02.12transformations

# Transformations

## Dilation

A dilation is a stretching or compressing of the graph.

For any relation (including functions) y = f(x), we can:

• dilate the graph in the y direction (away from the x-axis) or
• dilate the graph in the x direction (away from the y-axis)

### Dilation from x-axis (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 in the y-direction (from the x-axis)

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.

### Dilation from y- axis (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 1/n 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.

## 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 (in y-direction):

… … \$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 (in x-direction):

… … \$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.

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.

## Examples

Worksheet: The following graphs can be printed using the Word Document here:

Given y = f(x) looks like the following shape, sketch:

a) y = –3f(x)

b) y = f(0.5x) + 2

c) y = f(x + 3) + 1

d) y = f(0.5x + 2)

Solution

a) y = –3f(x)

b) y = f(0.5x) + 2

c) y = f(x + 3) + 1

d) y = f(0.5x + 2)

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