Which Parent Function is Represented by the Graph
Which Parent Function is Represented by the Graph
Bones Parent Functions  Writing Transformed Equations from Graphs 
Generic Transformations of Functions  Rotational Transformations 
Vertical Transformations  Transformations of Changed Functions 
Horizontal Transformations  Applications of Parent Function Transformations 
Mixed Transformations  More Exercise 
Transformations Using Functional Notation 
For Absolute Value Transformations, see theAbsolute Value Transformations section. Here are links to
Parent Part Transformations
in other sections:
Transformations of Quadratic Functions
(quick and piece of cake way);Transformations of Radical Functions;Transformations of Rational Functions;
Transformations of Exponential Functions;Transformations of Logarithmic Functions;
Transformations of Piecewise Functions;
Transformations of Trigonometric Functions;
Transformations of Inverse Trigonometric Functions
You may non be familiar with all the functions and characteristics in the tables; here are some topics to review:
 Whether functions are
fiftyfifty,
odd, or
neither, discussed hither in the
Advanced Functions: Compositions, Even and Odd, and Extrema. 
End behavior
and
asymptotes, discussed in the
Asymptotes and Graphing Rational Functions
and
Graphing Polynomials
sections 
Exponential
and
Logarithmic
Functions 
Trigonometric
Functions
Basic Parent Functions
Y’all’ll probably written report some “pop”
parent functions
and work with these to learn how to
transform functions
– how to move and/or resize them. We telephone call these basic functions “parent” functions since they are the simplest form of that type of office, meaning they are as close as they tin can get to the
origin
\(\left( {0,0} \correct)\).
The chart below provides some basic parent functions that yous should be familiar with. I’ve too included the
significant
points, or
critical points, the points with which to graph the parent function. I likewise sometimes telephone call these the “reference points” or “ballast points”.
Know the shapes of these parent functions well! Fiftyfifty when using
t
charts, you must know the general shape of the parent functions in lodge to know how to transform them correctly!
Parent Function  Graph  Parent Function  Graph 
\(y=x\) Domain: \(\left( {\infty ,\infty } \correct)\) Terminate Beliefs**: Disquisitional points: \(\displaystyle \left( {1,i} \correct),\,\left( {0,0} \right),\,\left( {1,1} \right)\) 
\(y=\left x \right\) Domain: \(\left( {\infty ,\infty } \correct)\) End Beliefs: Disquisitional points: \(\displaystyle \left( {one,ane} \right),\,\left( {0,0} \right),\,\left( {i,1} \right)\) 

\(y={{x}^{2}}\) Domain: \(\left( {\infty ,\infty } \right)\) End Behavior: Critical points: \(\displaystyle \left( {one,i} \right),\,\left( {0,0} \correct),\,\left( {ane,i} \correct)\) 
\(y=\sqrt{x}\) Domain: \(\left[ {0,\infty } \right)\) Finish Beliefs: \(\displaystyle \begin{assortment}{50}ten\to 0,\,\,\,\,y\to 0\\10\to \infty \text{,}\,\,y\to \infty \end{assortment}\) Critical points: \(\displaystyle \left( {0,0} \right),\,\left( {1,1} \correct),\,\left( {four,2} \right)\) 

\(y={{x}^{3}}\) Domain: \(\left( {\infty ,\infty } \right)\) Finish Behavior: Critical points: \(\displaystyle \left( {ane,ane} \right),\,\left( {0,0} \right),\,\left( {1,one} \right)\) 
\(y=\sqrt[3]{10}\) Domain: \(\left( {\infty ,\infty } \right)\) End Behavior: Critical points: \(\displaystyle \left( {i,1} \right),\,\left( {0,0} \right),\,\left( {i,i} \right)\) 

\(\begin{array}{c}y={{b}^{ten}},\,\,\,b>ane\,\\(\text{Example:}\,\,y={{two}^{x}})\stop{array}\) Exponential, Neither Domain: \(\left( {\infty ,\infty } \right)\) Stop Beliefs: Disquisitional points: \(\displaystyle \left( {1,\frac{1}{b}} \right),\,\left( {0,1} \right),\,\left( {1,b} \right)\) Asymptote: \(y=0\) 
\(\begin{assortment}{c}y={{\log }_{b}}\left( ten \right),\,\,b>1\,\,\,\\(\text{Example:}\,\,y={{\log }_{2}}x)\end{array}\) Log, Neither Domain: \(\left( {0,\infty } \correct)\) Cease Behavior: Disquisitional points: \(\displaystyle \left( {\frac{1}{b},1} \right),\,\left( {1,0} \right),\,\left( {b,i} \right)\) Asymptote: \(10=0\) 

