03.3transformations

# 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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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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