This Crash Course focuses on the roles of parameters a, b, h and k in functions where the rule is in standard (transformed) form. Its interactive video, key takeaways section and summary exercise will give you a quick refresher on the subject.
This Crash Course will help you understand the role of the 4 parameters used in the standard (transformed) equation of a function: parameters |a,| |b,| |h| and |k.|
To do this, you should be able to work with the rules of certain functions. For example, you should be able to transform a rule by factoring out a GCF. It helps to know what the graphs of some functions look like, such as the quadratic (2nd degree) function, or the greatest integer function. Last but not least, you should know how to work with a Cartesian plane.
Identifying the Parameters in a Rule
Here's a table illustrating the basic and standard forms of various functions:
| Basic form | Standard form |
|---|---|
|
||f(x)=x^2|| |
||f(x)=\color{#3A9A38}a(\color{#EC0000}b(x-\color{#51B6C2}h))^2+\color{#FA7921}k|| |
|
||f(x)=[x]|| |
||f(x)=\color{#3A9A38}a[\color{#EC0000}b(x-\color{#51B6C2}h)]+\color{#FA7921}k|| |
|
||f(x)=\sqrt x|| |
||f(x)=\color{#3A9A38}a\sqrt{\color{#EC0000}b(x-\color{#51B6C2}h)}+\color{#FA7921}k|| |
|
||f(x)=\cos(x)|| |
||f(x)=\color{#3A9A38}a\cos \big(\color{#EC0000}b(x-\color{#51B6C2}h)\big)+\color{#FA7921}k|| |
|
||f(x)=\vert x\vert || |
||f(x)=\color{#3A9A38}a|\color{#EC0000}b(x-\color{#51B6C2}h)|+\color{#FA7921}k|| |
The Role of Each Parameter and Their Effect on the Graph
The following table summarizes the role of each parameter:
Parameter |h| is the number to the right of the minus sign in the brackets only when the equation is in standard form. If the equation is not in standard form, factoring out the GCF |b| is necessary.
Example: |h(x)=2\vert 3x-12\vert | is not in standard form. By factoring out a GCF, we get: ||\begin{align}h(x)&=2|3x-12|\\&=2|3(x\color{#EC0000}{-}\color{#51B6C2}{4})|\\\\&\Rightarrow \color{#51B6C2}{h}=\color{#51B6C2}{4}\end{align}||
Animated Manipulation of the Parameters
In the following animation, you can select the function of your choice and modify the parameters |a,| |b,| |h| and |k.| You can see the changes that take place on each function compared to the basic one. For exponential and logarithmic functions, there is a 5th parameter (parameter |c|) that can also be modified.
From Basic to Transformed Form
The transformation that associates a point from the basic form of a function to its corresponding point from the transformed form is given by: ||(x,y)\mapsto \left(\frac{x}{\color{#EC0000}b}+\color{#51B6C2}h,\color{#3A9A38}ay+\color{#FA7921}k\right)||We can therefore take any point |(x,y)| from the basic form of the function and, by applying standard form parameters to it, obtain its associated point in the transformed function.