\(\displaystyle y=\frac{1}{x}\) Rational (Changed), Odd Domain: \(\left( {\infty ,0} \correct)\cup \left( {0,\infty } \right)\) Terminate Beliefs: Critical points: \(\displaystyle \left( {1,1} \right),\,\left( {ane,ane} \right)\) Asymptotes: \(y=0,\,\,x=0\) **Note that this function is the 
\(\displaystyle y=\frac{one}{{{{x}^{two}}}}\) Rational (Inverse Squared), Fiftyfifty Domain: \(\left( {\infty ,0} \right)\loving cup \left( {0,\infty } \right)\) End Behavior: Critical points: \(\displaystyle \left( {1,\,1} \right),\left( {i,1} \right)\) Asymptotes: \(x=0,\,\,y=0\) 

\(y=\text{int}\left( x \right)=\left\lfloor x \correct\rfloor \)
Greatest Integer^{*} Domain: \(\left( {\infty ,\infty } \correct)\) End Beliefs: Disquisitional points: \(\displaystyle \begin{array}{50}x:\left[ {1,0} \right)\,\,\,y:ane\\x:\left[ {0,1} \right)\,\,\,y:0\\x:\left[ {1,2} \right)\,\,\,y:1\cease{array}\) 
\(y=C\) (\(y=2\)) Abiding, Even Domain: \(\left( {\infty ,\infty } \right)\) End Behavior: Critical points: \(\displaystyle \left( {ane,C} \right),\,\left( {0,C} \correct),\,\left( {one,C} \correct)\) 

*The
Greatest Integer
Function, sometimes called the
Footstep Function, returns the greatest integer less than or equal to a number (think of rounding downwards to an integer). There’due south as well a
Least Integer
Role, indicated past \(y=\left\lceil x \right\rceil \), which returns the least integer greater than or equal to a number (recall of rounding upwards to an integer).
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**Notes on End Behavior: To go theend behavior of a office, nosotros just expect at thesmallest andlargest
values of \(ten\), and encounter which way the \(y\) is going. Non all functions take end behavior defined; for example, those that get dorsum and forth with the \(y\) values (called “periodic functions”) don’t have end behaviors.
Nearly of the time, our end behavior looks something similar this: \(\displaystyle \begin{array}{fifty}x\to \infty \text{, }\,y\to \,\,?\\x\to \infty \text{, }\,\,\,y\to \,\,?\cease{array}\) and nosotros have to fill in the \(y\) office. For example, the end behavior for a line with a positive gradient is: \(\begin{assortment}{l}10\to \infty \text{, }\,y\to \infty \\10\to \infty \text{, }\,\,\,y\to \infty \end{array}\), and the end behavior for a line with a negative slope is: \(\begin{assortment}{l}x\to \infty \text{, }\,y\to \infty \\x\to \infty \text{, }\,\,\,y\to \infty \end{assortment}\). One way to think of terminate behavior is that for \(\displaystyle x\to \infty \), we look at what’south going on with the \(y\) on the lefthand side of the graph, and for \(\displaystyle x\to \infty \), we expect at what’s happening with \(y\) on the righthand side of the graph.
There are a couple of exceptions; for example, sometimes the \(10\) starts at
(such as in theradical function), we don’t have the negative portion of the \(10\) finish behavior. As well, when \(x\) starts very close to
(such as in in thelog function), we indicate that \(10\) is starting from the positive (right) side of
(and the \(y\) is going down); nosotros indicate this by \(\displaystyle ten\to {{0}^{+}}\text{, }\,y\to \infty \).
Generic Transformations of Functions
Again, the “parent functions” presume that we have the simplest grade of the office; in other words, the function either goes through the origin \(\left( {0,0} \right)\), or if it doesn’t go through the origin, information technology isn’t shifted in whatever way. When a function is
shifted, stretched
(or
compressed), or flipped in any style from its “parent part“, information technology is said to be
transformed, and is a
transformation of a function.
Tcharts
are extremely useful tools when dealing with transformations of functions. For example, if y’all know that the quadratic parent function \(y={{ten}^{two}}\) is being transformed
ii
units to the right, and
ane
unit of measurement down
(merely a shift, non a stretch or a flip), we can create the original
t
nautical chart, following by the transformation points on the outside of the original points. Then we can plot the “exterior” (new) points to become the newly transformed function:
Transformation 
Tchart 
Graph  
Quadratic Office \(y={{10}^{two}}\)
Transform function This turns into the function \(y={{\left( {x2} \right)}^{two}}1\), oddly enough! 
Transformed: Domain: \(\left( {\infty ,\infty } \correct)\) Range: \(\left[ {1,\,\,\infty } \right)\) 
When looking at
the equation of the transformed part, notwithstanding, we have to be careful. When functions are transformed on the
exterior
of the \(f(x)\) part, you motility the function up and down and do the “regular” math, as we’ll see in the examples below. These are
vertical
transformations
or
translations, and touch the \(y\) part of the function. When transformations are made on the
within
of the \(f(ten)\) part, y’all move the function
back and forth
(but practice the “contrary” math – since if you were to isolate the \(x\), you lot’d move everything to the other side). These are
horizontal transformations
or
translations, and affect the \(x\) office of the part.
There are several means to perform transformations of parent functions; I like to apply
t
charts
, since they work consistently with ever function. And note that in most
tcharts,
I’ve included more just the
critical points
to a higher place, just to evidence the graphs ameliorate.
Vertical Transformations
Here are the rules and examples of when functions are transformed on the “outside” (find that the \(y\)values are affected). The
tcharts
include the points (ordered pairs) of the original parent functions, and also the transformed or shifted points. The first 2 transformations are
translations, the third is a
dilation, and the last are forms of
reflections. Absolute value transformations will be discussed more expensively in the
Absolute Value Transformations section!
Trans formation 
What Information technology Does  Example  Graph  
\(f\left( x \right)+b\) Translation 
Move graph Every signal on the graph is shifted up \(b\) units. The \(x\)’s stay the aforementioned; add together \(b\) to the \(y\) values. 
Parent:
Transformed:

Domain: \(\left( {\infty ,\infty } \right)\) 

\(f\left( x \correct)b\) Translation 
Move graph Every point on the graph is shifted down \(b\) units. The \(x\)’s stay the same; subtract \(b\) from the \(y\) values. 
Parent: \(y=\sqrt{10}\) Transformed: \(y=\sqrt{x} \,iii\)



\(a\,\cdot f\left( x \right)\) Dilation 
Stretch Every betoken on the graph is stretched \(a\) units. The \(x\)’s stay the aforementioned; multiply the \(y\) values past \(a\). 
Parent: \(y={{10}^{three}}\) Transformed: \(y={{4x}^{3}}\)



\(f\left( x \right)\) Reflection 
Flip Every point on the graph is flipped vertically. The \(x\)’s stay the aforementioned; multiply the \(y\) values by \(i\). 
Parent: \(y=\left x \right\) Transformed: \(y=\left 10 \right\)



\(\left {f\left( ten \right)} \correct\)
Absolute Value (More examples here in the 
Reflect part of graph underneath the \(x\)centrality (negative \(y\)’due south) across the \(x\)axis. Leave positive \(y\)’south the same.
The \(x\)’s stay the same; 
Parent: \(y=\sqrt[3]{x}\) Transformed: \(y=\left {\sqrt[3]{x}} \right\)

Domain: \(\left( {\infty ,\infty } \right)\) Range:\(\left[ {0,\infty } \correct)\) 
Horizontal Transformations
Hither are the rules and examples of when functions are transformed on the “inside” (notice that the \(x\)values are affected). Find that when the \(ten\)values are affected, you
do the math in the “opposite” manner from what the function looks like: if you lot’re adding on the inside, you subtract from the \(x\); if you’re subtracting on the within, you lot add to the \(ten\); if you’re multiplying on the inside, you divide from the \(x\); if you’re dividing on the inside, y’all multiply to the \(10\). If yous accept a negative value on the inside, you
flip across the
\(\boldsymbol{y}\) axis
(notice that you nevertheless multiply the \(x\) by \(one\) simply like you exercise for with the \(y\) for vertical flips). The first two transformations are
translations, the third is a
dilation, and the last are forms of
reflections.
Absolute value transformations will be discussed more than expensively in the
Absolute Value Transformations
department!
(Yous may observe it interesting is that a vertical stretch behaves the aforementioned mode as a horizontal compression, and vice versa, since when stretch something upward, we are making it skinnier.)
Trans formation 
What It Does  Example  Graph  
\(f\left( {10+b} \right)\)
Translation 
Move graph (Exercise the “contrary” when change is within the parentheses or underneath radical sign.) Every signal on the graph is shifted left \(b\) units. The \(y\)’s stay the same; subtract \(b\) 
Parent: \(y={{ten}^{2}}\) Transformed: \(y={{\left( {x+two} \right)}^{2}}\)



\(f\left( {xb} \right)\)
Translation 
Move graph Every point on the graph is shifted correct \(b\) units. The \(y\)’southward stay the aforementioned; add \(b\) 
Parent: \(y=\sqrt{x}\) Transformed: \(y=\sqrt{{x \,3}}\)



