Math 30
Math 30 Transformations
Function [pic]
Replacement Transformation Image Point
[pic] if [pic] [pic]
[pic] if [pic] [pic]
[pic]
Replacement Transformation Image Point
[pic] if [pic] [pic]
[pic] if [pic] [pic]
[pic]
Replacement Transformation Image Point
[pic] vertical stretch about the x-axis [pic]
by a factor of [pic]
Replacement Transformation Image Point
[pic] horizontal stretch about the [pic]
y-axis by a factor of [pic]
Replacement Transformation Image Point
[pic] reflection in the x-axis [pic]
(vertically)
Replacement Transformation Image Point
[pic] reflection in the y-axis [pic]
(horizontally)
Replacement Transformation Image Point
[pic] reflection in the line [pic] [pic]
Transformations
[pic] [pic]
Function has moved h units left and k units up.
Image point [pic]
[pic] [pic]
Function has been stretched vertically by a factor of 2
Image point [pic]
[pic] [pic]
Function has been stretched vertically by a factor of [pic], reflected about the x-axis, stretched horizontally by a factor of 3, horizontally translated 3 units left and vertically translated 2 units down.
Image point [pic]
[pic] Transformation Replacement Image Point
[pic] [pic] [pic]
[pic] [pic] [pic]
[pic] [pic] [pic]
[pic] [pic] [pic]
[pic] [pic] [pic]
[pic] [pic] [pic]
[pic] [pic] [pic]
Domain and Range
Recall the definitions for domain and range:
Domain: the set of x-values represented by the graph or equation of the function.
Range: the set of y-values represented by the graph or the equation of the function
The following table shows how transformations may affect the domain and range of a function.
|Transformation |Image |Domain |Range |
|[pic] |[pic] |No |Yes |
|[pic] |[pic] |Yes |No |
|[pic] |[pic] |No |Yes |
|[pic] |[pic] |Yes |No |
|[pic] |[pic] |No |Yes |
|[pic] |[pic] |Yes |No |
The graph has a domain of [pic]
and range of [pic]. State the new
domain and range of the function
[pic].
The function is stretched vertically
by a factor of [pic] and translated horizontally 2 units left.
So, [pic]
New domain :[pic], new range: [pic]
Intercepts
Recall the definition of the x-intercept and y-intercept.
x-intercept: the x-coordinate where the graph of a line or a function intersects the x-axis. [pic]
y-intercept: the y-coordinate where the graph of a line or a function intersects the y-axis. [pic]
To determine the intercepts algebraically:
x-intercept: set [pic] y-intercept: set [pic]
Graphs of Transformations
[pic] The graph of [pic]is shown below:
Key Image Points:
[pic]
Sketch the graph of :
[pic] [pic]
translated horizontally or shifted vertically stretched by a factor of
2 units left and translated 2 and reflected about the y-
vertically or shifted 3 units axis (i.e. horizontally)
down
Image points changes:
[pic] [pic]
Given the graph, determine the equation of [pic].
[pic] the graph of [pic], where [pic]is a quadratic function is shown below. The y-intercept of [pic] is [pic]and the maximum point of [pic]. What is the equation of [pic]?
[pic]
Solution:
The reciprocal of [pic]is 9, so the y-intercept will be 9. The parabola will have a minimum point at [pic]which is the vertex. This illustrates that there is a horizontal shift of 2 and a vertical shift of 1; therefore the equation of this function is [pic] or [pic]. To find the value of a, substitute [pic] the new y-intercept into the equation. The value of [pic], so the final equation is [pic]
-----------------------
Vertical Translations [pic]
Horizontal Translations [pic]
Vertical Stretches [pic]
Horizontal Stretches [pic]
Vertical Reflections [pic]
Horizontal Reflections [pic]
Inverse Functions [pic]
Remember to factor out b and move the parameters associated with y back to the left side.
Remember the ordering of transformations:
1. Stretches/Reflections
2. Translations
For stretches, the replacement and the factor are reciprocals of each other.
Remember the notation for inverse [pic]
[pic]
[pic]
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