Elementary Functions Function Transformations Part 1 ... · linear functions, quadratic and cubic functions, general polynomial and rational functions, exponential and logarithmic

Elementary FunctionsPart 1, Functions

Lecture 1.3a, Transformations of Functions: Shifts

Dr. Ken W. Smith

Sam Houston State University

2013

Smith (SHSU) Elementary Functions 2013 1 / 35

Function Transformations

In this course we learn to identify a variety of functions:

linear functions,

quadratic and cubic functions,

general polynomial

and rational functions,

exponential

and logarithmic functions,

trigonometric functions

and inverse trig functions.

Many of these functions can be identified by their “shape”.We will identify some basic functions and then learn transformations of thefunctions that give the same shape.


Function shape

For example, the graphs of the functions

f(x) = x2 and f(x) = 3(x− 5)2 + 7

are the same shape.

If one plots them on the x-interval [−1000, 1000] one gets the followingpictures.

The graph of y = x2 is on the left; the graph of y = 3(x− 5)2 + 7 on theright. Only the labels on the y-axis have changed!


Four types of transformations

There are four types of transformations we will study in this section.

In the first two types, we simply shift the graph by a fixed amount, eithervertically or horizontally.

In the last two types of transformations, we expand/shrink the graph by afixed ratio, either vertically or horizontally.

In this presentation, we concentrate on shifting (translating) a functionvertically (up or down) and horizontally (to the left or right.)


Vertical shifts

It is easy to shift the graph y = f(x) up (vertically) by a fixed positiveamount c.

Just add c to the y-value, that is, create the graph of y = f(x) + c.

If we can shift up by a fixed amount then shifting down is also easy – justmake c negative.

If c is negative then the graph of y = f(x) + c shifts the graph down by|c|. (For example, y = f(x)− 2 will shift the graph down by 2.)

Consider the graph of y = x2.The graph of y = x2 + 1 shifts the graph of y = x2 up one unit.The graph of y = x2 + 3 shifts the graph of y = x2 up three units.The graph of y = x2 − 2 shifts the graph down by two units.


Vertical shifts

Let’s graph these all on one plane to show the effect of the shifting.

Here are graphs of y = x2, y = x2 + 1,y = x2 + 3, y = x2 − 2,Smith (SHSU) Elementary Functions 2013 6 / 35

Horizontal shifts

Horizontal shifts are very similar, but there is a subtlety here.

Because the horizontal x-axis represents inputs to the function, if we wantto shift the curve to the right (in the positive x-direction) by a positiveamount c then we need to “prepare” the input by subtracting the amountc from x before it is inserted into the function.

This may be the opposite of what one expects, but by subtracting c fromx, we make an input x− c on the left of x act like the input x and thisshift, moving x− c to x is a shift to the right.

In the next slide we graph y = x2, y = (x− 1)2 and y = (x− 3)2.


Horizontal shifts

The graph of y = x2


Horizontal shifts

The graph of y = x2 and y = (x− 1)2


Horizontal shifts

The graph of y = x2 and y = (x− 1)2 and y = (x− 3)2


Horizontal shifts

The graph of y = x2 and y = (x− 1)2 and y = (x− 3)2

y = (x− 1)2 shifts the parabola 1 to the right; y = (x− 3)2 shifts it 3 tothe right.


Horizontal shifts

The graph of y = x2


Horizontal shifts

The graph of y = x2 and y = (x+ 2)2


Horizontal shifts

The graph of y = x2 and y = (x+ 2)2

y = (x+ 2)2 shifts the parabola 2 to the left.Smith (SHSU) Elementary Functions 2013 14 / 35

Horizontal shifts

Here they are all together:

y = x2, y = (x− 1)2 y = (x− 3)2 y = (x+ 2)2Smith (SHSU) Elementary Functions 2013 15 / 35

Expansions & shrinks

In this presentation we examined shifts of functions, vertically andhorizontally.

In the next presentation, we will examine expansions & contractions offunctions

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Elementary FunctionsPart 1, Functions

Lecture 1.3b, Transformations of Functions: Expansions/contractions

Dr. Ken W. Smith

Sam Houston State University

2013


Four types of function transformations

There are four basic types of transformations of functions.

In the first two types, we simply shift the graph by a fixed amount, eithervertically or horizontally.

In the last two types of transformations, we expand/shrink the graph by afixed ratio, either vertically or horizontally.

In the previous presentation (1.3a) we looked at translations.

A vertical shift by c occurred when we simply replaced y = f(x) byy = f(x) + c.

A horizontal shift by c occurred when we replaced y = f(x) byy = f(x− c). (Note the subtraction of c!)

In this presentation we look at expansions & contractions.


Vertical expansions & shrinks

What if we want to expand or shrink the image of our graph? We can dothis in the vertical (y-direction) simply by multiplying our function by aconstant. For example, if we have the graph y = f(x) then the graph ofy = 3f(x) will stretch (expand) the graph by a factor of 3 in they-direction.

The graph of y =1

3f(x) will contract (shrink) the graph by a factor of 3.

