TransformationsTransformationsTransformations

Transformations

2.4: Transformations of Functions and Graphs

We will be looking at simple functions and seeing how various modifications to the functions transform them.

VERTIC

AL

TR

AN

SLA

TIO

NS

Above is the graph of 2xxf

x

y

What would f(x) + 1 look like? (This would mean taking all the function values and adding 1 to them).

x

y

11 2 xxf

What would f(x) - 3 look like? (This would mean taking all the function values and subtracting 3 from them).

x

y

33 2 xxf

2xxf As you can see, a number added or

subtracted from a function will

cause a vertical shift or

translation in the function.

VERTIC

AL

TR

AN

SLA

TIO

NS

Above is the graph of xxf What would f(x) + 2 look like?

22 xxfSo the graph

f(x) + k, where k is any real

number is the graph of f(x) but vertically

shifted by k. If k is positive it will shift up. If k is negative it will shift down

x

y

x

y

x

y

44 xxf

xxf

What would f(x) - 4 look like?

Above is the graph of 2xxf

x

y

What would f(x+2) look like? (This would mean taking all the x values and adding 2 to them before putting them in the function).

As you can see, a number added or

subtracted from the x will cause

a horizontal shift or

translation in the function but opposite way of the sign of the

number.

HORIZONTAL TRANSLATIONS

x

y

x

y 2xxf

211 xxf

222 xxf

What would f(x-1) look like? (This would mean taking all the x values and subtracting 1 from them before putting them in the function).

HORIZONTAL TRANSLATIONS

Above is the graph of 3xxf What would f(x+1) look like?

So the graph f(x-h), where h is

any real number is the graph of f(x) but horizontally

shifted by h. Notice the negative.

(If you set the stuff in parenthesis = 0 & solve it will tell you how to shift

along x axis).

x

y

x

y

x

y

311 xxf 3xxf

What would f(x-3) look like?

333 xxf

03 x

So the graph f(x-h), where h is

any real number is the graph of f(x) but horizontally

shifted by h. Notice the negative.

(If you set the stuff in parenthesis = 0 & solve it will tell you how to shift

along x axis).

3xSo shift along the x-axis by 3

shift right 3

x

y

x

y

x

y

We could have a function that is transformed or translated both vertically AND horizontally.

Above is the graph of xxf What would the graph of look like? 3)2( xxf

up

3

left 2

and

If we multiply a function by a non-zero real number it has the effect of either stretching or compressing the function because it causes the function value (the y value) to be multiplied by that number.

Let's try some functions multiplied by non-zero real numbers to see this.

DILATION:

Above is the graph of xxf

So the graph a f(x), where a

is any real number

GREATER THAN 1, is the graph of f(x) but vertically stretched or dilated by a factor of a.

x

y

x

y

x

y

xxf

xxf 22 xxf 44

What would 2f(x) look like?

What would 4f(x) look like?

Notice for any x on the graph, the new (red) graph has a y value that is 2 times as much as the original (blue) graph's y value.

Notice for any x on the graph, the new (green) graph has a y value that is 4 times as much as the original (blue) graph's y value.

Above is the graph of xxf

So the graph a f(x), where a

is 0 < a < 1, is the graph of

f(x) but vertically

compressed or dilated by a factor of a.

x

y

x

y

Notice for any x on the graph, the new (red) graph has a y value that is 1/2 as much as the original (blue) graph's y value.

x

y

Notice for any x on the graph, the new (green) graph has a y value that is 1/4 as much as the original (blue) graph's y value.

xxf4

1

4

1

What if the value of a was positive but less than 1?

xxf

xxf2

1

2

1

What would 1/4 f(x) look like?

What would 1/2 f(x) look like?

Above is the graph of xxf

So the graph - f(x) is a reflection about the

x-axis of the graph of f(x). (The new graph is obtained by "flipping“ or reflecting the

function over the x-axis)

x

y

What if the value of a was negative?

What would - f(x) look like?

x

y

xxf

xxf

Notice any x on the new (red) graph has a y value that is the negative of the original (blue) graph's y value.

x

y

Above is the graph of 3xxf

There is one last transformation we want to look at.

Notice any x on the new (red) graph has an x value that is the negative of the original (blue) graph's x value.

x

y

3xxf 3xxf

What would f(-x) look like? (This means we are going to take the negative of x before putting in the function)

So the graph f(-x) is a

reflection about the

y-axis of the graph of f(x). (The new graph is obtained by "flipping“ or reflecting the

function over the y-axis)

Summary of Transformations So Far

khxfa

horizontal translation of h (opposite sign of number with the x)

If a > 1, then vertical dilation or stretch by a factor of a

vertical translation of k

If 0 < a < 1, then vertical dilation or compression by a factor of a

f(-x) reflection about y-axis

**Do reflections and dilations BEFORE vertical and horizontal translations**

If a < 0, then reflection about the x-axis (as well as being dilated by a factor of a)

Graph using transformations 12

1

xxf

We know what the graph would look like if it wasfrom our library of functions.

x

xf1

x

y

moves up 1

moves right 2

reflects about the

x -axis

x

y

x

y

x

y

x

y

There is one more Transformation we need to know.

kb

hxfa

)(

horizontal translation of h (opposite sign of number with the x)

If a > 1, then vertical dilation or stretch by a factor of a

vertical translation of k

If 0 < a < 1, then vertical dilation or compression by a factor of a

f(-x) reflection about y-axis

Do reflections and dilations BEFORE vertical and horizontal translations

If a < 0, then reflection about the x-axis (as well as being dilated by a factor of a)

horizontal dilation by a factor of b

Acknowledgement

I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.

www.slcc.edu

Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified.