Parent Functions and Transformations

BasicParent Functions	Writing Transformed Equations from Graphs
Generic Transformations of Functions	Rotational Transformations
Vertical Transformations	Transformations of Inverse Functions
Horizontal Transformations	Applications of Parent Function Transformations
Mixed Transformations	More Practice
Transformations Using Functional Notation

ForAbsolute Value Transformations, see theAbsolute Value Transformationssection. Here arelinks to ParentFunction Transformations in other sections: Transformations of Quadratic Functions (quick and easy way);Transformations of Radical Functions;Transformations of Rational Functions; Transformations of ExponentialFunctions;Transformations of Logarithmic Functions; Transformations of Piecewise Functions;Transformations of Trigonometric Functions; Transformations of Inverse Trigonometric Functions

You may not be familiar with all the functions and characteristics in the tables; here are some topics to review:

Whether functions are even, odd, or neither, discussed here 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

You’ll probably study some “popular” parent functions and work with these to learn how to transform functions – how to move and/or resize them. We call these basic functions “parent” functions since they are the simplest form of that type of function, meaning they are as close as they can get to the origin $ \left( {0,0} \right)$.

The chart below provides some basic parent functions that you should be familiar with. I’ve also included the significant points, or critical points, the points with which to graph the parent function. I also sometimes call these the “reference points” or “anchor points”.

Know the shapes of these parent functions well! Even when using t-charts, you must know the general shape of the parent functions in order to know how to transform them correctly!

*The Greatest Integer Function, sometimes called the Step Function, returns the greatest integer less than or equal to a number (think of rounding down to an integer). There’s also a Least Integer Function, indicated by $ y=\left\lceil x \right\rceil $, which returns the least integer greater than or equal to a number (think of rounding up to an integer).

**Notes on End Behavior: To get theend behaviorof a function, we just look at thesmallestandlargest values of $ x$, and see which way the $ y$ is going. Not all functions have end behavior defined; for example, those that go back and forth with the $ y$ values (called “periodic functions”) don’t have end behaviors.
Most of the time, our end behavior looks something like this: $ \displaystyle \begin{array}{l}x\to -\infty \text{, }\,y\to \,\,?\\x\to \infty \text{, }\,\,\,y\to \,\,?\end{array}$ and we have to fill in the $ y$ part. For example, the end behavior for a line with a positive slope is: $ \begin{array}{l}x\to -\infty \text{, }\,y\to -\infty \\x\to \infty \text{, }\,\,\,y\to \infty \end{array}$, and the end behavior for a line with a negative slope is: $ \begin{array}{l}x\to -\infty \text{, }\,y\to \infty \\x\to \infty \text{, }\,\,\,y\to -\infty \end{array}$. One way to think of end behavior is that for $ \displaystyle x\to -\infty $, we look at what’s going on with the $ y$ on the left-hand side of the graph, and for $ \displaystyle x\to \infty $, we look at what’s happening with $ y$ on the right-hand side of the graph.

There are a couple of exceptions; for example, sometimes the $ x$starts at 0 (such as in theradical function), we don’t have the negative portion of the $ x$end behavior. Also, when $ x$starts very close to 0 (such as in in thelog function), we indicate that $ x$is starting from the positive (right) side of 0 (and the $ y$is going down); we indicate this by $ \displaystyle x\to {{0}^{+}}\text{, }\,y\to -\infty $.

Generic Transformations of Functions

Again, the “parent functions” assume that we have the simplest form of the function; in other words, the function either goes through the origin $ \left( {0,0} \right)$, or if it doesn’t go through the origin, it isn’t shifted in any way. When a function is shifted, stretched (or compressed), or flippedin any way from its “parent function“, it is said to be transformed, and is a transformation of a function.

