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.

You’ll learn about the **trigonometric** parent functions and transformations here in the **Graphs of Trig Functions** and **Transformations of Trig Functions** sections, respectively.

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

**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.**

*y*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

**end behavior. Also, when**

*x***starts very close to zero (such as in in the**

*x***log function**), we indicate that

**is starting from the positive (**

*x***right**) side of 0 (and the

**is going down); we indicate this by .**

*y*## 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

**, you’d move everything to the other side). These are**

*x***horizontal transformations**or

**translations**.

### 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.

Notice that the first two transformations are **translations**, the third is a **dilation**, and the last is a** reflection**.

### 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 you’re adding on the inside, you subtract from the

**; 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**

*x***flip across the**.

*y*axisI 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:

## 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

**changes first, but within the**

*y***’s and**

*x***’s, we need to perform the transformations in the following order. Note that this is sort of similar to the order with**

*y***PEMDAS**(parentheses, exponents, multiplication/division, and addition subtraction). Note that when using these rules, the coefficients of the inside

**’s must be 1; for example, we would need to have instead of .**

*x*- 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

**points, followed by addition/subtraction. It makes it much easier!**

*y***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:

We first need to get the ** x by itself** on the inside, so we can perform the horizontal translations. So this is what we end up with:

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

**’s) and make all the moves at once,**

*x***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

**(vertical) changes the regular way.**

*y*### 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

**, so the all the transformations originally performed on the**

*y***will be performed on the**

*x***:**

*y***Problem:**

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**?

**Solution:**

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

**’s). These are a little trickier.**

*x*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

**absolute value transformation first (before the shift).**

*x*With something like , you perform the ** y** absolute value function first (before the shift); with something like , you perform the

**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!**

*y*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!

Thank you so much. This helps a lot!

i had a huge project on this and your page is the only ting that really helped me understand parent functions and their translations thank you very much.

I’m so glad I could help! Thanks so much for your comment, and please spread the word about She Loves Math Lisa

This is incredibly helpful! I’ve been trying to understand mixed transformations for the past two days with minimal success. Your explanations are brilliantly clear and I get it! Thank you so much.

Excellent resource for my students! You cover the topic from Algebra 1 through PreCalc, and your explanations, notations, and graphic representations are clear, mathematically correct, and easy to follow. Girls in math ROCK! [Ok, so boys in math aren’t bad, either.]

this website dont work it drag me to get the wrong anser on my homework please check this website all over thankyou

Hi Nancy!

So sorry to hear that the site gave wrong answers. Could you please give me some examples, so I can check them?

Thanks,

Lisa

Hi Lisa, really I can’t explain how this informations are very helpful to me. I was really confusing and I couldn’t understand the parent functions and their relative. But now every thing is clear. I will keep this as a reference. thanks Lisa.

so for example what will be the parent function of g(x)= -2^x+1 and how is the graph transformed from its parent function?

The parent function would be g(x) = 2^x. It would be flipped across the x axis and then shifted up 1. Does this make sense?

Nice Website! I love it! I have a 1 month project on parent functions. We have to use at least 7 different parent functions that are consistent and we have to use at least 10 different parent functions, all of this to make an image of something we want. Ex: Use parent functions to draw a cow. *sigh, it will be very challenging, but I have this amazing website to help me out with some functions!

Thanks so much for the nice note – that sounds like a challenging assignment. Hope the website helps! Let me know if you have any suggestions for it. Thanks again Lisa

thankyouu soooo muchh for thisss !!!!!!!!!!!! bless you

I am a math/chem tutor and I am often confronted with some math problems I haven’t seen in awhile. Thanks for helping me brush up on function transformations. Easy to follow and very useful.

Thanks so much for writing! That’s actually why I started writing the site; I had all this stuff in my head, and wanted to get it all down “on paper” so I could remember it from year to year. I’ll keep writing Lisa

I love this website!

One thing that would help me in particular to visualize the transformations, would be if the parent graph was in one color, and the transformed graph was in another color!

Great suggestion – thanks!!! Lisa

This helped me so much! Thank you! I know what I’m doing now. I am able to do my math analysis homework!

I need help translating this f(x) of g(x)= 2|x+2|-3

Hello,

This is a transformation of the absolute value function y = |x| that is 2 to the left, 3 down, and stretched by a factor of 2:

Hope this helps!

Lisa

What is lower bound, upper bound, and bounded?

Does this help? http://en.m.wikipedia.org/wiki/Upper_and_lower_bounds

Lisa

My Algebra 2 teacher said that the graphing order is reflection over the y axis, then horizantal shifts, reflection over x axis , then stretch and compression and last is vertical shifts

Thanks for writing! This is actually correct and I should update accordingly. But I found that the x transformations and y transformations are independent (don’t really affect each other), so you should get the same result as what I said. But remember that if you are shifting on the x and you have something like y = (2x + 4)^2, if you don’t take the 2 out and make it y = (2(x + 2))^2, you can shift to the left 4 first and then compress by 1/2 (opposite of PEMDAS).

