:}\)

1. Identify the parent function. Parent functions which are known as the simplest form of functions because they are close to the origin (0,0).

2. Determine the transformations using the guidelines posted above.

Examples of some common parent functions and their transformations (in general):


\(\hspace{50 pt} \begin{array}{|l|c|r|}
\hline
\sf \large Name&\sf \large Equation&\sf \large Transformations\\
\hline
\sf Quadratic&y=x^2&y=\sf \color{blue}{a}(\sf \color{red}{k}(x-\sf \color{green}{d}))^2+\sf \color{orange}{c}\\
\hline
\sf Cubic&y=x^2&y=\sf \color{blue}{a}(\sf \color{red}{k}(x-\sf \color{green}{d}))^3+\sf \color{orange}{c}\\
\hline
\sf Exponential&y=m^x&y=\sf \color{blue}{a}m^{\sf \color{red}{k}(x-\sf \color{green}{d})}+\sf \color{orange}{c}\\
\hline
\sf Rational&y=\frac{1}{x}&y=\frac{\sf \color{blue}{a}}{\sf \color{red}{k}(x-\sf \color{green}{d})}+\sf \color{orange}{c}\\
\hline
\sf Absolute&y=|x|&y=\sf \color{blue}{a}|\sf \color{red}{k}(x-\sf \color{green}{d})|+\sf \color{orange}{c}\\
\hline
\sf Radical&y=\sqrt{x}&y=\sf \color{blue}{a}\sqrt{\sf \color{red}{k}(x-\sf \color{green}{d})}+\sf \color{orange}{c}\\
\hline
\end{array}\)

Now you can easily identify the transformations by yourself ^_^

\(\sf \huge \text{Sample Questions:}\)

Determine all the transformations applied in the following functions:

1. \(\sf \Large f(x)= \color{blue}{2} ( \color{red}{4}(x + \color{green}{3}))^3+ \color{orange}{5}\)

\(\sf \text{Parent Function:}\) Since the highest exponent of the variable 'x' is 1, the parent function will be the cubic function.

\(\sf \text{Transformation:}\) From the guidelines stated above, you'll see that the function will have the following changes:

\(\bf \hspace{25 pt } \text{Vertical Transformations} \)
• Vertical Stretch by a factor of \(\color{blue}{2}\), since \(\color{blue}{2}\) is greater than 1.
• No vertical reflection since \(\color{blue}{2}\) > 0
• Vertical shift 5 units up, since \(\color{orange}{5}\) > 0

\(\bf \hspace{25 pt} \text{Horizontal Transformations}\)
• Horizontal compression by a factor of \(\color{red}{4}\), since \(\color{red}{4}\) > 1
• No horizontal reflection since \(\color{red}{4}\) > 0
• Horizontal shift 3 units to the left, since \(\color{green}{-3}\) < 0

***NOTE: Why -3 and not 3 for the horizontal shift?
Let's look at the general formula again: \(\sf y= \color{blue}{a}f(\color{red}{k} \color{#999900}{(x-\color{green}{d})})+\color{orange}{c}\). It is \(\sf "\color{#999900}{(x \color{magenta}{-}\color{green}{d})}" \) , which means that the value of d should be negative to achieve the given equation. Here: \(\sf \color{#999900}{(x \color{magenta}{-}\color{green}{(-3)})} = \color{#999900}{(x \color{magenta}{+}\color{green}{3})} \)


2. \(\sf \Large f(x)= \color{blue}{-3} ( \color{red}{6}(x - \color{green}{16}))^2+ \color{orange}{12}\)

\(\sf \text{Parent Function:}\) Since the highest exponent of the variable 'x' is 2, the parent function will be the quadratic function.

\(\sf \text{Transformation:}\) From the guidelines stated above, you'll see that the function will have the following changes:

\(\bf \hspace{25 pt } \text{Vertical Transformations} \)
• Vertical Stretch by a factor of \(\color{blue}{3}\), since \(\color{blue}{|-3|}\) is greater than 1.
• There will be a Vertical reflection since \(\color{blue}{-3}\) > 0. Therefore the graph will open downwards.
• Vertical shift 12 units up, since \(\color{orange}{12}\) > 0

\(\bf \hspace{25 pt} \text{Horizontal Transformations}\)
• Horizontal compression by a factor of \(\color{red}{4}\), since \(\color{red}{6}\) > 1
• No horizontal reflection since \(\color{red}{6}\) > 0
• Horizontal shift 16 units to the right, since \(\color{green}{16}\) < 0


\(\sf \large \text{Additional Information}\)

Sinusoidal Functions' Transformations also determine other physical characteristics of these functions as follows:

•\(\sf \text{Amplitude}\) of the function's graph is given by the value of \(\sf \color{blue}{a}\).

• \(\sf \text{Period}\) of the function's graph is given by the value of \(\large \sf \frac{2 \pi}{\color{red}{k}}\) or \(\large \sf \frac{360}{\color{red}{k}}\)

•\(\sf \text{Maximum Value}\) of the function's graph is given by the value of \(\sf \color{orange}{c} + \color{blue}{a}\)

• \(\sf \text{Minimum Value}\) of the function's graph is given by the value of \(\sf \color{orange}{c} - \color{blue}{a}\)

•\(\sf \text{Equation of axis}\) of the function's graph is given by the value of \(\sf \color{orange}{c}\)


***Sinusoidal Function is a periodic function that is formed by the changes applied to \(\sf y=sin(x)\) or the sine function.

Answer 3

[Done with the editing, thanks for the corrections] ^_^

Answer 4

All the joy and fun....

Answer 5

oopsie! sorry! someone got confused with it, so i decided to follow what you said before--delete the previous ones and post the right one. Thank you e.mc for checking it:)