T-charts are extremely useful tools when dealing with transformations of functions. For example, if you know that the quadratic parentfunction $ y={{x}^{2}}$is being transformed 2 units to the right, and 1 unit down (only a shift, not a stretch or a flip), we can create the original t-chart, following by the transformation points on the outside of the original points. Then we can plot the “outside” (new) points to get the newly transformed function:

When looking at the equation of the transformed function, however, we have to be careful. When functions are transformed on the outside of the$ f(x)$ part, you move 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 affect the $ y$ part of the function. When transformations are made on the inside of the $ f(x)$part, you move the function back and forth (but do the “opposite” math – since if you were to isolate the $ x$, you’d move everything to the other side). These are horizontal transformations or translations, and affect the $ x$part of the function.

There are several ways to perform transformations of parent functions; I like to use t-charts, since they work consistently with ever function. And note that in most t-charts, I’ve included more than just the critical points above, just to show the graphs better.

Vertical Transformations

Here are the rules and examples of when functions are transformed on the “outside”(notice that the $ y$values are affected).The t-charts include the points (ordered pairs) of the original parent functions, and also the transformed or shifted points. 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 expensively in the Absolute Value Transformations section!

(More examples here in the Absolute Value Transformation section)

Horizontal Transformations

Here are the rules and examples of when functions are transformed on the “inside” (notice that the $ x$-values are affected). Notice that when the $ x$-values are affected, you do the math in the “opposite” way from what the function looks like: if you’re adding on the inside, you subtract from the $ x$; if you’re subtracting on the inside, you add to the $ x$; if you’re multiplying on the inside, you divide from the $ x$; if you’re dividing on the inside, you multiply to the $ x$. If you have a negative value on the inside, you flip across the $ \boldsymbol{y}$axis (notice that you still multiply the $ x$by $ -1$ just like you do 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 expensively in the Absolute Value Transformations section!

(You may find it interesting is that a vertical stretch behaves the same way as a horizontal compression, and vice versa, since when stretch something upwards, we are making it skinnier.)

Mixed Transformations

Most of the problems you’ll get will involve mixed transformations, or multiple transformations, and we do need to worry about the order in which we perform the transformations. It usually doesn’t matter if we make the $ x$ changes or the $ y$ changes first, but within the $ x$’s and $ y$’s, we need to perform the transformations in the order below. Note that this is sort of similar to the order with PEMDAS(parentheses, exponents, multiplication/division, and addition/subtraction). When performing these rules, the coefficients of the inside $ x$ must be 1; for example, we would need to have $ y={{\left( {4\left( {x+2} \right)} \right)}^{2}}$ instead of $ y={{\left( {4x+8} \right)}^{2}}$ (by factoring). If you didn’t learn it this way, see IMPORTANT NOTE below.

Here is the order. We can do steps 1 and 2 together (order doesn’t actually matter), since we can think of the first 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 like to take the critical points and maybe a few more points of the parent functions, and perform all thetransformations at the same time with a t-chart! We just do the multiplication/division first on the $ x$ or $ y$ points, followed by addition/subtraction. It makes it much easier!Note again that since we don’t have an $ \boldsymbol {x}$ “by itself” (coefficient of 1) on the inside, we have to get it that way by factoring!For example,we’d have to change$ y={{\left( {4x+8} \right)}^{2}}\text{ to }y={{\left( {4\left( {x+2} \right)} \right)}^{2}}$.

Let’s try to graph this “complicated” equation and I’ll show you how easy it is to do with a t-chart: $ \displaystyle f(x)=-3{{\left( {2x+8} \right)}^{2}}+10$. (Note that for this example, we could move the $ {{2}^{2}}$ to the outside to get a vertical stretch of $ 3\left( {{{2}^{2}}} \right)=12$, but we can’t do that for many functions.) We first need to get the $ x$by itself on the inside by factoring, so we can perform the horizontal translations. This is what we end up with: $ \displaystyle f(x)=-3{{\left( {2\left( {x+4} \right)} \right)}^{2}}+10$. Look at what’s done on the “outside” (for the $ y$’s) and make all the moves at once, by following the exact math. Then look at what we do on the “inside” (for the $ x$’s) and make all the moves at once, but do the opposite math. We do this with a t-chart.

