Grade 11 Math 1 Ch-3 (Notes)
Chapter 3: Elementary Functions and Transformations
In Grade 10, students learned basic functions such as linear, quadratic, absolute value, square root, and rational functions. Chapter 3 extends these ideas by introducing more elementary functions and studying how their graphs can be transformed. (Education for Myanmar)
3.1 Elementary Functions
An elementary function is a basic function whose graph is used as a foundation for studying more complicated functions.
1. Constant Function
Function:
Characteristics
Graph is a horizontal line.
Domain: All real numbers.
Range: {1}
Graph Features
Parallel to the x-axis.
Every point has y-coordinate 1.
2. Identity Function
Function:
Characteristics
Straight line through the origin.
Gradient (slope) = 1.
Domain and Range
Domain: ℝ
Range: ℝ
Graph Features
Makes an angle of 45° with the positive x-axis.
3. Square Root Function
Function:
Characteristics
Defined only for x ≥ 0.
Domain
x ≥ 0
Range
y ≥ 0
Graph Features
Begins at the origin.
Increases gradually.
4. Quadratic Function
Function:
Characteristics
Graph is a parabola.
Opens upward.
Domain
All real numbers
Range
y ≥ 0
Graph Features
Vertex at (0,0)
Symmetric about the y-axis.
5. Absolute Value Function
Function:
Definition
Characteristics
V-shaped graph.
Vertex at origin.
Domain
All real numbers
Range
y ≥ 0
6. Rational Function
Function:
Characteristics
Undefined at x = 0.
Domain
x ǂ 0.
Range
y ǂ 0.
Graph Features
Two branches.
Symmetric about the origin.
x-axis and y-axis are asymptotes.
7. Cubic Function
Function:
Characteristics
Passes through origin.
Increasing function.
Domain
All real numbers
Range
All real numbers
Graph Features
Symmetric about the origin.
8. Cube Root Function
Function:
Characteristics
Inverse of cubic function.
Domain
All real numbers
Range
All real numbers
Graph Features
Passes through origin.
Increasing function.
Summary of Elementary Functions
| Function | Equation | Shape |
|---|---|---|
| Constant | y = 1 | Horizontal line |
| Identity | y = x | Straight line |
| Square Root | y = √x | Increasing curve |
| Quadratic | y = x² | Parabola |
| Absolute Value | y = | x |
| Rational | y = 1/x | Hyperbola |
| Cubic | y = x³ | S-shaped curve |
| Cube Root | y = ∛x | Sideways S-curve |
3.2 Transformations
A transformation changes the position, orientation, or size of a graph without changing its basic shape.
Three main transformations:
Translation
Reflection
Scaling (Stretching/Compression) (Scribd)
A. Translation
Translation means moving a graph from one position to another.
1. Vertical Translation
Given:
y=f(x)
New function:
Effect
Up k units when k > 0
Down k units when k < 0
Point transformation:
(x,y)→(x,y+k)
Example
Graph of y = x² moves upward by 3 units.
2. Horizontal Translation
Given:
y=f(x)
New function:
Effect
Right h units when h > 0
Left h units when h < 0
Point transformation:
(x,y)⟶(x+h,y)
Example
y=(x-2)²
Graph moves 2 units to the right.
3. Combined Translation
New function:
Effect
Right h units
Up k units
Point transformation:
(x,y)→(x+h,y+k)
Example
y=(x-1)²+4
Moves:
1 unit right
4 units up
B. Reflection
Reflection flips a graph.
1. Reflection in the x-axis
Function:
Effect
(x,y)→(x,-y)
Example
y=-x²
Parabola opens downward.
2. Reflection in the y-axis
Function:
Effect
(x,y)→(-x,y)
Example
y=(-x)3
Graph reflected across y-axis.
3. Reflection in Both Axes
Function:
y=-f(-x)
Effect
(x,y)→(-x,-y)
C. Scaling (Stretching and Compression)
Scaling changes the size of a graph.
1. Vertical Stretch
Function:
Effect
Graph becomes taller.
Example
y=2x²
All y-values double.
2. Vertical Compression
Function:
y=af(x), 0<a<1
Effect
Graph becomes shorter.
Example
y=1/2 x²
3. Horizontal Stretch
Function:
y=f (x/a). a > 1
Effect
Graph becomes wider.
4. Horizontal Compression
Function:
y=f(ax), a>1
Effect
Graph becomes narrower.
Combination of Transformations
Several transformations may occur together.
Example:
Transformations of y = x²:
Move right 3 units.
Stretch vertically by factor 2.
Reflect in x-axis.
Move up 4 units.
Important Rules for Transformations
| Transformation | Formula |
|---|---|
| Up k units | y = f(x) + k |
| Down k units | y = f(x) − k |
| Right h units | y = f(x − h) |
| Left h units | y = f(x + h) |
| Reflect in x-axis | y = −f(x) |
| Reflect in y-axis | y = f(−x) |
| Vertical stretch | y = af(x), a > 1 |
| Vertical compression | y = af(x), 0 < a < 1 |
| Horizontal stretch | y = f(x/a) |
| Horizontal compression | y = f(ax) |
Chapter Review
Elementary Functions
Constant Function
Identity Function
Square Root Function
Quadratic Function
Absolute Value Function
Rational Function
Cubic Function
Cube Root Function
Transformations
Translation
Reflection
Scaling (Stretching and Compression)
Combination of Transformations
These transformations allow complicated graphs to be obtained from simple elementary functions and form the foundation for later studies of calculus, trigonometry, and advanced algebra. (Education for Myanmar)
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