Shifting of Graphs

8/9/2019 Shifting of Graphs

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1.5 - Shifting, Reflecting, and StretchingGraphs

Definitions

Abscissa

The x-coordinateOrdinate

The y-coordinateShift

A translation in which the size and shape of a graph of a function is not changed,but the location of the graph is.

Scale

A translation in which the size and shape of the graph of a function is changed.

ReflectionA translation in which the graph of a function is mirrored about an axis.

Common Functions

Part of the beauty of mathematics is that almost everything builds upon something else,and if you can understand the foundations, then you can apply new elements to old. It isthis ability which makes comprehension of mathematics possible. If you were tomemorize every piece of mathematics presented to you without making the connectionto other parts, you will 1) become frustrated at math and 2) not really understand math.

There are some basic graphs that we have seen before. By applying translations tothese basic graphs, we are able to obtain new graphs that still have all the properties ofthe old ones. By understanding the basic graphs and the way translations apply tothem, we will recognize each new graph as a small variation in an old one, not as acompletely different graph that we have never seen before. Understanding thesetranslations will allow us to quickly recognize and sketch a new function without havingto resort to plotting points.

These are the common functions you should know the graphs of at this time:

y

Constant Function: y = cy Linear Function: y = xy Quadratic Function: y = x2y Cubic Function: y = x3y Absolute Value Function: y = |x|y Square Root Function: y = sqrt(x)y Greatest Integer Function: y = int(x) was talked about in the last section.


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Constant Function Linear Function Quadratic Function

Cubic function Absolute Value function Square Root function

Your text calls the linear function the identity function and the quadratic function thesquaring function.

Translations

There are two kinds of translations that we can do to a graph of a function. They areshifting and scaling. There are three if you count reflections, but reflections are just aspecial case of the second translation.

Shifts


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A shift is a rigid translation in that it does not change the shape or size of the graph ofthe function. All that a shift will do is change the location of the graph. A vertical shiftadds/subtracts a constant to/from every y-coordinate while leaving the x-coordinateunchanged. A horizontal shift adds/subtracts a constant to/from every x-coordinatewhile leaving the y-coordinate unchanged. Vertical and horizontal shifts can be

combined into one expression.

Shifts are added/subtracted to the x or f(x) components. If the constant is grouped withthe x, then it is a horizontal shift, otherwise it is a vertical shift.

Scales (Stretch/Compress)

A scale is a non-rigid translation in that it does alter the shape and size of the graph ofthe function. A scale will multiply/divide coordinates and this will change the appearanceas well as the location. A vertical scaling multiplies/divides every y-coordinate by aconstant while leaving the x-coordinate unchanged. A horizontal scaling

multiplies/divides every x-coordinate by a constant while leaving the y-coordinateunchanged. The vertical and horizontal scalings can be combined into one expression.

Scaling factors are multiplied/divided by the x or f(x) components. If the constant isgrouped with the x, then it is a horizontal scaling, otherwise it is a vertical scaling.

Reflections

A function can be reflected about an axis by multiplying by negative one. To reflectabout the y-axis, multiply every x by -1 to get -x. To reflect about the x-axis, multiply f(x)by -1 to get -f(x).

Putting it all together

Consider the basic graph of the function: y = f(x)

All of the translations can be expressed in the form:

y = a * f [ b (x-c) ] + d

Vertical Horizontal

Scale a b

Shift d c

acts normally acts inversely

Digression


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Understanding the concepts here are fundamental to understanding polynomial andrational functions (ch 3) and especially conic sections (ch 8). It will also play a very bigroll in Trigonometry (Math 117) and Calculus (Math 121, 122, 221, or 190).

Earlier in the text (section 1.2, problems 61-64), there were a series of problems which

wrote the equation of a line as:

x/a + y/b = 1

Where a was the x-intercept and b was the y-intercept of the line. The "a" could reallybe thought of how far to go in the x-direction (an x-scaling) and the "b" could be thoughtof as how far to go in the "y" direction (a y-scaling). So the "a" and "b" there are actuallymultipliers (even though they appear on the bottom). What they are multiplying is the 1which is on the right side. x+y=1 would have an x-intercept and y-intercept of 1.