This helped me a lot with my Algebra Two test tomorrow! Thank you

Please help!! Project directions says ” write a T chart for each transformation:reflections,translation, rotation, and dilation. Label the vertices of your original figure and image”. I’m the students aunt and I’m so lost!! Thank you..

Hello, and thanks for writing! I actually just added information about reflections, translations, rotations, and dilations in the Parent Functions and Transformations Section, since you ask great questions! See if it makes sense and let me know if you have any questions Lisa

Excellent! Thanks for organizing and publishing!

I’d like to know the process whereby you have points, and you know the general transformations are going to be as yours, i.e., how to do these problems in reverse.

Here’s an example with the answer. Given original points for f(x) are (-10,2), (-5,8), (0,2), (2,5) and (7,5) and transformed points are (-7, -3), (-2,9), (3,-3), (5,3) and (10,3).

The answer is the new function h(x) = 2f(x-3) -7. What’s a systematic way to do this?

Thanks for writing! You can find the linear relationship between the 2 sets of data. So, for the x values, let’s take a few points to see if we see a relationship: -10 and -7, -5 and -2. We can sort of see that the second is just the first + 3 (-10 + 3 = -7). But we could also get the “line” between the 2 points by using y = mx + b, getting slope first (-7 – -2)/(-10 – -5) = 1. Then you can get b by putting in a point: (-7 = 1x -10) : b = 3. So the first relationship (of the x’s is x + 3). Do the same for the y’s to get “2x – 7″ (use 2 points and equation for a line).

So now we have the x’s are 3 more, and the y’s are 7 less than 2 times.

So the transformation would be 2(f-3) – 7 (remember that we do the opposite math for the x’s, so we subtract 3 instead of add it).

Does this make sense? Lisa

f(x) = x – 5. Write the function rule for g(x), which is f(x) translated two units upward.

Type your answer as a function rule for g(x), like this: g(x) = 14x – 12 how can i do this ?

Thanks for writing! For this problem, I got g(x) = f(x) + 2. When you move a function up two units, you add 2 to the y, so g(x) would be g(x) = x – 5 + 2, or g(x) = x – 3. Does this make sense? Lisa

is the ∞ the same as saying all real numbers

Yes, the interval (-infinity, infinity) is the same as all real numbers. Lisa

I am a learning differences teacher, so I have to find other ways for my students to ‘see’ the big idea. This site is so neatly presented and clearly defined that I am bookmarking it for MY use. I am trying to make visual aids to support their understanding of transformations and how the changes can be understood by where the equation is changed. The classroom teacher is using linear equations first for transformations and will build in quadratic and exponential functions. Thanks!

Hello, I need to identify the transformations of the graph f(x)=x^3.

1. g(x) = 2(x-3)^4 +4

I wrote the following: ]

– up 4 units

– right 3 units

– horizontal compression of 2 ( the teacher marked this part wrong or crossed it out)

2. g(x) = -1/3 (x + 5) – 1

I wrote: down 1 unit, left 5 units, facing opposite because of negative,

3. g(x) = (5 (x + 6)^3 )

I wrote: left 6 units

vertical compression (the teacher crossed this part out)

For the first problem, since the 2 is on the outside, your teacher probably wanted “a vertical stretch of 2″. For the second problem, you need to add “vertical compression of 1/3″. For the third problem, since the 5 is on the inside, it would be a “horizontal compression of 1/5″. Does this make sense?

I need help with this problem:

The parent function, f(x)=|x|, has been transformed into g(x)=¼|x-6|+10. Describe the transformation of f(x). (In correct order).

Thanks for writing! The transformations on the y are to shrink the absolute value function by a factor of 4 (or stretch by 1/4) and add 10. The transformation on the x is to move the graph 6 units to the right. The order would be move the graph to the right, shrink the y, and then add 10 to the y. Does that make sense? Lisa

I understand that transformations are done in PEMDAS order, but my textbook has the following examples that seem to contradict this:

1. f(x) = 3^(3x-5); the given solution is to translate 5 units right followed by a horizontal shrink by factor of 1/3.

2. h(x) = 3^[3(x-5)]; the given solution is a vertical shrink by a factor of 1/3 followed by a translation 5 units right .

Any help would be greatly appreciated.

Bryan

Bryan,

Thanks for writing! You have a solved a great mystery for me! Yes, your book is correct. If you don’t take the 3 out (like make it 3(x – 5/3), like I suggest), THEN you move to the right first, and do the horizontal shrink by 1/3. I advise to make the coefficient 1 of the x like I do with 3(x – 5/3), so you can follow PEMDAS. But now I understand why some say to do it your way – and I’ve added a note about this in my web page. THANKS SO MUCH!! Lisa (does this make sense?)

Beautiful! Thanks for the quick and helpful response.