Parent Functions and Transformations

Before we get started, here are links to Parent Function Transformations in other sections:

This section covers:

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

Notes on End Behavior: To get the end behavior of a function, 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:\(\displaystyle \begin{array}{l}x\to -\infty \text{, }\,y\to ?\\x\to \infty \text{, }\,\,\,y\to ?\end{array}\) and we have to fill in the y part.  So the end behavior for a line with a positive slope is: \(\begin{array}{l}x\to -\infty \text{, }\,y\to -\infty \\x\to \infty \text{, }\,\,\,y\to \infty \end{array}\).

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 \(\displaystyle x\to {{0}^{+}}\text{, }\,y\to -\infty \).

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 \(\left( {0,\,0} \right)\). 

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

Parent Function


Parent Function


Linear, Odd


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


End Behavior:
\(\begin{array}{l}x\to -\infty \text{, }\,y\to -\infty \\x\to \infty \text{, }\,\,\,y\to \infty \end{array}\)

Critical points: \(\displaystyle \left( {-1,\,-1} \right),\,\left( {0,\,0} \right),\,\left( {1,\,1} \right)\)


\(y=\left| x \right|\)
Absolute Value, Even


Domain: \(\left( {-\infty ,\infty } \right)\)
Range: \(\left[ {0,\infty } \right)\)


End Behavior:
\(\begin{array}{l}x\to -\infty \text{, }\,y\to \infty \\x\to \infty \text{, }\,\,\,y\to \infty \end{array}\)

Critical points: \(\displaystyle \left( {-1,\,1} \right),\,\left( {0,\,0} \right),\,\left( {1,\,1} \right)\)

 Y=absolute Value

Quadratic, Even


Domain: \(\left( {-\infty ,\infty } \right)\)
Range: \(\left[ {0,\infty } \right)\)


End Behavior:
\(\begin{array}{l}x\to -\infty \text{, }\,y\to \infty \\x\to \infty \text{, }\,\,\,y\to \infty \end{array}\)

Critical points: \(\displaystyle \left( {-1,\,1} \right),\,\left( {0,\,0} \right),\,\left( {1,\,1} \right)\)

 Y = X Squared

Radical (Square Root), Neither


Domain: \(\left[ {0,\infty } \right)\)
Range: \(\left[ {0,\infty } \right)\)


End Behavior:
\(\displaystyle \begin{array}{l}x\to 0,\,\,\,\,y\to 0\\x\to \infty \text{, }\,\,\,y\to \infty \end{array}\)

Critical points: \(\displaystyle \left( {-1,\,1} \right),\,\left( {0,\,0} \right),\,\left( {1,\,1} \right)\)

 Y=square Root

Cubic, Odd


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


End Behavior:
\(\begin{array}{l}x\to -\infty \text{, }\,y\to -\infty \\x\to \infty \text{, }\,\,\,y\to \infty \end{array}\)

Critical points: \(\displaystyle \left( {-1,\,-1} \right),\,\left( {0,\,0} \right),\,\left( {1,\,1} \right)\)

 Y = X Cubed

Cube Root, Odd


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


End Behavior:
\(\begin{array}{l}x\to -\infty \text{, }\,y\to -\infty \\x\to \infty \text{, }\,\,\,y\to \infty \end{array}\)

Critical points: \(\displaystyle \left( {-1,\,-1} \right),\,\left( {0,\,0} \right),\,\left( {1,\,1} \right)\)

 Y=cube Root


Exponential, Neither


Domain: \(\left( {-\infty ,\infty } \right)\)
Range: \(\left( {0,\infty } \right)\)


End Behavior:
\(\begin{array}{l}x\to -\infty \text{, }\,y\to 0\\x\to \infty \text{, }\,\,\,y\to \infty \end{array}\)

Critical points: \(\displaystyle \left( {-1,\,\frac{1}{b}} \right),\,\left( {0,\,1} \right),\,\left( {1,\,b} \right)\)

Asymptote:  \(y=0\)

 Y=2 To The X

\(\begin{array}{c}y={{\log }_{b}}\left( x \right),\,\,b>1\,\,\,\\(y={{\log }_{2}}x)\end{array}\)

Log, Neither


Domain: \(\left( {0,\infty } \right)\)
Range: \(\left( {-\infty ,\infty } \right)\)


End Behavior:
\(\begin{array}{l}x\to {{0}^{+}}\text{, }\,y\to -\infty \\x\to \infty \text{, }\,y\to \infty \end{array}\)


Critical points: \(\displaystyle \left( {\frac{1}{b},\,-1} \right),\,\left( {1,\,0} \right),\,\left( {b,\,1} \right)\)

Asymptote: \(x=0\)


\(\displaystyle y=\frac{1}{x}\)


Rational (Inverse), Odd


Domain: \(\left( {-\infty ,0} \right)\cup \left( {0,\infty } \right)\)
Range: \(\left( {-\infty ,0} \right)\cup \left( {0,\infty } \right)\)


End Behavior:
\(\begin{array}{l}x\to -\infty \text{, }\,y\to 0\\x\to \infty \text{, }\,\,\,y\to 0\end{array}\)

Critical points: \(\displaystyle \left( {-1,\,-1} \right),\,\left( {1,\,1} \right)\)

Asymptotes: \(y=0,\,\,x=0\)

 1 Over X

\(\displaystyle y=\frac{1}{{{{x}^{2}}}}\)


Rational (Inverse Squared), Even


Domain: \(\left( {-\infty ,0} \right)\cup \left( {0,\infty } \right)\)
Range: \(\left( {0,\infty } \right)\)


End Behavior:
\(\begin{array}{l}x\to -\infty \text{, }\,y\to 0\\x\to \infty \text{, }\,\,\,y\to 0\end{array}\)

Critical points: \(\displaystyle \left( {-1,\,1} \right),\,\left( {1,\,1} \right)\)

Asymptotes: \(x=0,\,\,y=0\)

 1 Over X Squared

\(y=\text{int}\left( x \right)\text{=}\left[ \text{x} \right]\)


Greatest Integer, Neither

Domain:\(\left( {-\infty ,\infty } \right)\)
Range: \(\{y:y\in \mathbb{Z}\}\text{ (integers)}\)

End Behavior:
\(\begin{array}{l}x\to -\infty \text{, }\,y\to -\infty \\x\to \infty \text{, }\,\,\,y\to \infty \end{array}\)

Critical points: \(\displaystyle \begin{array}{l}x:\left[ {-1,\,0} \right)\,\,\,y:-1\\x:\left[ {0,\,1} \right)\,\,\,y:0\\x:\left[ {1,\,2} \right)\,\,\,y:1\end{array}\)



\(y=C\)   (\(y=2\))


Constant, Even

Domain: \(\left( {-\infty ,\infty } \right)\)
Range: \(\{y:y=C\}\)


End Behavior:
\(\begin{array}{l}x\to -\infty \text{, }\,y\to C\\x\to \infty \text{, }\,\,\,y\to C\end{array}\)

Critical points: \(\displaystyle \left( {-1,\,C} \right),\,\left( {0,\,C} \right),\,\left( {1,\,C} \right)\)


Constant Function




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 flipped 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 quadratic parent function \(y={{x}^{2}}\) 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” (new) points to get the newly transformed function:




Quadratic Function




Transform function 2 units to the right, and 1 unit down.


First T Chart


Transformation Of Function T Chart Squared


When looking at the equation of the moved 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.

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.

Vertical Transformations

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 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 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.)

Horizontal Transformations

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 following order. (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:

  1. Perform Flipping across the axes first  (negative signs).
  2. Perform Stretching and Shrinking next (multiplying and dividing).
  3. Perform Horizontal and Vertical shifts last (adding and subtracting).

But 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.”

I like to take the critical points and maybe a few more points of the parent functions, and perform 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 again 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 \(y={{\left( {4x+8} \right)}^{2}}\text{ to }y={{\left( {4\left( {x+2} \right)} \right)}^{2}}\).

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 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: 

\(\displaystyle f(x)=-3{{\left( {2\left( {x+4} \right)} \right)}^{2}}+10\)

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 \(f(x)={{x}^{2}}\). 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 (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 of \(\frac{1}{2}\)), 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).

Note: You might see mixed transformations in the form \(g\left( x \right)=a\cdot f\left( {\left( {\frac{1}{b}} \right)\left( {x-h} \right)} \right)+k\), or with a coordinate rule \(\left( {x,\,y} \right)\to \left( {\frac{1}{b}x+h,\,\,ay+k} \right)\), 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. Our transformation is \(g\left( x \right)=-3f\left( {\left( {\frac{1}{2}} \right)\left( {x-4} \right)} \right)+10\), or, using coordinates, \(\boldsymbol{{\left( {x,\,y} \right)\to \left( {.5x-4,\,-3y+10} \right)}}\).

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: 


Transformed Graph

Parent:  \(y={{x}^{2}}\)  (Quadratic)

 Transformed:  \(\displaystyle f(x)=-3{{\left( {2\left( {x+4} \right)} \right)}^{2}}+10\)

 y changes:      \(\displaystyle f(x)=\color{blue}{{-3}}{{\left( {2\left( {x+4} \right)} \right)}^{2}}\color{blue}{+10}\)  

 x changes:    \(\displaystyle f(x)=-3{{\left( {\color{blue}{2}\left( {x\text{ }\color{blue}{{+\text{ }4}}} \right)} \right)}^{2}}+10\)


Opposite for x, “regular” for y:

\(\left( {x,\,y} \right)\to \left( {.5x-4,\,-3y+10} \right)\)

Tchart Mixed Trans Example

Mixed Transformation Example

Domain:    \(\left( {-\infty ,\infty } \right)\)   Range: \(\left( {-\infty ,10} \right]\)


Note: Since this is a parabola, we could have also graphed the transformation by noticing (by the vertex form) that the vertex is \(\left( {-4,\,10} \right)\), like we did in the Introduction to Quadratics section here. Then, by moving the \({{2}^{2}}\) to the outside to make a vertical stretch of 12, we could go over (and back) 1 and down 12 from the vertex to get other points. Cool!

IMPORTANT NOTE In some books, for \(\displaystyle f\left( x \right)=-3{{\left( {2x+8} \right)}^{2}}+10\), 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. For example, for this problem, you would move to the left 8 first for the x, and then compress with a factor of ½ for the x (which is opposite of PEMDAS). Then you would perform the y (vertical) changes the regular way – reflect and stretch by 3 first, and then shift up 10.

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 Mixed Tranformation Function Form, 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 takes 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.

Transformation of Graph

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 \(f\left( x \right)={{x}^{2}}+4\) 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: \(\displaystyle -2f\left( {x-1} \right)+3=-2\left( {{{{\left( {x-1} \right)}}^{2}}+4} \right)+3\) \(\displaystyle =-2\left( {{{x}^{2}}-2x+1+4} \right)+3=-2{{x}^{2}}+4x-7\).

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, and their transformations are discussed in more detail in the Transformations, Inverses, Compositions, and Inequalities of Exponents/Logs section.

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

Mixed Transformations of Functions

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:

Transformation of Greatest Integer Function

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.

Note that when figuring out the transformations from a graph, it’s difficult to know whether you have an “a” or a “b” in the equation \(g\left( x \right)=a\cdot f\left( {\left( {\frac{1}{b}} \right)\left( {x-h} \right)} \right)+k\). Since a vertical stretch (“a”) is really the same thing as a horizontal compression (\(\frac{1}{b}\)), for simpler graphs like cubics and quadratics below, we can just look for an “a” (vertical stretch), and ignore the “b” part (by pulling the “a” outside of the function). Or sometimes, you’ll be asked to write a function from a transformed graph, but they will tell you what parameters (a, b, and so on) to look for. For more complicated graphs, you may want to take several points and perform a regression in your calculator to get the function.

We will find a transformed equation from an absolute value graph in the Absolute Value section below:

Getting Equation from Graph


Rotational Transformations

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:

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:

Inverse 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:

Absolute Value Transformation

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.

Note that with the absolute value on the outside (affecting the y’s), we just take all negative y values and make them positive, and with absolute value on the inside (affecting the x’s), we take all the 1st and 4th quadrant points and reflect them over the y axis, so that the new graph is symmetric to the y axis.

Absolute Value Tranformation Chart

Here’s an example of a mixed absolute value transformation; you can see that this can get complicated.  It looks like when we have absolute values on the inside (affecting the x’s), we do those first:

Mixed Transformation with Absolute Value

Here’s an example of writing an absolute value function from a graph:


Here are more absolute value examples with parent functions:

Mixed Transformations with Absolute Value

Note:  These mixed transformations with absolute value are very tricky; it’s really difficult to know what order to use to perform them.  The general rule of thumb is to perform the absolute value first for the absolute values on the inside, and the absolute value last for absolute values on the outside (work from the inside out).  The best thing to do is to play around with them on your graphing calculator to see what’s going on.

For example, with something like Absolute Value Exponential 3, you perform the y absolute value function first (before the shift); with something like Absolute Value Exponential 4, you perform the y absolute value last (after the shift).   (These two make sense, when you look at where the absolute value functions are.)  But we saw that with Exponential Function, we performed the x absolute value function last (after the shift).  I also noticed that with Absolute Value Exponential 2, you perform the x absolute value transformation first (before the shift).

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?

Absolute Value Parent Function

More Absolute Value Transformations

What about Absolute Value of both x and y?  Play around with this in your calculator with Mixed Absolute Value Transformation, 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 image114, 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!

More Absolute Value Transformations

Applications of Parent Function Transformations

You may see a “word problem” that used Parent Function Transformations, and you may just have to use what you know about how to shift the functions (instead of coming up with the solution off the top of your head).

Here is an example:

Parent Function Transformation Profit Problem

Learn these rules, and practice, practice, practice!

For Practice: Use the Mathway widget below to try a Transformation problem. Click on Submit (the blue arrow to the right of the problem) and click on Describe the Transformation to see the answer.

You can also type in your own problem, or click on the three dots 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 get any type of math problem solved!).

On to Piecewise Functions – you are ready!

88 thoughts on “Parent Functions and Transformations

  1. 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.

  2. 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.

  3. 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.]

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

  4. 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.

  5. 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?

  6. 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

  7. 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

  8. 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!

    • 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!

  9. 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).

  10. 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..

  11. 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

  12. 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

  13. 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!

  14. 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

  15. 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,
      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?)

        • Thank you for putting this together!! Very helpful.

          One thing just so I am clear, let’s say the equation is y=-2sqrt(-3x+9)-5. As long as I factor out the -3 to make it y=-2sqrt(-3(x-3))-5, then it doesn’t matter what order I write reflections and stretches/shrinks in, just that the translations are last?

          So I can write the order as:
          reflect x-axis, reflect y-axis, vertical stretch by 2, horizontal shrink by 1/3, 3 right, 5 down

          and this would be the same as saying, for instance,

          horizontal shrink by 1/3, reflect y-axis, reflect x-axis, vertical stretch by 2, 3 right, 5 down.

          In other words, provided I factor out as I did, the first four transformations I have written can be written in any order, just make sure to put the two translations last?

          • YES! GOOD observation – you are absolutely correct. It worked when you tried these two ways, right? Thanks for writing – I think I’ll add this observation to my website. Lisa

  16. This website gave me the wrong answer to my homework! I need help, would you help me…..please
    I need help with this question:
    f(x) = x – 5. Write the function rule for g(x), which is f(x) translated two units upward.
    -as a function rule for g(x), like this: g(x) = 14x – 12

    • Thanks for writing! Could you please tell me where the website was wrong? For your problem, it would be f(x) = (x – 5) + 2 = x – 3. So g(x) = x – 3. Does that make sense? Lisa

    • Thanks for writing! Here’s what an exponential parent function looks like – see the third one on this chart:

  17. Lisa, you ROCK! Absolute Value of functions (vs absolute value functions) can be really challenging, especially for those who have a hard time manipulating graphics mentally. Your explanations are clear, detailed and awesome. The visuals are particularly useful. What software do you use? I tutor students and always offer your site as a reference. I think it’s interesting that you always ask Does this make sense? Because I find myself doing the same in all my tutoring sessions. I wish teachers had the time to ask that too — but that seems one of the luxuries of being a tutor!

    • Thanks so much for writing – comments like these make me want to keep working!! I use WordPress, but I do the work in a word document where I use Mathtype and Graph programs. Please let me know if you can think of any improvements to the site, or if you see any errors 😉 We tutors have to stick together 😉 Lisa

  18. I work in Learning Strategies, a program to assist students with learning challenges through a rigorous prep curriculum for high school. I found your site last year, but with Math 2 (Integrated) or Algebra 2 we used Desmos for visualizing. I will be working to use the Mathways software as my 3H math student has abysmal handwriting and could use a way to type his work so his teacher won’t make him rewrite it. Is Mathtype another way to type math? “We” are working on Math 3 (pre-calc) and recursive equations, but I need the big picture with visuals in order to help my students “see” the method before they can understand.
    Many thanks,

    • Thanks for writing! Mathway is more like a program to enter problems and see the steps. I would look into Wolfram Alpha – and it’s also a free app! I think MatLab might be another program for inputting. Hope that helps! Lisa

  19. When you are looking at the graph and trying to write the function, how do you know if if the dilation affects a or b? I usually see that the default function used is for finding a.

    • Thanks for writing! That’s a GREAT question – for a lot of functions (like quadratics and cubics), you can just sort of ignore the “b” part (horizontal compression) and try to find the “a” part (vertical stretch) – which will be the same thing. I added a note about this on my website, under Writing Transformed Equations from Graphs. Does that help? Lisa

  20. SUPER job! Your website is an excellent resource for parents AND students. I’ll definitely be recommending this site. You did an amazing job. MANY thanks. Keep up the outstanding work!!!

  21. hey i have a question so if i were to try to find the parent function of sqrt x-2 +4 the parent function is simply sqrt x right?

  22. Hi lisa. I need help on this problem,
    Describe how y=3/2x+4 is a transformation of the parent function y=x.

    Plz help, I dont know how to answer it.

    • Thanks for writing! What you are describing is a line that is stretched by (3/2) (so it’s 1 and 1/2 times steeper, since the slope is 3/2 instead of 1), and shifted up 4. Does that make sense? Lisa

  23. Thank You so much Lisa. You are a big help too me and other students too. Keep up this website. I will surely come back with questions. Good job!

  24. This website is so helpful! I’m still slightly confused on how to find the parent equation if the transformed function is given. If the given transformed function is f2(x) = (2/3)x -4, how to i find the parent function f1(x)?

    • Thanks for writing! I would just look at the “x” part – the variable – to see what it is (like a square root, squared, cubed, etc). This is just an “x”, so the parent function is just y = x. Does that make sense? Lisa

  25. Thanks for writing. I would say that the parent function is 1/x. Then the function is shifted to the left 1 unit, stretched vertically by a factor of 2, and shifted up 2 units. Does that make sense? Lisa

    • Thanks so much for writing! That’s a great idea, but sorry, I don’t have poster sizes of my charts. I should think about it though! If you’d like to create one, feel free, as long as you credit it to Thanks, Lisa

  26. Thank you SOOO much for your graphics about functions and other confusing math concepts. I copy your graphs, giving plugs for this site, and keep them in plastic sleeves for reference for all levels of high school math. I am visual and need these reminders so I can help guide my students through their assignments. They would look great on a math classroom wall!

    • Nancy,
      Thanks you SO MUCH for writing, and for all your kind words! Comments like yours make me want to keep writing and finish the site! And please let me know if there are any graphics that you feel are missing, or you find any mistakes in the tables. Thanks again, Lisa

  27. Cool guys this really helped when I was writing my Paper 1 of grade 12 Mathematics on the 2nd of June this week!I’m expecting good results when we reopen!!! -)

  28. This has been a great help understanding this concept. But I am still a little confused on how to describe a transformation for a sine graph. I know the parent function would be y = sin(x)
    My equation is y= 0.6 sin(x) + 3
    How do I describe the transformation?

    • Thanks for writing! This sin graph would be moved up 3 and it would be “squished” (compressed) by a factor of 3/5. So the amplitude would be 3/5 (or .6) instead of 1. Does that make sense? Lisa

  29. Dear Lisa

    It has been a blessing to find this site. I have been teaching Middle School Math for four years and declined teaching A Level standard Math as I felt rusty on “some” details of “some” topics and didn’t want to disadvantage the students.

    However, many students this year petitioned that they wanted me to teach this and I eventually was convinced. It has been taking quite some time to prepare for them and I felt the responsibility and concern of making the “odd” small error.

    But, now having seen “just” this one page of your website….I feel it will give me the clarity on those subtle details to help me teach my students and then continue on for many years and beyond.

    Thank you so much….I hope all your pages are as detailed as this and I can’t imagine how long it must take to offer such a spending free service..

    They say a teacher doesn’t after one person…but rather generations to come…I think you are certainly doing that

    • Thanks so much for your nice note! Notes like yours make me want to keep writing and finish the site. Let me know how I can improve it. Thanks again 😉 Lisa

  30. Dear Lisa,

    Can I just check if the transcript on this page below should read “left” 8 instead of “right” 8?

    (NOTE: In some books, for \displaystyle f\left( x \right)=-3{{\left( {2x+8} \right)}^{2}}+10, 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 for the x, and then compress with a factor of ½ for the x (which is opposite of PEMDAS). Then you would perform the y (vertical) changes the regular way – reflect and stretch by 3 first, and then shift up 10.)

  31. Dear Lisa, I have to describe the transformations of the quadratic when compared to the parent function. Do you think you could help?

    (x) = x2 – 8x + 9
    f(x) = ( x2 – 8x + 16 ) – 9 – 16
    f(x) = ( x – 4 )2 – 25

    • Thanks for writing! This function would be moved to the right 4 and down 25. So the new vertex would be at (4, 25). Does that make sense? Lisa

  32. Hi,
    I have an assignment where we have to write an equation for a function that goes through a few purple points. Is there any way you can help?

    • Thanks for writing! You got this correct – but the first part of the second term just integrate to (1/2)ln(x^2 + 1) – you don’t need to introduce trig. Does that make sense? Lisa

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