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Exploring Transformations
Resources:
• Graphing Calculator
• – don’t use Internet Explorer
Goal:
• To understand how adding a constant to the value of a function affects its graph [pic]
• To understand how changing the value of x in a function affects its graph. [pic]
Compare the following graphs:
[pic] [pic] [pic] [pic] [pic] [pic]
[pic] [pic]
Describe the transformations in words: Describe the transformations in words:
A mapping is a formula that tells you how to change a set of coordinates.
-For example, the mapping [pic] would move the point [pic].
[pic]Write the mappings for the previous graphs you worked with.
[pic]If [pic] is a point on [pic], what is the mapping when the function is transformed to: [pic]?
Without using your graphing calculator sketch the graph after the following transformations have been applied and express the transformation as a mapping:
[pic] i) [pic] [pic] [pic]
[pic] [pic] [pic]
[pic] ii) [pic] [pic] [pic]
[pic] [pic] [pic]
Determine what transformation has been applied to the graph on the left to obtain the graph on the right. Answer as both a function and map.
[pic][pic] [pic][pic]
Goal: To understand how to reflect the graph of a function, [pic]
Use your graphing calculator to graph the following functions and record them on the graphs provided below:
i) [pic] vs. [pic] ii) [pic] vs.[pic]
[pic] [pic]
[pic]How does [pic] relate to the graph [pic]?
[pic]What is the mapping for [pic]
Goal: To understand how to reflect the graph of a function, [pic]
Use your graphing calculator to graph the following functions and record them on the graphs provided below:
i) [pic]vs. [pic] ii) [pic]vs. [pic]
[pic] [pic]
[pic]How does [pic] relate to the graph [pic]?
[pic]What is the mapping for [pic]
Test yourself:
[pic] vs. [pic] [pic] vs. [pic]
Goal: To understand how a vertical stretch factor effects the graph of a function, [pic]
Using your graphing calculator, sketch the following graphs:
[pic]
[pic]
[pic]
Describe the transformations in words:
Describe the transformations as a map:
[pic]How does [pic] relate to the graph [pic]? [pic]What is the mapping for [pic]
Goal: To understand how a horizontal stretch factor affects the graph of a function, [pic]
Using your graphing calculator, sketch the following graphs:
[pic]
[pic]
[pic]
Describe the transformations in words:
Describe the transformations as a map:
[pic]How does [pic] relate to the graph [pic]? [pic]What is the mapping for [pic]
Test Yourself:
[pic] [pic]
[pic] [pic]
Describe the following transformations in words and as a mapping:
[pic] [pic]
[pic]What do you think an invariant point would be? Are there any in the exercises you’ve worked on?
Is there a difference between[pic]and[pic]?
[pic]Take a look at their graph(s) and note the change from the original [pic] in words below:
[pic]Which form [pic]or[pic] makes it easier to determine the transformation?
Why do you think that is?
So far we’ve isolated the pieces of transformations…
Goal: Learn how to combine the pieces of transformations we’ve learned about so far
[pic]What do you think the mapping is for [pic]?
[pic]If you’ve been successful in finding the correct mapping, record the rules for the order in which transformations must be applied to a graph:
Describe the transformations in words, in an appropriate order and use your description to create a mapping.
[pic]
[pic]
[pic]
[pic]
Create transformation described below in function notation and give its mapping
i) If the original function is[pic], write a transformation for the function after it has been:
Reflected in the x and y axis, then expanded vertically by a factor of 2 and compressed horizontally by a factor of [pic] and finally translated 2 units to the left and 4 units down.
ii) If the original function is[pic], write a transformation for the function after it has been:
Reflected in the y axis, then expanded vertically by a factor of 5 and expanded horizontally by a factor of 2 and finally translated 6 units to the right and 3 units up.
Test Yourself:
Find the point after the transformation has been applied:
(3,-7) [pic] (5,2) [pic]
Sketch the transformation given:
i) [pic] [pic]
ii) [pic] [pic]
If [pic], what will the equation look like after it has been transformed into [pic]?
Describe the transformation shown in words, with function notation, and a mapping:
a. [pic] [pic]
b. [pic] [pic]
[pic]Graph the following functions, what do you notice about their symmetry?
[pic] [pic] [pic] [pic]
| | |
How could you explain the relationship between points on inverse graphs in English?
[pic]Formal definition:
[pic]What is the mapping of a point from [pic] to [pic]?
Graph the inverse functions:
[pic][pic]
Another notation also describes inverse functions: [pic]. Try it on Desmos… Graph the following
[pic] [pic] [pic]
[pic]
This is particularly helpful because it gives us a way to find inverse functions algebraically:
Solve algebraically for the inverse of [pic] Solve algebraically for the inverse of [pic]
*AP
If you are given a graph, what’s a quick way to determine if that graph is a function or not?
If you are given a graph, what’s a quick way to determine whether or not its inverse graph will be a function?
How could you restrict the domain into two pieces so [pic] has an inverse function?
[pic]
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