\(f\left( {a\cdot 10} \right)\)
Dialation 
Compress Every bespeak on the graph is compressed \(a\) The \(y\)’southward stay the same; multiply the \(x\)values by \(\displaystyle \frac{1}{a}\). 
Parent: \(y={{x}^{iii}}\) T



\(f\left( {10} \right)\)
Reflection 
Flip Every betoken on the graph is flipped around the \(y\) axis. The \(y\)’south stay the same; multiply the \(10\)values by \(1\). 
Parent: \(y=\sqrt{10}\) Transformed: \(y=\sqrt{{x}}\)



\(f\left( {\left x \right} \right)\)
Absolute Value (More examples here in the 
“Throw abroad” the negative \(ten\)’due south; reflect the positive \(x\)’s beyond the \(y\)axis.
The positive \(x\)’southward stay the aforementioned; the 
Parent: \(y=\sqrt{x}\) Transformed: \(y=\sqrt{{\left 10 \right}}\)

Domain: \(\left( {\infty ,\infty } \right)\)Range:\(\left[ {0,\infty } \correct)\) 
Mixed Transformations
Most of the bug you’ll get volition involve
mixed transformations, or multiple transformations, and we practise need to worry about
the order
in which we perform the transformations. It commonly doesn’t matter if we make the \(x\) changes or the \(y\) changes beginning, merely within the \(x\)’s and \(y\)’due south, we need to perform the transformations in the order below. Note that this is sort of similar to the social club with
PEMDAS(parentheses, exponents, multiplication/partitioning, and improver/subtraction). When performing these rules, the coefficients of the within \(x\) must exist
1
; for instance, nosotros would need to have \(y={{\left( {4\left( {x+2} \correct)} \right)}^{2}}\) instead of \(y={{\left( {4x+eight} \right)}^{2}}\) (by factoring). If yous didn’t learn it this way, meet
IMPORTANT NOTE
below.
Here is the order. We can do
steps i and 2 together
(society doesn’t really matter), since nosotros can recall of the commencement two steps as a “negative stretch/compression.”
 Perform
Flipping across the axes first (negative signs).  Perform
Stretching and Shrinking next
(multiplying and dividing).  Perform
Horizontal and Vertical shifts last
(adding and subtracting).
I similar to take the
disquisitional points
and maybe a few more points of the parent functions, and perform all thetransformations at the same time
with a
tchart! We merely exercise the multiplication/division first on the \(x\) or \(y\) points, followed by addition/subtraction. It makes information technology much easier!Note again that since we don’t accept an
\(\boldsymbol {ten}\)
“by itself” (coefficient of
one) on the inside, we have to get it that mode by factoring!
For instance,we’d have to change \(y={{\left( {4x+8} \right)}^{2}}\text{ to }y={{\left( {iv\left( {x+ii} \right)} \right)}^{two}}\).
Let’south try to graph this “complicated” equation and I’ll evidence y’all how piece of cake it is to do with a
tchart:
\(\displaystyle f(ten)=three{{\left( {2x+8} \right)}^{2}}+10\).
(Note that for this example, we could motion the \({{two}^{two}}\) to the outside to get a vertical stretch of \(iii\left( {{{2}^{ii}}} \correct)=12\), but we can’t exercise that for many functions.) We first need to go the \(x\) by itself
on the inside past
factoring, so we tin perform the horizontal translations. This is what nosotros end upward with:
\(\displaystyle f(x)=3{{\left( {2\left( {x+4} \right)} \right)}^{2}}+10\).
Wait at what’south done on the “outside” (for the \(y\)’due south) and make all the moves at once, by following the
exact math. Then look at what we do on the “inside” (for the \(ten\)’due south) and make all the moves at once,
merely exercise the
opposite math. We do this with a
t
chart.
Beginning with the parent function \(f(x)={{x}^{ii}}\). If we await at what we’re doing on the
outside
of what is being squared, which is the \(\displaystyle \left( {two\left( {ten+4} \correct)} \right)\), we’re
flipping
it across the \(x\)axis (the minus sign),
stretching
information technology by a
factor of
three
, and
adding
10
(shifting upward
10
). These are the things that we are doing
vertically, or to the \(y\). If nosotros look at what we are doing on the inside of what we’re squaring, we’re multiplying it by
ii
, which means we have to
divide by
2
(horizontal compression by a factor of \(\displaystyle \frac{i}{2}\)), and we’re adding
4
, which means nosotros have to
decrease
4
(a left shift of
4
). Remember that we do the
contrary
when we’re dealing with the \(ten\). Also recollect that nosotros always have to do the
multiplication or segmentation first
with our points, and so the
adding and subtracting
(sort of like
PEMDAS).
Hither is the
tchart
with the original function, and then the transformations on the outsides. Now we tin graph the outside points (points that aren’t crossed out) to get the graph of the transformation. I’ve also included an explanation of how to transform this parabola
without a
tnautical chart, as we did in the here in the
Introduction to Quadratics
section.
tchart 
Transformed Graph  
Parent: \(y={{x}^{ii}}\) (Quadratic) Transformed:\(\displaystyle f(x)=iii{{\left( {ii\left( {10+4} \right)} \right)}^{2}}+10\)
Opposite for \(x\), “regular” for \(y\), multiplying/dividing outset: Coordinate Rule: \(\left( {10,\,y} \right)\to \left( {.5xfour,3y+ten} \right)\)

