Graph Transformation
The book will guide you through the exploration of functional aspect and graphical aspect of transformation, limited to the scope of IB Higher Level Mathematics.
Table of Contents
- Revision
- Function Substitution
- Function Mapping
- Families of Functions
- Polynomial function:
- Reciprocal Function
- Modulus Function
- Translation
- Vertical Translation
- Horizontal Translation
- Combined Translation
- Stretching
- Vertical Stretching
- Horizontal Stretching
- Combined Transformation
- Combined Transformation (Part 1)
- Combined Transformation (Part 2)
- Reflection
- Modulus Function
- Reciprocal Function
- Square Function
- Squaring the function
Function Substitution
Revision 1:
Given that[math]f\left(x\right)=x^2[/math] , write down the expression for the following composite functions.
(a) [math]f\left(-x+1\right)[/math]
(b)[math]f\left(-\left(x+1\right)\right)[/math]
Revision 2:
Given that [math]f\left(x\right)=\frac{1}{x}[/math] , which of the following composite functions gives the expression [math]\frac{2}{3\left(x-2\right)}-2[/math]?
Polynomial function:
Polynomial functions are of the form [br][br][math]f\left(x\right)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0[/math][br]where[math]a_n,a_{n-1},...,a_0[/math] are constants, and [math]n[/math] is a positive integer[br][math]a_n\ne0[/math][br]The degree of a polynomial is[math]n[/math] , which is the highest power of [math]x[/math] in the function.[br]They can be categorized into two types: odd power function and even power function.
Example of even power polynomial function:
Quadratic function, [math]n=2[/math]
Example of Odd Power Polynomial Function:
Cubic function, [math]n=3[/math]
Vertical Translation
Given that[math]f\left(x\right)=x^2[/math] , investigate the following graphs in the applet below. [br]
(i) [math]g\left(x\right)=f\left(x\right)+2[/math][br](ii)[math]g\left(x\right)=f\left(x\right)-2[/math]
Vertical Stretching
Use the applet below to explore the behavior of [math]y=pf\left(x\right).[/math][br](i) [math]y=2f\left(x\right),[/math][br](ii)[math]y=\frac{1}{2}f\left(x\right)[/math][br](iii) [math]y=-f\left(x\right),[/math][br](iv)[math]y=-2f\left(x\right),[/math][br](v)[math]y=-\frac{1}{2}f\left(x\right).[/math]
Combined Transformation (Part 1)
Recall from revision,
Given that[math]f\left(x\right)=\frac{1}{x},[/math] , the composite functions gives the expression[br][math]g\left(x\right)=2f\left(3\left(x-2\right)\right)-2[/math] [math]=\frac{2}{3\left(x-2\right)}-2[/math][br]What is the order of translation and stretching that will help us obtain the graph of [math]g\left(x\right)[/math] from [math]f\left(x\right)[/math] ?
First let's consider what happens in the bracket of the function (which affects x only)
Which of the following order is correct?[br][br]1. [math]f\left(x\right)=\frac{1}{x}\Longrightarrow h\left(x\right)=f\left(x-2\right)=\frac{1}{x-2}\Longrightarrow[/math]Horizontal translation of 2 units to the right[br]2.[math]p\left(x\right)=h\left(3x\right)=\frac{1}{3x-2}\Longrightarrow[/math] Horizontal stretching of scale factor [math]\frac{1}{3}[/math]
OR [br][br]1.[math]f\left(x\right)=\frac{1}{x}\Longrightarrow h\left(x\right)=f\left(3x\right)=\frac{1}{3x}\Longrightarrow[/math] Horizontal stretching of scale factor[math]\frac{1}{3}[/math] [br]2.[math]p\left(x\right)=h\left(x-2\right)=\frac{1}{3\left(x-2\right)}\Longrightarrow[/math] Vertical translation of 2 units to the right
Hence,
the graph of[math]f\left(x\right)=\frac{1}{x}[/math] is given above, pls sketch the image of [math]f\left(x\right)[/math]after the two horizontal transformation steps.
Squaring the function
Take note of the smoothing at the turning points after the transformation.