Start with the parent function $ f(x)={{x}^{2}}$. If we look at what we’re doing on the outside of what is being squared, which is the $ \displaystyle \left( {2\left( {x+4} \right)} \right)$, we’re flipping it across the $ x$-axis (the minus sign), stretching it by a factor of 3, and adding 10 (shifting up 10). These are the things that we are doing vertically, or to the $ y$. If we look at what we are doing on the inside of what we’re squaring, we’re multiplying it by 2, which means we have to divide by 2(horizontal compression by a factor of $ \displaystyle \frac{1}{2}$), and we’re adding 4, which means we have to subtract 4 (a left shift of 4). Remember that we do the opposite when we’re dealing with the $ x$. Also remember that we always have to do the multiplication or division first with our points, and then the adding and subtracting (sort of like PEMDAS).

Here is the t-chart with the original function, and then the transformations on the outsides. Now we can 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 t-chart, as we did in the here in the Introduction to Quadratics section.

IMPORTANT NOTE:In some books, for$ \displaystyle f\left( x \right)=-3{{\left( {2x+8} \right)}^{2}}+10$, they may NOT have you factor out the2on the inside, but just switch the order of the transformation on the $ \boldsymbol{x}$.

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 Functionsand Logarithmic Functions sections. Also, the last type of function is a rational function that will be discussed in the Rational Functions section.

Here’s a mixed transformation with the Greatest Integer Function (sometimes called the Floor Function). Note how we can use intervals as the $ x$ values to make the transformed function easier to draw:

Transformations Using Functional Notation

You might see mixed transformations in the form $ \displaystyle g\left( x \right)=a\cdot f\left( {\left( {\frac{1}{b}} \right)\left( {x-h} \right)} \right)+k$, 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, we have the coordinate rule $ \displaystyle \left( {x,y} \right)\to \left( {bx+h,\,ay+k} \right)$. For example, for the transformation $ \displaystyle f(x)=-3{{\left( {2\left( {x+4} \right)} \right)}^{2}}+10$, we have $ a=-3$, $ \displaystyle b=\frac{1}{2}\,\,\text{or}\,\,.5$, $ h=-4$, and $ k=10$. Our transformation $ g\left( x \right)=-3f\left( {2\left( {x+4} \right)} \right)+10$ $ \displaystyle =-3f\left( {\left( {\frac{1}{{\frac{1}{2}}}} \right)\left( {x-\left( {-4} \right)} \right)} \right)+10$ would result in a coordinate rule of $ {\left( {x,\,y} \right)\to \left( {.5x-4,-3y+10} \right)}$. (You may also see this as $ g\left( x \right)=a\cdot f\left( {b\left( {x-h} \right)} \right)+k$, with coordinate rule $ \displaystyle \left( {x,\,y} \right)\to \left( {\frac{1}{b}x+h,\,ay+k} \right)$; the end result 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( x \right)$, give the transformed coordinates for the point on the graph $ y=-6g\left( {-2x} \right)-2$. To do this, to get the transformed $ y$, multiply the $ y$ part of the point by –6 and then subtract 2. To get the transformed $ x$, multiply the $ x$ part of the point by $ \displaystyle -\frac{1}{2}$ (opposite math). The new point is $ \left( {-4,10} \right)$. Let’s do another example: If the point $ \left( {-4,1} \right)$ is on the graph $ y=g\left( x \right)$, the transformed coordinates for the point on the graph of

$ \displaystyle \begin{align}y&=2g\left( {-3x-2} \right)+3\\&=2g\left( {-3\left( {x+\frac{2}{3}} \right)} \right)+3\end{align}$ is $ \displaystyle \begin{align}&\left( {-4,1} \right)\to \left( {\left( {-\frac{1}{3}} \right)\left( {-4} \right)-\frac{2}{3},2\left( 1 \right)+3} \right)\\&=\left( {\frac{2}{3},5} \right)\end{align}$ , using coordinate rules $ \displaystyle \begin{align}&\left( {x,\,y} \right)\to \left( {\frac{1}{b}x+h,\,\,ay+k} \right)\\\,\,\,&=\left( {-\frac{1}{3}x-\frac{2}{3},\,\,2y+3} \right)\end{align}$

