Transformations
Dilation
A dilation is a stretching or compressing of the graph.
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For any relation (including functions) $y = f(x)$, we can:
- dilate the graph away from the x-axis (in the y direction) or
- dilate the graph away from the y-axis (in the x direction)
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Dilation in the y direction: y = af(x)
If we multiply the rule by a {eg $y = af(x)$ }
- it will cause a dilation by a factor of a from the x-axis (in the y-direction).
- Each y coordinate is multiplied by a.
If |a| > 1, the graph will be stretched away from the x-axis.
If |a| < 1, the graph will be compressed into the x-axis.
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Dilation in the x direction: y = f(nx)
If we multiply every x value within the rule by n {ie $y = f(nx)$ }
- it will cause a dilation by a factor of \frac{1}{n} from the y-axis (in the x direction).
- Each x coordinate is divided by n.
If |n| > 1, the graph will be compressed into the y-axis.
If |n| < 1, the graph will be stretched away from the y-axis.
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Reflection
A reflection creates a "mirror image" of the original graph.
We can reflect the graph across either the x-axis or the y-axis.
- later, we will reflect across the line $y = x$ (in Chapter 5: Inverse Functions)
Reflection across the x-axis: y = –f(x)
If we put a minus in front of the rule (multiply by –1) {ie $y = -f(x)$ }
- it will cause a reflection across the x-axis (in the y direction).
- The sign of each y-coordinate is reversed.
Reflection across the y-axis: y = f(–x)
If we put a minus in front of every x-value within the rule (multiply x by –1) {ie $y = f(-x)$ }
- it will cause a reflection across the y-axis (in the x direction).
- The sign of each x-coordinate is reversed.
Reflection in both axes: y = –f(–x)
The basic graph is reflected across the x-axis then the y-axis
- or vice-versa, the order doesn't matter
- Both x-coordinates and y-coordinates have their signs reversed.
Translations
A translation involves the graph sliding (or shifting) vertically or horizontally (or a mixture of both).
For a standard graph, y = f(x)
- y = f(x – h) shifts the graph to the right (positive direction) by a distance of h (opposite to the sign)
- y = f(x) + k shifts the graph up (positive direction) by a distance of k (same direction as sign)
Note that
- the value inside the rule (with the x) translates in the x direction and
- the value outside the rule translates in the y direction.
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Examples
For an activity using the above transformations, go here
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Note about translations
When we write $y = f(x – h) + k$, this causes a translation sideways by +h and up by +k.
The rule about the direction (+ or –) might appear arbitrary but there is a reason.
Notice that $y = f(x – h) + k$ can be rearranged to form
… … $(y – k) = f(x – h)$
In this form, we can see that:
- the value in the bracket with the x translates the function in the x-direction, opposite to the sign.
- the value in the bracket with the y translates the functin in the y-direction, opposite to the sign.
Now we can see that both h and k obey the same rule.
This same rule applies to graphs of many different relationships.
For example, the equation of a circle centered on the origin with a radius of r is:
… … $x^2 + y^2 = r^2$
For a circle centered at (+h, +k) and a radius of r, the equation is:
… … $(x - h)^2 + (y - k)^2 = r^2$
Again,
- the value in the bracket with the x translates the circle in the x-direction, opposite to the sign.
- the value in the bracket with the y translates the circle in the y-direction, opposite to the sign.
Rules in mathematics always have a reason. The reason isn't always obvious but there is always a reason.
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