Okay. Consider the equation: y = f(x)

This is the most basic graph of the function. But transformations can be applied to it,too. It can be written in the format shown to the below.

In this format, the "a" is a vertical multiplier and the "b" is a horizontal multiplier. Weknow that "a" affects the y because it is grouped with the y and the "b" affects the xbecause it is grouped with the x.

The "d" and "c" are vertical and horizontal shifts, respectively. We know that they areshifts because they are subtracted from the variable rather than being divided into thevariable, which would make them scales.

In this format, all changes seem to be the opposite of what you would expect. If youhave the expression (y-2)/3, it is a vertical shift of 2 to the right (even though it says yminus 2) and it is a vertical stretching by 3 (even though it says y divided by3). It isimportant to realize that in this format, when the constants are grouped with the variablethey are affecting, the translation is the opposite (inverse) of what most people think itshould be.

However, this format is not conducive to sketching with technology, because we likefunctions to be written as y =, rather than (y-c)/d =. So, if you take the notation aboveand solve it for y, you get the notation below, which is similar, but not exactly our basicform state above.

y = a * f( (x-c) / b ) + d


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Note that to solve for y, you have had to inverse both the "a" and "d" constants. Insteadof dividing by "a", you are now multiplying by "a". Well, it used to be that you had toapply the inverse of the constant anyway. When it said "divide by a", you knew that itmeant to "multiply each y by a". When it said "subtract d", you knew that you really hadto "add d". You have already applied the inverse, so don't do it again! With the

constants affecting the y, since they have been moved to the other side, take them atface value. If it says multiply by 2, do it, don't divide by 2.

However, the constants affecting the x have not been changed. They are still theopposite of what you think they should be. And, to make matters worse, the "x dividedb" that really means multiply each x-coordinate by "b" has been reversed to be writtenas "b times x" so that it really means divide each x by "b". The "x minus c" really meansadd c to each x-coordinate.

So, the final form (for technology) is as above:

y = a * f [ b (x-c) ] + d

Ok, end of digression.

Normal & Inverse Behavior

You will notice that the chart says the vertical translations are normal and the horizontaltranslations are inversed. For an explanation of why, read the digression above. Theconcepts in there really are fundamental to understanding a lot of graphing.

Examples

y=f(x)No translation

y=f(x+2)

The +2 is grouped with the x, therefore it is a horizontal translation. Since it isadded to the x, rather than multiplied by the x, it is a shift and not a scale. Since itsays plus and the horizontal changes are inversed, the actual translation is tomove the entire graph to the left two units or "subtract two from every x-coordinate" while leaving the y-coordinates alone.

y=f(x)+2The +2 is not grouped with the x, therefore it is a vertical translation. Since it is

added, rather than multiplied, it is a shift and not a scale. Since it says plus andthe vertical changes act the way they look, the actual translation is to move theentire graph two units up or "add two to every y-coordinate" while leaving the x-coordinates alone.

y=f(x-3)+5This time, there is a horizontal shift of three to the right and vertical shift of fiveup. So the translation would be to move the entire graph right three and up five or"add three to every x-coordinate and five to every y-coordinate"


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y=3f(x)The 3 is multiplied so it is a scaling rather than a shifting. The 3 is not groupedwith the x, so it is a vertical scaling. Vertical changes are affected the way youthink they should be, so the result is to "multiply every y-coordinate by three"while leaving the x-coordinates alone.

y=-f(x)The y is to be multiplied by -1. This makes the translation to be "reflect about thex-axis" while leaving the x-coordinates alone.

y=f(2x)The 2 is multiplied rather than added, so it is a scaling instead of a shifting. The 2is grouped with the x, so it is a horizontal scaling. Horizontal changes are theinverse of what they appear to be so instead of multiplying every x-coordinate bytwo, the translation is to "divide every x-coordinate by two" while leaving the y-coordinates unchanged.

