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.

02.12dilate01.gif

If |a| < 1, the graph will be compressed into the x-axis.

02.12dilate02.gif

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.

02.12dilate03.gif

If |n| < 1, the graph will be stretched away from the y-axis.

02.12dilate04.gif

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.

02.12reflect01.gif

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.

02.12reflect02.gif

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.

02.12reflect03.gif

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)

02.12translate01.gif

$y = f(x) + k$ shifts the graph up (positive direction) by a distance of k (same direction as sign)

02.12translate02.gif

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.

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.

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)

02.12agraph1.gif

Solution

a) y = –3f(x)

02.12agraph2.gif
b) y = f(0.5x) + 2
02.12agraph3.gif

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

02.12agraph4.gif

d) y = f(0.5x + 2)

02.12agraph5.gif

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