Domain: 

How to graphwithout a tchart: \(\displaystyle f(ten)=3{{\left( {2\left( {x+4} \right)} \right)}^{two}}+x\) Since this is a Notice that the coefficient of is The parent graph quadratic goes up 
IMPORTANT NOTE:In some books, for
\(\displaystyle f\left( ten \right)=3{{\left( {2x+8} \right)}^{ii}}+x\)
, they may Non have you factor out thetwo on the inside, but just switch the guild of the transformation on the
\(\boldsymbol{x}\).
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In this case, the order of transformations would be horizontal shifts, horizontal reflections/stretches, vertical reflections/stretches, and and so vertical shifts. For example, for this problem, y’all would move to the left
eight
offset for the
\(\boldsymbol{x}\), and then compress with a factor of
\(\displaystyle \frac {1}{two}\)
for the
\(\boldsymbol{10}\) (which is contrary of PEMDAS). And so you would perform the
\(\boldsymbol{y}\)
(vertical) changes the regular way: reflect and stretch by
3
showtime, and so shift up
10. So, you would accept
\(\displaystyle {\left( {10,\,y} \right)\to \left( {\frac{one}{2}\left( {ten8} \correct),3y+ten} \right)}\). Try a
tchart; you’ll get the same
tnautical chart as above!
More Examples of Mixed Transformations:
Here are a couple more examples (using
t
charts), with different parent functions. Don’t worry if you are totally lost with the exponential and log functions; they will be discussed in the
Exponential Functions and
Logarithmic Functions
sections. Also, the concluding type of function is a rational part that will be discussed in the
Rational Functions
section.
Transformation 
Tchart/Domain and Range 
Graph  
\(\displaystyle y=\frac{three}{two}{{\left( {x} \right)}^{3}}+ii\) Parent function: \(y={{x}^{3}}\) For this part, note that could accept also put the negative sign on the 
Domain: \(\left( {\infty ,\infty } \right)\) Range:\(\left( {\infty ,\infty } \right)\) 


\(\displaystyle y=\frac{1}{2}\sqrt{{ten}}\) Parent function: \(y=\sqrt{x}\) 
Domain: \(\left( {\infty ,0} \right]\) Range:\(\left[ {0,\infty } \correct)\) 

\(y={{2}^{{x4}}}+iii\) Parent function: \(y={{two}^{x}}\) For 
Domain: \(\left( {\infty ,\infty } \correct)\) Range:\(\left( {3,\infty } \right)\) Asymptote: \(y=3\) 

\(\begin{array}{l}y=\log \left( {2xtwo} \correct)1\\y=\log \left( {2\left( {xi} \correct)} \correct)1\end{array}\) Parent part: \(y=\log \left( ten \correct)={{\log }_{{10}}}\left( x \correct)\) For 
Domain: \(\left( {1,\infty } \correct)\) Range:\(\left( {\infty ,\infty } \right)\) Asymptote: \(x=1\) 

\(\displaystyle y=\frac{3}{{2x}}\,\,\,\,\,\,\,\,\,\,\,y=\frac{3}{{\left( {xii} \right)}}\) Parent role: \(\displaystyle y=\frac{ane}{x}\) For this function, notation that could take as well put the negative sign on the 
Domain:\(\left( {\infty ,ii} \right)\loving cup \left( {2,\infty } \right)\) Range:\(\left( {\infty ,0} \right)\loving cup \left( {0,\infty } \correct)\) Asymptotes: \(y=0\) and \(x=2\) 
Here’due south a mixed transformation with the
Greatest
Integer Function
(sometimes called the
Flooring Role). Note how we can employ intervals every bit the \(ten\) values to make the transformed function easier to draw:
Transformation 
Tchart/Domain and Range 
Graph  
\(\displaystyle y=\left[ {\frac{i}{2}xii} \right]+3\) \(\displaystyle y=\left[ {\frac{i}{ii}\left( {xfour} \right)} \right]+3\) Parent function: \(y=\left[ x \right]\) Annotation how we had to have out the \(\displaystyle \frac{i}{two}\) to make information technology in the correct course. 
Domain:\(\left( {\infty ,\infty } \right)\) Range:\(\{y:y\in \mathbb{Z}\}\text{ (integers)}\) 

