Function Transformations

Transformation of functions means that the curve representing the graph either "moves to left/right/up/down" or "it expands or compresses" or "it reflects". For example, the graph of the function f(x) = x 2 + 3 is obtained by just moving the graph of g(x) = x 2 by 3 units up. Function transformations are very helpful in graphing the functions just by moving/expanding/compressing/reflecting the curve without actually needing to graph it from scratch.

In this article, we will see what are the rules of function transformations and we will see how to do transformations of different types of functions along with examples.

What are Function Transformations?

A function transformation either "moves" or "resizes" or "reflects" the graph of the parent function. There are mainly three types of function transformations :

  • Translation

Among these transformations, only dilation changes the size of the original shape but the other two transformations change the position of the shape but not the size of the shape. We can see what each of these transformations of functions mean in the table below.

In math words, the transformation of a function y = f(x) typically looks like y = a f(b(x + c)) + d. Here, a, b, c, and d are any real numbers and they represent transformations. Note that all outside numbers (that are outside the brackets) represent vertical transformations and all inside numbers represent horizontal transformations. Also, note that addition/subtraction indicates translation and multiplication/division represents dilation. Any minus sign multiplies means that it is a reflection. Here,

  • 'a' represents the vertical dilation
  • 'b' represents the horizontal dilation
  • 'c' represents the horizontal translation
  • 'd' represents the vertical translation

Let us learn each of these function transformations in detail.

Translation of Functions

A translation occurs when every point on a graph (representing a function) moves by the same amount in the same direction. There are two types of translations of functions.

  • Horizontal translations
  • Vertical translations

Horizontal Translation of Functions :

In this translation, the function moves to the left side or right side. This changes a function y = f(x) into the form y = f(x ± k), where 'k' represents the horizontal translation . Here,

  • if k > 0, then the function moves to the left side by 'k' units.
  • if k < 0, then the function moves to the right by 'k' units.

Horizontal translation of functions

Here, the original function y = x 2 (y = f(x)) is moved to 3 units right to give the transformed function y = (x - 3) 2 (y = f(x - 3)).

Vertical Translation of Functions :

In this translation, the function moves to either up or down. This changes a function y = f(x) into the form f(x) ± k, where 'k' represents the vertical translation . Here,

  • if k > 0, then the function moves up by 'k' units.
  • if k < 0, then the function moves down by 'k' units.

vertical translation of functions

Here, the original function y = x 2 (y = f(x)) is moved to 2 units up to give the transformed function y = x 2 + 2 (y = f(x) + 2).

Dilation of Functions

A dilation is a stretch or a compression. If a graph undergoes dilation parallel to the x-axis, all the x-values are increased by the same scale factor. Similarly, if it is dilated parallel to the y-axis, all the y-values are increased by the same scale factor. There are two types of dilations.

Horizontal Dilation

Vertical Dilation

The horizontal dilation (also known as horizontal scaling ) of a function either stretches/shrinks the curve horizontally. It changes a function y = f(x) into the form y = f(kx), with a scale factor '1/k', parallel to the x-axis. Here,

  • If k > 1, then the graph shrinks.
  • If 0 < k < 1, then the graph stretches.

In this dilation, there will be changes only in the x-coordinates but there won't be any changes in the y-coordinates. Every old x-coordinate is multiplied by 1/k to find the new x-coordinate. In the following graph, the original function y = x 3 is stretched horizontally by a scale factor of 3 to give the transformed function graph y = (x/3) 3 . For example, the point (1,1) of the original graph is transformed to (3, 1) of the new graph.

horizontal dilation of functions

The vertical dilation (also known as vertical scaling ) of a function either stretches/shrinks the curve vertically. It changes a function y = f(x) into the form y = k f(x), with a scale factor 'k', parallel to the y-axis. Here,

  • If k > 1, then the graph stretches.
  • If 0 < k < 1, then the graph shrinks.

In this dilation, there will be changes only in the y-coordinates but there won't be any changes in the x-coordinates. Every old y-coordinate is multiplied by k to find the new y-coordinate. In the following graph, the original function y = x 3 is stretched vertically by a scale factor of 3 to give the transformed function graph y = 3x 3 . For example, the point (1, 1) (on the original graph) corresponds to (1, 3) on the new graph.

Vertical dilation of functions

Reflections of Functions

A reflection of a function is just the image of the curve with respect to either x-axis or y-axis. This occurs whenever we see the multiplication of a minus sign happening somewhere in the function. Here,

  • y = - f(x) is the reflection of y = f(x) with respect to the x-axis.
  • y = f(-x) is the reflection of y = f(x) with resepct to the y-axis.

Observe the graph below where the original graph y = (x + 2) 2 is reflected with respect to each of the x and y axes.

Reflections of Functions

Here, note that when the function is reflected

  • with respect to the x-axis, only the signs of the y-coordinates are changed and there is no change in x-coordinates.
  • with respect to the y-axis, only the signs of the x-coordinates are changed and there is no change in y-coordinates.

Function Transformation Rules

So far we have understood the types of transformations of functions and how do addition/subtraction/multiplication/division of a number and the multiplication of a minus sign would reflect a graph. Let us tabulate all function transformation rules together.

Are the above rules are confusing and difficult to remember? Let us see some important tips to remember these rules.

Tips and Tricks to Remember Function Transformations:

  • If some operation is inside the bracket, note that it is related to "horizontal" and in this case, things would happen reverse than what we think. For example, we may think f(x + 2) transforms f(x) to the right because it is + but it actually moves left by 2 units. In the same way, we may think f(3x) stretches f(x) but no, it shrinks f(x) by a scale factor of 1/3.
  • If some operation is outside the bracket, note that it is related to "vertical" and in this case, things would happen straight (not reverse). For example, f(x) + 2 moves f(x) "up" it is a "+" symbol there. In the same way, 3 f(x) stretches f(x) by a scale factor of 3 as 3 > 1.
  • If some number is being added / subtracted , then its related to "translation". For example, f(x + 2) is a horizontal translation and f(x) + 2 is a vertical translation.
  • If some number is being multiplied / divided , then its related to "dilation". For example, f(2x) is a horizontal dilation and 2 f(x) is a vertical dilation.
  • Just in case of reflection, it is just the opposite of the first and second tricks here. If the minus sign is inside the bracket, it is with respect to the y-axis and if the minus sign is outside the bracket, it is with respect to the x-axis.

Describing Function Transformations

We can use the above rules to describe any function transformation. For example, if the question is what is the effect of transformation g(x) = - 3f(x + 5) + 2 on y = f(x), then first observe the sequence of operations that had to be applied on f(x) to get g(x) and then use the above rules to define the transformations. Here, to get g(x) from f(x)

  • first f(x) changes into f(x + 5). i.e., horizontal translation by 5 units to the left.
  • Then it changes into 3 f(x + 5). i.e., vertical dilation by a scale factor of 3.
  • Then it changes into -3 f(x + 5). i.e., reflection about the x-axis.
  • Finally, it changes into -3 f(x + 5) + 2. i.e., vertical translation by 2 units up.

Thus, g(x) is obtained from f(x) by horizontal translation by 5 units to the left, vertical dilation by a scale factor of 3, reflection about the x-axis, and vertical translation by 2 units up. We can describe the transformations of functions by using the above tricks also. Give it a try now.

Graphing Transformations of Functions

Identifying the transformation by looking at the original and transformed graphs is easy because just by looking at the graph, we can say that the graph moves up by 2 units or left by 3 units, etc. But when a graph is given, graphing the function transformation is sometimes difficult. The following steps make graphing transformations so easier. Here, we are transforming the function y = f(x) to y = a f(b (x + c)) + d.

  • Step 1: Note down some coordinates on the original curve that define its shape. i.e., we now know the old x and y coordinates .
  • Step 2: To find the new x-coordinate of each point just set "b (x + c) = old x-coordinate" and solve this for x.
  • Step 3: To find the new y-coordinate of each point, just apply all outside operations (of brackets) on the old y-coordinate. i.e., find ay + d to find each new y-coordinate where 'y' is the old y-coordinate.

We can understand these steps better by using the example below.

Example: The following graph represents f(x). Graph the function transformation y = 2 f(x/2) + 3.

Graphing Transformations of Functions

We can clearly see that (-3, 2), (-1, 2), (2, -1) and (6, 1) are defining the shape of the graph. Let us find the new x and y coordinates of each of these points.

Now, we will plot all old points and new points on the coordinate plane and observe the transformations.

Graph Transformations

☛ Related Topics:

  • Transformation Matrix
  • Linear Fractional Transformation

Function Transformations Examples

Example 1: Describe the transformations of quadratic function g(x) = x 2 + 4x + 5 by comparing it to its parent function f(x) = x 2 .

To identify the transformation of quadratic functions, we have to convert it into vertex form . Then we can write g(x) = x 2 + 4x + 5 can be written as g(x) = (x + 2) 2 + 1.

Now we will compare the original function f(x) = x 2 with g(x) = (x + 2) 2 + 1 and apply the function transformation rules.

  • x converted to x + 2 and it corresponds to the horizontal translation of 2 units to the left.
  • 1 is added to the function and it corresponds to the vertical translation of 1 unit upwards.

Answer: 2 units to left and 1 unit to up.

Example 2: State the combination of transformations applied on the function f(x) to obtain g(x): f(x) = -3x - 6 and g(x) = x + 2.

We have g(x) = x + 2 = -1/3 (-3x - 6) = -1/3 f(x)

Thus, the combinations of transformations applied on f(x) are:

  • Vertical dilation by a scale factor of 1/3 and
  • reflection with respect to the x-axis.

Answer: Vertical dilation and reflection.

Example 3: Write the function corresponding to the graph of g(x) that transformed from the graph f(x) by using the function transformation rules.

Function Transformations Example

Take f(x) as the original function and observe how it is moving/transforming to give g(x). Observe the vertex of both graphs to get an idea. It is very clear that

  • it moved 6 units to the left and so the function is f(x + 6).
  • it then reflected with respect to the x-axis, so the function is - f(x + 6).

Answer: g(x) = - f(x + 6).

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homework 6 parent functions & transformations

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Practice Questions on Function Transformations

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FAQs on Function Transformations

What are transformations of functions.

The transformations of functions define how to graph a function is moving and how its shape is being changed. There are basically three types of function transformations: translation, dilation, and reflection.

How Do You Find the Function Transformations?

To find the function transformations we have to identify whether it is a translation, dilation, or reflection or sometimes it is a mixture of some/all the transformations. For a function y = f(x),

  • if a number is being added or subtracted inside the bracket then it is a horizontal translation . If the number is negative then the horizontal transformation is happening to the right side. If the number is positive then the horizontal transformation is happening to the left side.
  • If a number is being added or subtracted outside the bracket then it is a vertical translation. If the number is positive then the vertical translation is happening toward up. If the number is negative then the vertical translation is happening to the downside.
  • If a number is being multiplied or divided inside the brackets then it is horizontal dilation. If the number is > 1, then it is a horizontal shrink. If the number is between 0 and 1, then it is a horizontal stretch.
  • If a number is being multiplied or divided outside the brackets then it is vertical dilation. If the number is > 1, then it is a vertical stretch. If the number is between 0 and 1, then it is a vertical shrink.
  • If the function is multiplied by the minus sign inside the bracket, then it is a reflection with respect to the y-axis.
  • If the function is multiplied by the minus sign outside the bracket, then it is a reflection with respect to the x-axis.

How to Explain the Function Transformations?

To explain the function transformations we have to apply the rules of transformations of functions. For example, 3 f(x + 2) - 5 is obtained by applying the following function transformations on f(x):

  • horizontal translation by 2 units left.
  • Vertical dilation by a scale factor of 3.
  • Vertical translation by 5 units down.

What are the Rules of Transformations of Functions?

The rules of function transformations for each of the translation, dilation, and reflection:

  • Horizontal translation: it is of the form f(x + k) and it moves f(x) to k units left if k > 0 and k units right if k < 0. Vertical translation: it is of the form f(x) + k and it moves f(x) to k units up if k > 0 and k units down if k < 0.
  • Horizontal dilation : It is of the form f(kx) and it shrinks f(x) if k > 1 and stretches f(x) if 0 < k < 1. Vertical dilation: It is of the form k f(x) and it shrinks f(x) if 0 < k < 1 and stretches f(x) if k > 1.
  • Reflection with respect to the x-axis is of the form - f(x). Reflection with respect to the y-axis is of the form f(-x).

What are Different Types of Function Transformations?

There are mainly three types of function transformations.

  • Translation: it moves the graph of the original function to either left, right, up, or down.
  • Dilation: it either shrinks or stretches the graph of the original function horizontally or vertically.
  • Reflection: it reflects the graph of the original function ( in other words it creates the mirror image of the original function) with respect to x or y axes.

What is the Easiest Way of Remembering Function Transformations?

Here is the easiest way of remembering the function transformations. If something is happening inside the bracket then it corresponds to the horizontal transformations. If something is happening outside the brackets then it corresponds to the vertical transformations. If a minus sign is being multiplied either outside or inside the bracket then it corresponds to the reflection.

Parent Functions and Transformations

For Absolute Value Transformations, see the  Absolute Value Transformations  section . Here are links to Parent Function Transformations in other sections: Transformations of Quadratic Functions (quick and easy way);  Transformations of Radical Functions ;  Transformations of Rational Functions ; Transformations of Exponential Functions ;  Transformations of Logarithmic Functions ; Transformations of Piecewise Functions ;  Transformations of Trigonometric Functions ; Transformations of Inverse Trigonometric Functions

You may 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 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 (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 the  radical 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 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 $.

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 .

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 (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:

Transformed :

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

Range:    $ \left[ {-1,\,\,\infty } \right)$

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 s ection !

Domain:   $ \left( {-\infty ,\infty } \right)$    Range:   $ \left[ {2,\infty } \right)$

$ f\left( x \right)-b$

Translation

Move graph down $ b$ units

Every point on the graph is shifted down $ b$ units.

The $ x$’s stay the same; subtract $ b$ from the $ y$ values.

Parent : $ y=\sqrt{x}$

Transformed : $ y=\sqrt{x}- \,3$

$ a\,\cdot f\left( x \right)$

Stretch graph vertically by a scale factor of $ a$ (sometimes called a dilation ). Note that if $ a<1$, the graph is compressed or shrunk.

Every point on the graph is stretched $ a$ units.

The $ x$’s stay the same; multiply the $ y$ values by $ a$.

Parent : $ y={{x}^{3}}$

Transformed : $ y={{4x}^{3}}$

$ -f\left( x \right)$

Flip graph around the $ x$-axis.

Every point on the graph is flipped vertically.

The $ x$’s stay the same; multiply the $ y$ values by $ -1$.

Parent : $ y=\left| x \right|$

Transformed : $ y=-\left| x \right|$

Absolute Value on the $ y$

(More examples here in the Absolute Value Transformation section )

The $ x$’s stay the same; take the absolute value of the $ y$ ’s .

Parent :  $ y=\sqrt[3]{x}$

Transformed : $ y=\left| {\sqrt[3]{x}} \right|$

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.)

Move graph right $ b$ units

Every point on the graph is shifted right $ b$ units.

The $ y$’s stay the same; add $ b$ to the $ x$ values.

Transformed : $ y=\sqrt{{x- \,3}}$

Compress graph horizontally by a scale factor of $ a$ units (stretch or multiply by $ \displaystyle \frac{1}{a}$)  

Every point on the graph is compressed $ a$ units horizontally.

The $ y$’s stay the same; multiply the $ x$-values by $ \displaystyle \frac{1}{a}$.

  T ransformed : $ y={{\left( {4x} \right)}^{3}}$

Flip graph around the $ y$- axis

Every point on the graph is flipped around the $ y$ axis.

The $ y$’s stay the same; multiply the $ x$-values by $ -1$.

Transformed : $ y=\sqrt{{-x}}$

$ f\left( {\left| x \right|} \right)$

Absolute Value on the $ x$

The positive $ x$’s stay the same; the negative $ x$ ’s take on the $ y$ ’s of the positive $ x$ ’s.

Parent : $ y=\sqrt{x}$

Transformed : $ y=\sqrt{{\left| x \right|}}$

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

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 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 $ \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 .

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

How to graph   without a t -chart: $ \displaystyle f(x)=-3{{\left( {2\left( {x+4} \right)} \right)}^{2}}+10$

Since this is a parabola and it’s in vertex form ($ y=a{{\left( {x-h} \right)}^{2}}+k,\,\,\left( {h,k} \right)\,\text{vertex}$), the vertex of the transformation is $ \left( {-4,10} \right)$.

Notice that the coefficient of  is –12 (by moving the $ {{2}^{2}}$ outside and multiplying it by the –3 ); this is an easier way to deal with the transformation, but it doesn’t work with every function. Then the vertical stretch is 12 , and the parabola faces down because of the negative sign. The parent graph quadratic goes up 1 and over (and back) 1 to get two more points, but with a vertical stretch of 12 , we go over (and back) 1 and down 12 from the vertex. Now we have two points from which you can draw the parabola from the vertex.

Note that without multiplying out to get $ -12$, we could have also gone over $ 1$, down $ 3$ from the vertex for the outside $ -3$ (vertical flip/stretch), and then squeeze back in $ \frac{1}{2}$ for the inside $ 2$ (horizontal compression).

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 the  2  on the inside, but just switch the order of the transformation on the $ \boldsymbol{x}$ .

In this case, 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 $ \boldsymbol{x}$ , and then compress with a factor of $ \displaystyle \frac {1}{2}$ for the $ \boldsymbol{x}$  (which is opposite of PEMDAS). Then you would perform the $ \boldsymbol{y}$ (vertical) changes the regular way: reflect and stretch by 3 first, and then shift up 10 . So, you would have $ \displaystyle {\left( {x,\,y} \right)\to \left( {\frac{1}{2}\left( {x-8} \right),-3y+10} \right)}$ . Try a t -chart; you’ll get the same t -chart as above!

More Examples of Mixed Transformations:

Here are a couple more examples (using t -charts), with different parent functions. Don’t worry if you are totally lost with the exponential and log functions; they will be discussed in the Exponential Functions  and Logarithmic Functions sections. Also, the last type of function is a rational function that will be discussed in the Rational Functions section.

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

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

$ \displaystyle y=\frac{1}{2}\sqrt{{-x}}$

Transformation : Reflection across the $ y$-axis, vertical compression of $ \displaystyle \frac{1}{2}$

Parent function: $ y=\sqrt{x}$

Domain:   $ \left( {-\infty ,0} \right]$

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

$ y={{2}^{{x-4}}}+3$

Transformation : Shift right $ 4$ and up $ 3$

Parent function: $ y={{2}^{x}}$

For exponential functions, use –1 , 0 , and 1 for the $ x$-values for the parent function. (Easy way to remember: ex ponent is like $ x$).

Range:   $ \left( {3,\infty } \right)$

Asymptote :  $ y=3$

$ \begin{array}{l}y=\log \left( {2x-2} \right)-1\\y=\log \left( {2\left( {x-1} \right)} \right)-1\end{array}$

Transformation : Horizontal compression of $ \displaystyle \frac{1}{2}$, shift right $ 1$ and down $ 1$

Parent function: $ y=\log \left( x \right)={{\log }_{{10}}}\left( x \right)$

For log and ln functions, use – 1 , 0 , and 1 for the $ y$-values for the parent function For example, for $ y={{\log }_{3}}\left( {2\left( {x-1} \right)} \right)-1$ , the $ x$ values for the parent function would be $ \displaystyle \frac{1}{3},\,1,\,\text{and}\,3$.)

Domain:   $ \left( {1,\infty } \right)$

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

Asymptote :  $ x=1$

$ \displaystyle y=\frac{3}{{2-x}}\,\,\,\,\,\,\,\,\,\,\,y=\frac{3}{{-\left( {x-2} \right)}}$

Transformation : Reflection across the $ y$-axis, vertical stretch of $ 3$, shift right $ 2$

Parent function: $ \displaystyle y=\frac{1}{x}$

For this function, note that could have also put the negative sign on the outside (thus, used $ x+2$ and $ -3y$).

Domain:  $ \left( {-\infty ,2} \right)\cup \left( {2,\infty } \right)$

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

Asymptotes : $ y=0$ and $ x=2$

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:

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

Range:   $ \{y:y\in \mathbb{Z}\}\text{ (integers)}$

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 $ \displaystyle g\left( x \right)=-3f\left( {2\left( {x+4} \right)} \right)+10=g\left( x \right)=-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 y=2g\left( {-3x-2} \right)+3=2g\left( {-3\left( {x+\frac{2}{3}} \right)} \right)+3$ is $ \displaystyle \left( {-4,1} \right)\to \left( {-4\left( {-\frac{1}{3}} \right)-\frac{2}{3},2\left( 1 \right)+3} \right)=\left( {\frac{2}{3},5} \right)$ (using coordinate rules $ \displaystyle \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)$).

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 -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$. 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.

Remember to draw the points in the same order as the original to make it easier! 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!

Transformation :   $ \displaystyle f\left( {-\frac{1}{2}\left( {x-1} \right)} \right)-3$

$ y$ changes:  $ \displaystyle f\left( {-\frac{1}{2}\left( {x-1} \right)} \right)\color{blue}{{-\text{ }3}}$

$ x$ changes:   $ \displaystyle f\left( {\color{blue}{{-\frac{1}{2}}}\left( {x\text{ }\color{blue}{{-\text{ }1}}} \right)} \right)-3$

Note that this transformation flips around the $ \boldsymbol{y}$ – axis , has a horizontal stretch of 2 , moves right by 1 , and  down by 3 .

Key Points Transformed:

(we do the “opposite” math with the “$ x$”)

Transformed Function:

Domain:   $ \left[ {-9,9} \right]$   Range:  $ \left[ {-10,2} \right]$

Transformation:  $ \displaystyle f\left( {\left| x \right|+1} \right)-2$

$ y$ changes:  $ \displaystyle f\left( {\left| x \right|+1} \right)\color{blue}{{\underline{{-\text{ }2}}}}$

$ x$ changes:   $ \displaystyle f\left( {\color{blue}{{\underline{{\left| x \right|+1}}}}} \right)-2$:

Note that this transformation moves down by 2 , and left 1 . Then, for the inside absolute value, we will “get rid of” any values to the left of the $ y$-axis and replace with values to the right of the $ y$-axis, to make the graph symmetrical with the $ y$-axis. We do the absolute value part last, since it’s only around the $ x$ on the inside.

Let’s just do this one via graphs. First, move down 2 , and left 1 :

Then reflect the right-hand side across the $ y$-axis to make symmetrical.

Domain:   $ \left[ {-4,4} \right]$  Range:   $ \left[ {-9,0} \right]$

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 a  transformed equation from an absolute value graph is in the  Absolute Value Transformations  section .

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 $ \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:

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

Transformations of Inverse 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 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 Absolute Value Transformations – you are ready!

Function Transformations

Let us start with a function, in this case it is f(x) = x 2 , but it could be anything:

Here are some simple things we can do to move or scale it on the graph:

We can move it up or down by adding a constant to the y-value:

Note: to move the line down , we use a negative value for C.

  • C > 0 moves it up
  • C < 0 moves it down

We can move it left or right by adding a constant to the x-value:

Adding C moves the function to the left (the negative direction).

Why? Well imagine you will inherit a fortune when your age=25 . If you change that to (age+4) = 25 then you will get it when you are 21. Adding 4 made it happen earlier.