You may also be asked to transform a parent or non-parent equation to get a new equation. We can do this without using a t-chart, but by using substitution and algebra. For example, if we want to transform $ f\left( x \right)={{x}^{2}}+4$ using the transformation $ \displaystyle -2f\left( {x-1} \right)+3$, we can just substitute “$ x-1$” for “$ x$” in the original equation, multiply by –2, and then add 3. For example: $ \displaystyle \begin{align}-2f\left( {x-1} \right)+3&=-2\left[ {{{{\left( {x-1} \right)}}^{2}}+4} \right]+3\\&=-2\left( {{{x}^{2}}-2x+1+4} \right)+3\\&=-2{{x}^{2}}+4x-7\end{align}$. We used this method to help transform a piecewise function here.

Transformations in Function 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 case, you’ll probably be 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 correct “shape”.

Here are some examples; the second example is the transformation with an absolute value on the $ x$; see the Absolute Value Transformations section for more detail.

(More examples here in the Absolute Value Transformation section)

Writing Transformed Equations from Graphs

You might be asked to write a transformed equation, give a graph. A lot of times, you can just tell by looking at it, but sometimes you have to use a point or two. And you do have to be careful and check your work, since the order of the transformations can matter.

Note that when figuring out the transformations from a graph, it’s difficult to know whether you have an “$ a$” (vertical stretch) or a “$ b$” (horizontal stretch) in the equation $ \displaystyle g\left( x \right)=a\cdot f\left( {\left( {\frac{1}{b}} \right)\left( {x-h} \right)} \right)+k$. Sometimes the problem will indicate what parameters ($ a$, $ b$, and so on)to look for. For others, like polynomials (such as quadratics and cubics), a vertical stretch mimics a horizontal compression, so it’s possible to factor out a coefficient to turn a horizontal stretch/compression to a vertical compression/stretch. (For more complicated graphs, you may want to take several points and perform a regression in your calculator to get the function, if you’re allowed to do that).

Here are some problems. Note that atransformed equation from an absolute value graph is in theAbsolute Value Transformationssection.

Note that there are more examples of exponential transformations here in the Exponential Functions section, and logarithmic transformations here in the Logarithmic Functions section.

Rotational Transformations

You may be asked to perform a rotationtransformation on a function (you usually see these in Geometry class). A rotation of 90° counterclockwise involves replacing $ \left( {x,y} \right)$ with $ \left( {-y,x} \right)$, a rotation of 180° counterclockwise involves replacing $ \left( {x,y} \right)$ with $ \left( {-x,-y} \right)$, and a rotation of 270° counterclockwise involves replacing $ \left( {x,y} \right)$ with $ \left( {y,-x} \right)$. Here is an example:

Transformations of Inverse Functions

We learned about Inverse Functions here, and you might be asked to compare original functions and inverse functions, as far as their transformations are concerned. Remember that an inverse function is one 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 down 2 units, what transformations are done to its inverse function?
Solution:
We need to do transformations on the opposite variable. Thus, the inverse of this function will be horizontally stretched by a factor of 3, reflected over the $ \boldsymbol {x}$-axis, and shifted to the left 2 units. Here is a graph of the two functions:

Note that examples of Finding Inverses with Restricted Domains can be found here.

Applications of Parent Function Transformations

You may see a “word problem” that used Parent Function Transformations, and you can use what you know about how to shift a function. Here is an example:

Learn these rules, and practice, practice, practice!

For Practice: Use the Mathwaywidget below to try aTransformation problem. Click on Submit (the blue arrow to the right of the problem) and click on Describe the Transformationto see the answer.

You can also type in your own problem, or click on the threedots in the upper right hand corner and click on “Examples” to drill down by topic.

If you click on Tap to view steps, or Click Here, you can register at Mathway for a free trial, and then upgrade to a paid subscription at any time (to getany type of math problem solved!).

On to Absolute Value Transformations – you are ready!

Parent Functions and Transformations | mathhints.com (2024)