Transformations Using Functional Note
Yous might run into mixed transformations in the class \(\displaystyle g\left( x \right)=a\cdot f\left( {\left( {\frac{ane}{b}} \correct)\left( {xh} \right)} \correct)+g\), where \(a\) is the vertical stretch, \(b\) is the horizontal stretch, \(h\) is the horizontal shift to the right, and \(k\) is the vertical shift upwards. In this case, nosotros have the coordinate dominion \(\displaystyle \left( {ten,y} \right)\to \left( {bx+h,\,ay+k} \right)\). For example, for the transformation
\(\displaystyle f(ten)=3{{\left( {two\left( {x+iv} \right)} \right)}^{2}}+10\), we take \(a=3\), \(\displaystyle b=\frac{ane}{2}\,\,\text{or}\,\,.five\), \(h=4\), and \(1000=10\). Our transformation \(\displaystyle g\left( ten \correct)=3f\left( {two\left( {x+four} \correct)} \right)+10=thousand\left( x \correct)=3f\left( {\left( {\frac{ane}{{\frac{1}{2}}}} \right)\left( {x\left( {iv} \correct)} \right)} \right)+x\) would result in a coordinate rule of \({\left( {x,\,y} \correct)\to \left( {.5x4,3y+10} \right)}\).
(Y’all may besides see this every bit \(m\left( x \right)=a\cdot f\left( {b\left( {xh} \right)} \right)+thou\), with coordinate rule \(\displaystyle \left( {x,\,y} \right)\to \left( {\frac{i}{b}x+h,\,ay+1000} \right)\); the end effect will be the same.)
You may be given a
random point
and give the transformed coordinates for the point of the graph. For example, if the point \(\left( {8,2} \right)\) is on the graph \(y=g\left( ten \right)\), requite the transformed coordinates for the betoken on the graph \(y=6g\left( {2x} \correct)ii\). To do this, to get the transformed \(y\), multiply the \(y\) part of the point by
–6
and and then subtract
2
. To get the transformed \(ten\), multiply the \(ten\) part of the betoken by \(\displaystyle \frac{1}{2}\) (contrary math). The new bespeak is \(\left( {4,10} \correct)\). Let’s do another instance: If the point \(\left( {iv,1} \right)\) is on the graph \(y=thousand\left( x \right)\), the transformed coordinates for the signal on the graph of \(\displaystyle y=2g\left( {3xtwo} \right)+three=2g\left( {3\left( {x+\frac{2}{three}} \right)} \right)+3\) is \(\displaystyle \left( {iv,one} \right)\to \left( {iv\left( {\frac{1}{three}} \right)\frac{2}{three},2\left( 1 \right)+3} \right)=\left( {\frac{two}{iii},5} \correct)\) (using coordinate rules \(\displaystyle \left( {ten,\,y} \correct)\to \left( {\frac{1}{b}ten+h,\,\,ay+k} \correct)=\left( {\frac{1}{3}x\frac{2}{iii},\,\,2y+3} \right)\)).
You may likewise be asked to transform a parent or nonparent equation
to get a new equation. Nosotros can exercise this
without using a
tchart, only by using
substitution
and
algebra. For case, if we desire to transform \(f\left( x \right)={{ten}^{2}}+4\) using the transformation \(\displaystyle 2f\left( {xi} \right)+3\), nosotros can only substitute “\(x1\)” for “\(x\)” in the original equation, multiply by
–ii
, and then add
3
. For example: \(\displaystyle 2f\left( {xane} \correct)+iii=ii\left[ {{{{\left( {xane} \right)}}^{2}}+4} \right]+3=ii\left( {{{x}^{two}}2x+ane+4} \right)+3=2{{ten}^{2}}+4x7\). We used this method to assist transform a
piecewise role
here.
Transformations in Role Notation (based on Graph and/or Points).
You may also be asked to perform a transformation of a function
using a graph and individual points; in this instance, you’ll probably exist given the transformation in
function notation. Note that we may need to use several points from the graph and “transform” them, to make sure that the transformed function has the right “shape”.
Here are some examples; the 2d instance is the transformation with an absolute value on the \(x\); see the
Absolute Value Transformations
section for more item.
Original Graph and Points of Function 
Transformation Example 
Transformation Example 

