DefinitionsAbscissa The x-coordinate Ordinate The y-coordinate Shift A translation in which the size and shape of a graph of a function is not changed, butthe location of the graph is. Scale A translation in which the size and shape of the graph of a function is changed. Reflection A translation in which the graph of a function is mirrored about an axis.
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Part of the beauty of mathematics is that almost everything builds upon something else, and ifyou can understand the foundations, then you can apply new elements to old. It is this abilitywhich makes comprehension of mathematics possible. If you were to memorize every piece ofmathematics presented to you without making the connection to other parts, you will 1) becomefrustrated at math and 2) not really understand math.
There are some basic graphs that we have seen before. By applying translations to these basicgraphs, we are able to obtain new graphs that still have all the properties of the old ones. Byunderstanding the basic graphs and the way translations apply to them, we will recognize eachnew graph as a small variation in an old one, not as a completely different graph that we havenever seen before. Understanding these translations will allow us to quickly recognize andsketch a new function without having to resort to plotting points.
These are the common functions you should know the graphs of at this time:Constant Function: y = c Linear Function: y = x Quadratic Function: y = x2 Cubic Function: y = x3 Absolute Value Function: y = |x| Square Root Function: y = sqrt(x)
Absolute Value function
Square Root function
Your text calls the linear function the identity function and the quadratic function the squaringfunction.
There are two kinds of translations that we can do to a graph of a function. They are shifting andscaling. There are three if you count reflections, but reflections are just a special case of thesecond translation.
A shift is a rigid translation in that it does not change the shape or size of the graph of thefunction. 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-coordinate unchanged. A horizontal shift adds/subtracts a constant to/from every x-coordinate while 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 with the x,then it is a horizontal shift, otherwise it is a vertical shift.
A scale is a non-rigid translation in that it does alter the shape and size of the graph of thefunction. A scale will multiply/divide coordinates and this will change the appearance as well asthe location. A vertical scaling multiplies/divides every y-coordinate by a constant while leavingthe x-coordinate unchanged. A horizontal scaling multiplies/divides every x-coordinate by aconstant while leaving the y-coordinate unchanged. The vertical and horizontal scalings can becombined into one expression.
Scaling factors are multiplied/divided by the x or f(x) components. If the constant is groupedwith the x, then it is a horizontal scaling, otherwise it is a vertical scaling.
A function can be reflected about an axis by multiplying by negative one. To reflect about 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
|acts normally||acts inversely|
Understanding the concepts here are fundamental to understanding polynomial and rationalfunctions (ch 3) and especially conic sections (ch 8). It will also play a very big roll inTrigonometry (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 theequation 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 really bethought of how far to go in the x-direction (an x-scaling) and the "b" could be thought of as howfar to go in the "y" direction (a y-scaling). So the "a" and "b" there are actually multipliers (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 of1.
Okay. Consider the equation: y = f(x)
This is the most basic graph of the function. But transformations canbe 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. We know that "a" affects the y because it is grouped with the y and the "b" affects the x because it is grouped with the x.
The "d" and "c" are vertical and horizontal shifts, respectively. We know that they are shifts because they are subtracted from the variable rather than being divided into the variable, which would make them scales.
In this format, all changes seem to be the opposite of what you would expect. If you have theexpression (y-2)/3, it is a vertical shift of 2 to the right (even though it says y minus 2) and it is avertical stretching by 3 (even though it says y divided by 3). It is important to realize that in thisformat, when the constants are grouped with the variable they are affecting, the translation is theopposite (inverse) of what most jajalger2018.org think it should be.
However, this format is not conducive to sketching with technology,because we like functions to be written as y =, rather than (y-c)/d =. So, if you take the notation above and solve it for y, you get the notation below, which issimilar, but not exactly our basic form state above.
y = a * f( (x-c) / b ) + d
Note that to solve for y, you have had to inverse both the "a" and "d" constants. Instead of dividing by "a", you are now multiplying by "a". Well, it used to be that you had to apply the inverse of the constant anyway. When it said "divide by a", you knew thatitmeant to "multiplyeach y by a". When it said "subtract d", you knew that you really had to "add d". You havealready applied the inverse, so don"t do it again! With the constants affectingthe y, since they have been moved to the other side, take them at face value.If it says multiply by 2, do it, don"tdivide by 2.