y=f(-x)The x is to be multiplied by -1. This makes the translation to be "reflect about the

y-axis" while leaving the y-coordinates alone.y=1/2 f(x/3)

The translation here would be to "multiply every y-coordinate by 1/2 and multiplyevery x-coordinate by 3".

y=2f(x)+5There could be some ambiguity here. Do you "add five to every y-coordinate andthen multiply by two" or do you "multiply every y-coordinate by two and then addfive"? This is where my comment earlier about mathematics building upon itselfcomes into play. There is an order of operations which says that multiplicationand division is performed before addition and subtraction. If you remember this,then the decision is easy. The correct transformation is to "multiply every y-coordinate by two and then add five" while leaving the x-coordinates alone.

y=f(2x-3)Now that the order of operations is clearly defined, the ambiguity here aboutwhich should be done first is removed. The answer is not to "divide each x-coordinate by two and add three" as you might expect. The reason is thatproblem is notwritten in standard form. Standard form is y=f[b(x-c)]. Whenwritten in standard form, this problem becomes y=f[2(x-3/2)]. This means that

the proper translation is to "divide every x-coordinate by two and add three-halves" while leaving the y-coordinates unchanged.

y=3f(x-2)The translation here is to "multiply every y-coordinate by three and add two toevery x-coordinate". Alternatively, you could change the order around. Changesto the x or y can be made independently of each other, but if there are scales andshifts to the same variable, it is important to do the scaling first and the shiftingsecond.

Translations and the Effect on Domain & Range


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Any horizontal translation will affect the domain and leave the range unchanged. Anyvertical translation will affect the range and the leave the domain unchanged.

Apply the same translation to the domain or range that you apply to the x-coordinates orthe y-coordinates. This works because the domain can be written in interval notation as

the interval between two x-coordinates. Likewise for the range as the interval betweentwo y-coordinates.

In the following table, remember that domain and range are given in interval notation. Ifyou're not familiar with interval notation, then please check the prerequisite chapter. The

first line is the definition statement and should be used to determine the rest of theanswers.

Graph Translation Domain Range

y=f(x) none (-2,5) [4,8]

y=f(x-2) right 2 (0,7) [4,8]

y=f(x)-2 down 2 (-2,5) [2,6]

y=3f(x) multiply each y by 3 (-2,5) [12,24]

y=f(3x) divide each x by 3 (-2/3,5/3) [4,8]

y=2f(x-3)-5 multiply each y by 2 and subtract 5;add 3 to every x

(1,8) [3,11]

y=-f(x) reflect about x-axis (-2,5) [-8,-4]

y=1/f(x) take the reciprocal of each y (-2,5) [1/8,1/4]

Notice on the last two that the order in the range has changed. This is because ininterval notation, the smaller number always comes first.

Really Good Stuff

Understanding the translations can also help when finding the domain and range of afunction. Let's say your problem is to find the domain and range of the function y=2-sqrt(x-3).

Begin with what you know. You know the basic function is the sqrt(x) and you know thedomain and range of the sqrt(x) are both [0,+infinity). You know this because you knowthose six common functions on the front cover of your text which are going to be usedas building blocks for other functions.

Function Translation Domain Range


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Begin with what youknow

y=sqrt(x) None [0,+infinity) [0,+infinity)

Apply the translations y=-sqrt(x) Reflect about x-axis [0,+infinity) (-infinity,0]

y=2-sqrt(x) Add 2 to eachordinate

[0,+infinity) (-infinity,2]

y=2-sqrt(x-3)

Add 3 to eachabscissa

[3,+infinity) (-infinity,2]

So, for the function y=2-sqrt(x-3), the domain is x3 and the range is y2.

And the bestpartof it is that you understood it! Not only did you understand it, but youwere able to do it without graphing it on the calculator.

There is nothing wrong with making a graph to see what's going on, but you should beable to understand what's going on without the graph because we have learned that thegraphing calculator doesn't always show exactly what's going on. It is a tool to assistyour understanding and comprehension, not a tool to replace it.

It is this cohesiveness of math that I want all of you to "get". It all fits together sobeautifully.

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Shifting of Graphs