transformations transformations of functions and graphs we will be looking at simple functions and...

TransformationsTransformationsTransformations

Transformations

Transformations of Functions and Graphs

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

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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

As you can see, a number added or

subtracted from a function will

cause a vertical shift or

translation in the function.

2xxf

y = f(x) + k

y = f(x) ̶ k

Transformation

k units

Down k units

UP

VERTICAL TRANSLATIONS

)(xg

)(xh

xf

3xf

3xf

what is the transformation?

y = f(x) +10

Parent function

y = f(x) Up 10 units

Down 9 units

Up 5 units

Down 7 unitsy = f(x)

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).

y = f(x + h)

y = f(x ̶ h)

Transformation

h units

Right h units

left

For h>0, and

HORIZONTAL TRANSLATIONS

x

y

xf

)3( xfshift right 3

1xf

)(xh

)(xgshift left 1

what is the transformation?

y = f(x+10)

Parent function

y = f(x) Left 10 units

Right 9 units

Left 5 units

Right 7 unitsy = f(x)

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

what is the transformation?

y = f(x+1)-6

Parent function

y = f(x) Left 1 and down 6

Right 3 and up 2

Left 5 and up 7

Right 8 and down 1 y = f(x-8)-1 y = f(x)

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)

x

y

x

y

x

y

xxf

xf2 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.

The bigger a is. The narrower

the graph is.

vertically

stretched by a factor of a.

x2

Above is the graph of xxf

So the graph a f(x), where a

is 0 < a < 1, is the graph of f(x)

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?

The smaller a is.

The wider the graph is.

verticallyCompressed

by a factor of a.

y = a f(x) Transformation

Stretched Vertically

By factor of a a>1

0<a<1 Compressed

Vertically

By factor of a

VERTICAL TRANSLATIONS

)(2 xf

xf

xf4/1

xf2/1

3. Horizontal translation

Procedure: Multiple Transformations (From left to right)

2. Stretching or shrinking

1. Reflecting

4. Vertical translation

what is the transformation?

y = 5f(x+10)-6

vertically stretched by factor of 5, Left 10, down 6

vertically compressed by factor of ¼ , Right 7, up 2

what is the transformation?

vertically compressed by factor of 1/5, Left 6, down 7

vertically stretched by factor of ¼ , Right 9, up 2

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

xf

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

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 xf

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)

3x

Summary of Transformations So Far

khxfa h >0

vertically stretched by a factor of a

Up k units

vertically compressed by a factor of a

reflected across y-axis

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

-f (x)

If a > 1,

If 0 < a < 1,

reflected across x-axis

f(-x)

(opposite sign of number with the x)

Left h units

h <0 Right h units

k>0

k<0 Down k units

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