However, the constants affecting the x have not been changed. They are still the opposite ofwhat you think they should be. And, to make matters worse, the "x divided b" that really meansmultiply each x-coordinate by "b" has been reversed to be written as "b times x" so that it reallymeans divide each x by "b". The "x minus c" really means add 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. The concepts inthere really are fundamental to understanding a lot of graphing.
Examplesy=f(x) No translation y=f(x+2) The +2 is grouped with the x, therefore it is a horizontal translation. Since it is addedto the x, rather than multiplied by the x, it is a shift and not a scale. Since it says plusand the horizontal changes are inversed, the actual translation is to move the entiregraph to the left two units or "subtract two from every x-coordinate" while leaving they-coordinates alone. y=f(x)+2 The +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 and the verticalchanges act the way they look, the actual translation is to move the entire graph twounits up or "add two to every y-coordinate" while leaving the x-coordinates alone. y=f(x-3)+5 This time, there is a horizontal shift of three to the right and vertical shift of five up. Sothe translation would be to move the entire graph right three and up five or "add threeto every x-coordinate and five to every y-coordinate" y=3f(x) The 3 is multiplied so it is a scaling rather than a shifting. The 3 is not grouped withthe x, so it is a vertical scaling. Vertical changes are affected the way you think theyshould 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 the x-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 2 isgrouped with the x, so it is a horizontal scaling. Horizontal changes are the inverse ofwhat they appear to be so instead of multiplying every x-coordinate by two, thetranslation is to "divide every x-coordinate by two" while leaving the y-coordinatesunchanged. 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)+5 There could be some ambiguity here. Do you "add five to every y-coordinate and thenmultiply by two" or do you "multiply every y-coordinate by two and then add five"? This is where my comment earlier about mathematics building upon itself comes intoplay. There is an order of operations which says that multiplication and division isperformed before addition and subtraction. If you remember this, then the decision iseasy. The correct transformation is to "multiply every y-coordinate by two and thenadd five" while leaving the x-coordinates alone. y=f(2x-3) Now that the order of operations is clearly defined, the ambiguity here about whichshould be done first is removed. The answer is not to "divide each x-coordinate by twoand add three" as you might expect. The reason is that problem is not written in standard form. Standard form is y=f. When written in standard form, thisproblem become 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 to every x-coordinate". Alternatively, you could change the order around. Changes to the x or ycan be made independently of each other, but if there are scales and shifts to the samevariable, it is important to do the scaling first and the shifting second.
Translations and the Effect on Domain & Range
Any horizontal translation will affect the domain and leave the range unchanged. Any verticaltranslation 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 or the y-coordinates. This works because the domain can be written in interval notation as the intervalbetween two x-coordinates. Likewise for the range as the interval between two y-coordinates.
In the following table, remember that domain and range are given in interval notation. If you"renot familiar with interval notation, then please check the prerequisite chapter. The first line is thedefinition statement and should be used to determine the rest of the answers.
|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 in intervalnotation, the smaller number always comes first.
Really Good Stuff
Understanding the translations can also help when finding the domain and range of a function. 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 the domainand range of the sqrt(x) are both <0,+infinity). You know this because you know those sixcommon functions on the front cover of your text which are going to be used as building blocksfor other functions.
|y=-sqrt(x)||Reflect about x-axis||<0,+infinity)||(-infinity,0>|
|y=2-sqrt(x)||Add 2 to each ordinate||<0,+infinity)||(-infinity,2>|
|y=2-sqrt(x-3)||Add 3 to each abscissa||<3,+infinity)||(-infinity,2>|
So, for the function y=2-sqrt(x-3), the domain is x≥3 and the range is y≤2.
And the best part of it is that you understood it! Not only did you understand it, but you wereable 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 be able tounderstand what"s going on without the graph because we have learned that the graphingcalculator doesn"t always show exactly what"s going on. It is a tool to assist your understandingand comprehension, not a tool to replace it.
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It is this cohesiveness of math that I want all of you to "get". It all fitstogether so beautifully.