Multiplying f(x) by −1 will flip the graph over, reflecting it across thex-axis, replacing positive y-values by negative ones and conversely,replacing negative y-values by positive ones. This is our first example of areflection.

The graph of y = −f(x) is a reflection of y = f(x) across the x-axis.


Horizontal expansions & shrinks

We can also expand or contract a graph in the horizontal direction, alongthe x-axis. But, just like horizontal shifts, because the horizontal axisrepresents the input variable, the action may be the reverse of what onemight expect.To expand the graph horizontally by a factor of 2, we must divide x by 2before inserting it into the function.

On the next slide, in thick black ink, is the graph of y = x2. In lighter blue

ink is the graph of y = (x

2)2.

By dividing x by two, we stretch the graph in the horizontal direction by afactor of 2.



The graph of y = x2

Graphs of y = x2 (thick black curve), y = (x

2)2 (thin blue),



The graph of y = x2 and y = (x

2)2

Graphs of y = x2 (thick black curve), y = (x

2)2 (thin blue),

If instead we multiply the input variable x by a constant, we will contract(shrink) the graph in the horizontal direction. In the picture below, thegraph of y = x2 is again a thick black curve; the graph of y = (2x)2 is thethinner green curve and if we graph y = (5x)2 we get the curve in red,shrunk even more in the horizontal direction.If we replace x by −x, we interchange the role of positive and negativex-values and so we reflect the graph across the y-axis. This is our secondexample of a reflection.



The graph of y = x2



The graph of y = x2 and y = (2x)2



The graph of y = x2 and y = (2x)2 and y = (5x)2



The graph of y = x2 and y = (2x)2 and y = (5x)2

By multiplying x by 2 or by 5, we shrunk (contracted) the x-axis.Smith (SHSU) Elementary Functions 2013 26 / 35

Putting it all together

In summary,

1 To shift a function up by c units, replace y = f(x) by y = f(x) + c.

2 To shift a function to the right by c units, replace y = f(x) byy = f(x− c).

3 To expand a function vertically by a factor of c, replace y = f(x) byy = cf(x).

4 To expand a function horizontally by a factor of c, replace y = f(x)

by y = f(x

c).

We can combine these transformations by creating a sequence oftransformations.For example, we could translate a function in a diagonal direction,over to the right by 2 and then up by 2 by replacing f(x) by f(x− 2) + 2.Replacing x by x− 2 moves the graph 2 units to the right; then adding 2to the entire function moves the graph up two units.


Some more examples

What transformation is required to map the graph on the left (in red) tothe graph on the right (in blue)?

Solution. Shift the red graph up by 3.So, if the red graph is y = f(x) then the blue graph is y = f(x) + 3.


Some more examples


Solution. Shift the red graph to the right by 2 and up by 4.So, if the red graph is y = f(x) then we replace x by x− 2 (inside thefunction) and add 4 on the outside so that the blue graph isy = f(x− 2) + 4.


Some more examples


Solution. We reflected the graph across the x-axis.So, if the red graph is y = f(x) then the blue graph is y = −f(x).


Some more examples


Solution. Expand the red graph by a factor of 2 in the horizontaldirection.So, if the red graph is y = f(x) then the blue graph is y = f(x2 ) sincedoubling horizontally requires dividing x by 2.


Moving graphs around

Examples.

1 Consider the graphs below. The one on the left is the graph off(x) = |x|.

On the right, the original graph has been contracted horizontally by afactor of two and then shifted 2 units to the right and then up 1 unit.What function is graphed on the right?Solution. First contract the graph horizontally by a factor of 2,replacing x by 2x. Then shift the graph to the right by 2, replacing xby x− 2. Finally add 1 to the result. So the expression |x| becomes|2(x− 2)|+ 1.


Moving graphs around

Examples.

1 Consider the graphs below. The one on the left is the graph off(x) = |x|.

What function is graphed on the right?

Solution. The graph on the right is y = |2(x− 2)|+ 1.


More moving graphs...

2 What transformations (in order) must be done to the graph of

y = f(x) to create the graph of y = 2f(x− 5

3)− 7 ?

Solutions. Think about the process of inputting an x-value. Do thefollowing steps, in this order:

1 Shift right by 52 Expand horizontally by a factor of 3 (about the line x = 5)3 Expand vertically by a factor of 24 Shift down 7.


A single form for all our transformations

These four types of transformations can be combined into a single form,changing f(x) into the expression af(b(x− c)) + d where

(first) subtracting c translates everything to the right by c units,(second) multiplying by b inside the function shrinks the graph (centeredon fixes (c, 0)) in the horizontal direction by a factor of bwhile (third) multiplying by a on the outside of the function expands thegraph vertically by a factor of a.Finally (fourth) adding d to the entire piece raises the graph d units up.

We will apply these transformations throughout our precalculus course.

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Elementary Functions Function Transformations Part 1 ... · linear functions, quadratic and cubic functions, general polynomial and rational functions, exponential and logarithmic