Original Part:
Domain: Key Points:
Remember to draw the points in the same order every bit the original to make information technology easier! If you lot’re having problem drawing the graph from the transformed ordered pairs, just take more points from the original graph to map to the new one! 
Transformation:\(\displaystyle f\left( {\frac{one}{2}\left( {tenane} \correct)} \right)three\) \(y\) changes:\(\displaystyle f\left( {\frac{ane}{2}\left( {ten1} \right)} \right)\color{blue}{{\text{ }3}}\) \(10\) changes:\(\displaystyle f\left( {\color{bluish}{{\frac{1}{2}}}\left( {x\text{ }\color{blue}{{\text{ }ane}}} \right)} \right)iii\) Note that this transformation Key Points Transformed: (we practise the “contrary” math with the “\(x\)”)
Transformed Function:
Domain: \(\left[ {9,9} \right]\) 
Transformation: \(y\) changes: \(x\) changes:\(\displaystyle f\left( {\colour{blue}{{\underline{{\left x \correct+i}}}}} \right)2\): Note that this transformation Allow’s just exercise this one via graphs. First, motility down
And then reflect the rightmanus side across the \(y\)axis to make symmetrical. Transformed Function:
Domain: \(\left[ {4,iv} \right]\) 
Writing Transformed Equations from Graphs
You might be asked to
write a transformed equation, give a graph. A lot of times, you lot can merely tell past looking at it, but sometimes yous take to use a bespeak or two. And you do have to be conscientious and check your work, since the order of the transformations tin matter.
Note that when figuring out the transformations from a graph, information technology’s difficult to know whether you have an “\(a\)” (vertical stretch) or a “\(b\)” (horizontal stretch) in the equation \(\displaystyle g\left( x \correct)=a\cdot f\left( {\left( {\frac{1}{b}} \correct)\left( {xh} \right)} \right)+k\). Sometimes the problem volition indicate what parameters (\(a\), \(b\), and then on) to await for. For others, similar polynomials (such as quadratics and cubics), a
vertical stretch
mimics a
horizontal compression, and then it’due south possible to factor out a coefficient to turn a horizontal stretch/compression to a vertical compression/stretch. (For more complicated graphs, yous may want to take several points and perform a
regression in your estimator
to become the office, if you’re allowed to do that).
Hither are some problems. Note that atransformed equation from an absolute value graph
is in the Absolute Value Transformationssection.
Transformed Graph  Getting Equation 
Write the full general equation for the cubic equation in the grade: \(\displaystyle y={{\left( {\frac{ane}{b}\left( {10h} \right)} \right)}^{3}}+k\).

We see that that the heart point, or critical betoken is at \(\left( {four,v} \right)\), so the cubic is in the course: \(\displaystyle y={{\left( {\frac{i}{b}\left( {x+four} \right)} \right)}^{3}}5\). Notice that to go dorsum and over to the next points, we get dorsum/over \(3\) and down/up \(1\), so we encounter in that location’s a horizontal stretch of \(three\), so \(b=3\). (We could have also used some other indicate on the graph to solve for \(b\)). We have \(\displaystyle y={{\left( {\frac{one}{three}\left( {10+4} \correct)} \correct)}^{3}}5\). Try information technology – it works! Annotation that if we wanted this function in the class \(\displaystyle y=a{{\left( {\left( {xh} \right)} \right)}^{three}}+k\), we could use the point \(\left( {seven,halfdozen} \correct)\) to become \(\displaystyle y=a{{\left( {\left( {x+four} \right)} \right)}^{3}}v;\,\,\,\,6=a{{\left( {\left( {vii+iv} \correct)} \correct)}^{3}}v\), or \(\displaystyle a=\frac{1}{{27}}\). This makes sense, since if we brought the \(\displaystyle {{\left( {\frac{1}{three}} \correct)}^{3}}\) out from above, it would exist \(\displaystyle \frac{1}{{27}}\)!) 
Find the equation of this graph in whatsoever course:

Here’south a generic method you can typically use: We see that this is a We demand to find \(a\); employ the bespeak \(\left( {i,10} \right)\): 
Find the equation of this graph in any grade:

The graph looks similar a quadratic with vertex \(\left( {ane,eight} \correct)\), which is a shift of \(8\) down and \(1\) to the left. This volition give united states an equation of the form \(y=a{{\left( {x+ane} \correct)}^{2}}eight\), which is (not and so coincidentally!) the vertex form for quadratics. We demand to detect \(a\); utilise the betoken \(\left( {1,0} \right)\):\(\begin{marshal}y&=a{{\left( {x+1} \correct)}^{2}}eight\\\,0&=a{{\left( {1+i} \right)}^{2}}8\\8&=4a;\,\,a=two\finish{align}\). Note: we could have also noticed that the graph goes over \(1\) and up \(2\) from the vertex, instead of over \(1\) and upwards \(one\) usually with \(y={{x}^{two}}\). This would hateful that our vertical stretch is \(2\). 
Find the equation of this graph in any course:

The graph looks like a rational with the “center” of asymptotes at \(\left( {2,iii} \right)\), which is a shift of 2 to the left and iii upwards. This will give usa an equation of the form \(\displaystyle y=a\left( {\frac{1}{{10+2}}} \right)+iii\), with asymptotes at \(ten=2\) and \(y=3\). We need to find \(a\); use the given bespeak \((0,4)\): \(\begin{align}y&=a\left( {\frac{1}{{x+2}}} \right)+3\\four&=a\left( {\frac{ane}{{0+2}}} \right)+3\\1&=\frac{a}{2};\,a=2\terminate{align}\). The equation of the graph is: \(\displaystyle y=2\left( {\frac{1}{{x+two}}} \right)+three,\,\text{or }y=\frac{two}{{x+2}}+iii\).
Note: we could have also noticed that the graph goes over 
Find the equation of this graph with a base of \(.5\) and horizontal shift of \(1\):

We see that thisexponential graph has a horizontal asymptote at \(y=3\), and with the horizontal shift, we have \(y=a{{\left( {.5} \right)}^{{x+i}}}3\) so far.
When you have a problem similar this, \(\begin{array}{c}y=a{{\left( {.5} \correct)}^{{x+i}}}3;\,\,one=a{{\left( {.5} \right)}^{{0+one}}}three;\,\,\,\,2=.5a;\,\,a=4\\y=four{{\left( {.five} \correct)}^{{ten+1}}}3\terminate{assortment}\) Note that there are more examples of exponential transformations hither in the 
Rotational Transformations
Y’all may be asked to perform a
rotation transformation on a function (you commonly run into these in
Geometry
grade). A rotation of
90°
counterclockwise involves replacing \(\left( {x,y} \right)\) with \(\left( {y,10} \correct)\), a rotation of
180°
counterclockwise involves replacing \(\left( {x,y} \right)\) with \(\left( {ten,y} \right)\), and a rotation of
270°
counterclockwise involves replacing \(\left( {x,y} \right)\) with \(\left( {y,x} \right)\). Here is an instance:
Transformation  Example  Graph  
Rotate graph 270° counterclockwise 
Parent: \(y={{ten}^{2}}\) Replace \((x,y)\) with \((y,–10)\)

Rotated Part Domain: 
Transformations of Inverse Functions
We learned about
Changed Functions
here, and you might be asked to compare original functions and inverse functions, as far as their transformations are concerned. Call back that an changed function is ane where the \(x\) is switched by the \(y\), so the all the transformations originally performed on the \(x\) will be performed on the \(y\):
Problem:
If a cubic function is
vertically stretched by a factor of
3
,
reflected over the
\(\boldsymbol {y}\)axis, and
shifted downwards
2
units, what transformations are washed to its
changed role?
Solution:
We demand to do transformations on the
contrary variable. Thus, the inverse of this office will be
horizontally stretched by a cistron of
iii
,
reflected over the
\(\boldsymbol {ten}\)axis, and
shifted to the left
ii
units. Here is a graph of the two functions:
Note that examples of
Finding Inverses with Restricted Domains
can be constitute
here.
Applications of Parent Function Transformations
You may run across a “word problem” that used Parent Function Transformations, and you tin use what you know nigh how to shift a function. Here is an example:
Transformation Application Trouble  Solution 
The following polynomial graph shows the profit that results from selling math books afterwards September 1. The polynomial is \(p\left( x \right)=5{{10}^{iii}}xx{{x}^{2}}+40x1\), where \(x\) is the number of weeks after September 1.
The publisher of the math books were

Since we’re moving the fourth dimension in weeks past i week, we are shifting the graph horizontally, or shifting the inside, or \(ten\) values. Since our showtime profits volition start a little after calendar week

Acquire these rules, and practise, practice, exercise!
For Practice: Use the
Mathway widget beneath to try aTransformation
trouble. Click on
Submit
(the blue arrow to the right of the problem) and click on
Describe the Transformation to run into the answer.
You tin can likewise type in your own problem, or click on the iii dots in the upper correct hand corner and click on “Examples” to drill downwardly past topic.
If y’all click on
Tap to view steps, or
Click Hither, yous can register at
Mathway
for a
free trial, and then upgrade to a paid subscription at any time (to get any type of math trouble solved!).
On to
Absolute Value Transformations
– you are gear up!
Which Parent Function is Represented by the Graph
Source: https://mathhints.com/parentgraphsandtransformations/