02Atransformations

toc =** 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 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. 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 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: **
**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.





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. Go to top of page: :flat

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