This section covers:
- Basic Parent Functions
- Generic Transformations of Functions
- Vertical Transformations
- Horizontal Transformations
- Mixed Transformations
- Transformations in Function Notation
- Writing Transformed Equations from Graphs
- Transformations of Inverse Functions
- Absolute Value Transformations
Note that examples of transformations of Rationals can be found here, and more examples of transformations of Exponential and Log Functions can be found here. Also, Transformations of Trig Functions can be found here, and Transformations of the Inverse Trig Functions can be found here.
Basic Parent Functions
You’ll probably study some “popular” parent functions and work with these to learn how to transform functions – how to move them around. 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 (0, 0).
The chart below provides some basic parent functions that you should be familiar with. It also includes the domain and range of each function, and if they are even, odd, or neither, which we learned here in the Compositions of Functions, Even and Odd, Increasing and Decreasing Section.
I know we haven’t covered all these yet (sorry!), especially the exponential and log functions, which you’ll learn about in the Exponential Functions and Logarithmic Functions sections. You’ll also learn about how to transform a piecewise function here in the Piecewise Functions section.
We haven’t really talked about end behavior, but we will in the Asymptotes and Graphing Rational Functions and Graphing Polynomials sections. To get the end behavior, we just look at the smallest and largest 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 and never really go way up or way down (called “periodic functions”) don’t have end behaviors.
Most of the time, our end behavior looks something like this: and we have to fill in the y part. So the end behavior for a line with a positive slope is:
There are a couple of exceptions; for example, sometimes the x starts at 0 (such as in the radical function), we don’t have the negative portion of the x end behavior. Also, when x starts very close to zero (such as in in the log function), we indicate that x is starting from the positive (right) side of 0 (and the y is going down); we indicate this by .
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 (0, 0), or if it doesn’t go through the origin, it isn’t shifted in any way.
When a function is shifted in any way from its “parent function“, it is said to be transformed, and is a transformation of a function. Functions are typically transformed either vertically or horizontally.
T-charts are extremely useful tools when dealing with transformations of functions. For example, if you know that the linear function y = x is being transformed 2 units to the right, and 1 unit down, 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” points to get the newly transformed function:
When looking at the equation of the function, however, we have to be careful.
When functions are transformed on the outside of the 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.
When transformations are made on the inside of the part, you move the function back and forth (but do the “opposite” math – basically since if you were to isolate the x, you’d move everything to the other side). These are horizontal transformations or translations.
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.
Notice that the first two transformations are translations, the third is a dilation, and the last is a reflection.
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 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 y axis.
I didn’t include the absolute value function for the horizontal flip, since it will just be the same function!
Notice that the first two transformations are translations, the third is a dilation, and the last is a reflection.
You may be asked to perform a rotation transformation on a function (you usually see these in Geometry class). A rotation of 90° counterclockwise involves replacing (x, y) with (–y, x), a rotation of 180° counterclockwise involves replacing (x, y) with (–x, –y), and a rotation of 270° counterclockwise involves replacing (x, y) with (y, –x).
Here is an example:
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 following order. Note that this is sort of similar to the order with PEMDAS (parentheses, exponents, multiplication/division, and addition subtraction). Note that when using these rules, the coefficients of the inside x’s must be 1; for example, we would need to have instead of .
- 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).
But we can do steps 1 and 2 together, since we can think of the first two steps as a “negative stretch.”
Now we can do all the transformations 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 that since we don’t have an 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
Let’s try to graph this “complicated” equation and I’ll show you how easy it is to do with a t-chart:
Now, what we need to do is to 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 we can look 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.
So we’re starting with the function . If we look at what we’re doing on the outside of what is being squared, which is the , we’re flipping it (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.
Now 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 ½), 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.
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:
NOTE: In some books, for , they may have you not factor out the 2 on the inside, but just switch the order of the transformation on the x. In these cases, the order of transformations would be horizontal shifts, horizontal reflections/stretches, vertical reflections/stretches, and then vertical shifts. So for this problem, you would move to the right 8 first, and then compress with a factor of ½ (which is opposite of PEMDAS). Then you would perform the y (vertical) changes the regular way.
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”.
Let’s do an example. Let’s use the transformation , given the following points and graph. Remember that the transformations inside the parentheses are done to the x (doing the opposite math), and outside are done to the y.
Note that this transformation take the original function, flips it around the y axis, performs a horizontal stretch by 2, moves it right by 1, and then down by 3.
Just remember if you’re having trouble drawing the graph from the transformed ordered pairs, just take more points from the original graph to map to the new one!
Functional Notation Transformation Using Algebra
Let’s say we want to use a “function notation” transformation to transform a parent or non-parent equation. We can do this without using a t-chart, but by using substitution and algebra.
For example, if we want to transform using the transformation , we can just substitute “x – 1” for “x” in the original equation, multiply by –2, and then add 3.
For example: .
We used this method to help transform a piecewise function here.
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 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:
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.
The second example was found here in the Solving Quadratics by Factoring and Completing the Square section; the last will be shown here in the Exponential Functions section.
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:
If a cubic function is vertically stretched by a factor of 3, reflected over the y axis, and shifted down 2 units, what transformations are done to its inverse function?
We need to do transformations on the opposite variable. So the inverse of this function will be horizontally stretched by a factor of 3, reflected over the 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.
Absolute Value Transformations
Now let’s try some of the absolute value shifts, first using the absolute value parent function. Here is an example:
Now let’s look at taking the absolute value of functions, both on the outside (affecting the y’s) and the inside (affecting the x’s). These are a little trickier.
Let’s look at a function of points, and see what happens when we take the absolute value of the function “on the outside” and then “on the inside”. Then we’ll show absolute value transformations using parent functions.
Here are absolute value examples with parent functions:
Note: These mixed transformations with absolute value are very tricky; it’s really difficult to know what order to use to perform them. The best thing to do is to play around with them on your graphing calculator to see what’s going on.
For example, we saw that with , we performed the x absolute value function last (after the shift). I also noticed that with , you perform the x absolute value transformation first (before the shift).
With something like , you perform the y absolute value function first (before the shift); with something like , you perform the y absolute value last (after the shift). (These two make sense, when you look at where the absolute value functions are.) I don’t think you’ll get this detailed with your transformations, but you can see how complicated this can get!
Here’s an example where we’re using what we know about the absolute value transformation, but we’re using it on an absolute value parent function! Pretty crazy, huh?
More Absolute Value Transformations
What about ? Play around with this in your calculator with , for example. You’ll see that it shouldn’t matter which absolute value function you apply first, but it certainly doesn’t hurt to work from the inside out. And with , it’s a good idea to perform the inside absolute value first, then the outside, and then the flip across the x axis. So the rule of thumb with these absolute value functions and reflections is to move from the inside out.
Let’s do more complicated examples with absolute value and flipping – sorry that this stuff is so complicated! Just be careful about the order by trying real functions in your calculator to see what happens. These are for the more advanced Pre-Calculus classes!
Learn these rules, and practice, practice, practice!
On to Piecewise Functions – you are ready!