Apply transformations to sketch functions of the form where is a polynomial, reciprocal, absolute value, exponential or logarithmic function and and are constants
Examine dilations and the graphs of and using technology
Given a function or a relation, we can swap and for new expressions involving and to stretch or shrink the graph in a particular direction:
Dilating any Relation
We can stretch the graph horizontally by units and vertically by units if we swap for the expressions:
This technique works for both functions and relations:
Translating a Function
Suppose we wanted to stretch a function like:
To half it’s horizontal size, but triple it’s vertical size, we would apply the swap:
So the new function would be:
We can then simplify this by moving the 3 across to the other side:
And as you can see, this has achieved the desired effect:
Translating a Relation
Suppose we had a relation like:
Then we wanted to triple its width but we want to squish it to half its vertical height. We would apply the following substitutions:
This gives us the new expression:
After this we can distribute the brackets and simplify:
We could simplify this if we like, but there’s no need to. Graphing this shows us that we did achieve the desired effect:
To understand why dividing by makes the graph bigger, we need to realize that squashing the axis is the same as stretching the axis:
Dilating an Axis or Relation
If we start with the graph of some function:
The expression implies that we are swapping the axis for a new axis that has doubled in size:
If we label the old axis and the new axis , we notice that our measuring stick has gotten twice as long, so . So any imagine attaching the graph to this new axis, and pulling the axis back to its original size, we notice that the graph has in fact gotten two times wider as shown:
So if we swap , the axis will grow by a factor of . Similarly, if we swap for the axis shrinks by a factor of . Which means that the graph grows by a factor of . So again, the main takeaway is:
Any transformation we apply to the axis causes the opposite transformation to the graph
So for the same reason, swapping will shrink the y axis by a factor of which causes the graph to stretch by a factor of .
Dilating Functions
Given a general function , we can use a shortcut to translate it:
Translating a Function
To stretch horizontally by and vertically by , use:
This can be easily applied to move a function around:
Translating a Function
Given a very simple function like:
Here, , so if we wanted to squish the function to half of its horizontal size and triple its vertical size , we would just apply the equation above:
As you can see, this had the desired effect:
This is equivalent to the substitution for a relation:
Functions Transform like Relations
If we start with the expression:
We can stretch a graph horizontally by a factor of by swapping and we can do the same vertically by a factor of by swapping , which gives us:
If we then move the to the other side:
We have the same expression we had above.
Reflecting Relations
Negative stretch factors change the direction of the axis. So if we stretch horizontally by , the axis will get three times bigger, but it will also flip along the horizontal direction as well:
Reflecting Functions
If we flip our relation horizontally
If we flip our relation vertically
Lesson Loading
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SOLUTION
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Time
The graph of usually repeats every radians as shown:
Find a new function that repeats every 1 radian instead
Find a new function by stretching that repeats at every 10 radian interval
What is the general formula if we wanted the graph to repeat after every radians?
SOLUTION
Part 1
If we wanted to repeat it every radian, we’d have to scale the graph down so that the point at maps to . To do this, we can see that:
So that means we need to make the replacement:
So the equation of our new formula is:
Part 2
Now to make the period 10 units long. We need to make the axis 10 times smaller, so we make the replacement , which gives us:
This produces the correct graph as shown.
Part 3
To change the width of the graph to any other width we like. We just replace the 10 in the previous step for , this gives us the replacement:
Which we apply to . This gives the final form:
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Reflecting And Dilating Graphs
Cambridge 2 Unit Year 12 - Chapter 2G
Question 1 (a, c, h)
Question 2
Question 3 (a, b)
Question 5 (a)
Question 6 (a)
Question 7 (b)
Question 10
Question 11
Lesson Loading
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Reflecting Functions
Review Questions
Given the function , what will the function look like ?
A reflection of around the line
A reflection of around the x axis
A reflection of around the y axis
Given the function , what will the function look like ?
A reflection of around the x and y axis
A reflection of around the x axis
It will look the same as
A reflection of around the y axis
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Reflecting Functions - Example
Review Questions
If we were asked to find , which is defined to be a reflection of about the y axis, we could do so by:
Trialing different coefficients in the function
Substituting into and simplifying the result.
Multiplying by
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Dilating Functions
Review Questions
What effect does the constant have in the following equation:
k acts a vertical scaling factor. When k > 1 it results in a squish and when 0 < k < 1 it results in a stretch.
k acts a vertical scaling factor. When k > 1 it results in a stretch and when 0 < k < 1 it results in a squish.
k acts a horizontal scaling factor. When k > 1 it results in a squish and when 0 < k < 1 it results in a stretch.
k acts a horizontal scaling factor. When k > 1 it results in a stretch and when 0 < k < 1 it results in a squish.
What effect does the constant have in the following equation:
k acts a horizontal scaling factor. When k > 1 it results in a stretch and when 0 < k < 1 it results in a squish.
k acts a vertical scaling factor. When k > 1 it results in a stretch and when 0 < k < 1 it results in a squish.
k acts a vertical scaling factor. When k > 1 it results in a squish and when 0 < k < 1 it results in a stretch.
k acts a horizontal scaling factor. When k > 1 it results in a squish and when 0 < k < 1 it results in a stretch.
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Dilating Functions - Vertical Example
Review Questions
Describe the transformation applied to the function in the following equation: .
The function has been squished to half its height
The function has been squished to half its width
The function has been stretched to double its height
The function has been stretched to double its width
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Dilating Functions - Horizontal Example
Review Questions
Which of the following options correctly describes the transformation applied to in the following equation:
The function has been squished by a factor of 3 in the y direction and stretched by a factor of 2 in the x direction
The function has been stretched by a factor of 3 in the y direction and squished by a factor of 2 in the x direction
The function has been stretched by a factor of 3 in the x direction and squished by a factor of 2 in the y direction
The function has been squished by a factor of 3 in the x direction and stretched by a factor of 2 in the y direction
