Here is a graph of . What transformations could you do to transform this function back to the original function ?
12.2
Activity
Card Sort: Function Moves
Your teacher will give you a set of cards that show equations and graphs.
Sort the cards into categories of your choosing. Be prepared to describe your categories.
Pause here for a class discussion.
Sort the cards into new categories in a different way. Be prepared to describe your new categories.
12.3
Activity
Same Transformation, Different Function
An original function is called . The function is transformed from using the following transformations, in this order:
Translate left 1
Stretch horizontally by a factor of 3
Reflect over the -axis
Stretch vertically by a factor of 6
Write an equation for if is:
How could you write an equation for in terms of an unknown function ?
Student Lesson Summary
We can examine the equation of a function to determine what transformations have taken it from an original function. Here is an example:
If the original function is , then we can identify the transformations from to :
Shift left 4
Horizontal stretch by a factor of , which compresses the graph
Vertical stretch by a factor of 2
Shift down 6
We can see that the stretch by a factor of and the shift by 4 must be horizontal transformations because they are grouped with the input of the function. The stretch by a factor of 2 and shift by 6 must be vertical transformations because they are affecting the output of the function.
Now let's consider a new function:
If the original function is , then we can identify the transformations from to for this pair of functions also:
Shift left 4
Horizontal stretch by a factor of
Vertical stretch by a factor of 2
Shift down 6
These are the exact same transformations as the first pair, even though the functions are very different! In fact, we can identify the transformations for an unknown original function using an equation in the same way. If is a transformation of an unknown function that has been shifted left 4, horizontally stretched by a factor of , vertically stretched by a factor of 2, and shifted down 6, we can write a general equation for :
