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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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