  • C > 0 moves it left
  • C < 0 moves it right

BUT we must add C wherever x appears in the function (we are substituting x+C for x).

Example: the function v(x) = x 3 - x 2 + 4x

To move C spaces to the left, add C to x wherever x appears :

w(x) = (x + C ) 3 − (x + C ) 2 + 4(x + C )

An easy way to remember what happens to the graph when we add a constant:

add to y to go high add to x to go left

We can stretch or compress it in the y-direction by multiplying the whole function by a constant.

  • C > 1 stretches it
  • 0 < C < 1 compresses it

We can stretch or compress it in the x-direction by multiplying x by a constant.

  • C > 1 compresses it
  • 0 < C < 1 stretches it

Note that (unlike for the y-direction), bigger values cause more compression .

We can flip it upside down by multiplying the whole function by −1:

This is also called reflection about the x-axis (the axis where y=0)

We can combine a negative value with a scaling:

Example: multiplying by −2 will flip it upside down AND stretch it in the y-direction.

We can flip it left-right by multiplying the x-value by −1:

It really does flip it left and right! But you can't see it, because x 2 is symmetrical about the y-axis . So here is another example using √(x) :

This is also called reflection about the y-axis (the axis where x=0)

Example: the function g(x) = 1/x

Here are some things we can do:

Example: the function v(x) = x 3 − 4x

All in one ... .

We can do all transformations on f()  in one go using this:

a is vertical stretch/compression

  • |a| > 1 stretches
  • |a| < 1 compresses
  • a < 0 flips the graph upside down

b is horizontal stretch/compression

  • |b| > 1 compresses
  • |b| < 1 stretches
  • b < 0 flips the graph left-right

c is horizontal shift

  • c < 0 shifts to the right
  • c > 0 shifts to the left

d is vertical shift

  • d > 0 shifts upward
  • d < 0 shifts downward

Example: 2√(x+1)+1

a=2, c=1, d=1

So it takes the square root function, and then

  • Stretches it by 2 in the y-direction
  • Shifts it left 1, and
  • Shifts it up 1

Play with this graph

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Homework 6 Parent Functions Transformations

Displaying top 8 worksheets found for - Homework 6 Parent Functions Transformations .

Some of the worksheets for this concept are Work parent functions transformations no, Parent functions and transformations work answers, Name date, Transformations work name date, The parent functions, Transforming functions work, Function parent graph characteristics name function, To of parent functions with their graphs tables and.

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1. Worksheet 2.4 Parent Functions & Transformations No ...

2. parent functions and transformations worksheet answers, 3. 2.5/2.6:transformationsworksheet name date, 4. transformations worksheet name: date, 5. the parent functions, 6. transforming functions worksheet, 7. function parent graph characteristics name function, 8. to of parent functions with their graphs, tables, and ....

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  6. Doing homework with a parent be like: (Credits to @TheManniiShow) #shorts

COMMENTS

  1. Solved Unit 2: Functions & Their Graphs Date: Homework 6 ...

    Question: Unit 2: Functions & Their Graphs Date: Homework 6: Parent Functions & Transformations ** This is a 2-page document ** Directions: Given each function, identify both the parent function and the transformations from the parent function. 2. f (x)= (-x)+7 1. S (x)--4 3.5 (x) = -1 4. f (x) = √ (x - 2) +7 5. 8 (x)=x+5 6. f (x) = 7.

  2. Transformations of functions

    We can think graphs of absolute value and quadratic functions as transformations of the parent functions |x| and x². Importantly, we can extend this idea to include transformations of any function whatsoever! This fascinating concept allows us to graph many other types of functions, like square/cube root, exponential and logarithmic functions.

  3. PDF ew. +6x-4 6. 8. . —x +6x—10

    Unit 3: Parent Functions & Transformations Homework 6: Converting to Vertex Form ** This is a 2-page document! ** Write each function in vertex form. Give the vertex. -12x+36 2X+5 5. +7x-10 7. +42x 2. f (x) = x 2 +4x—l vex\ew. ... Unit 3 Homework #6 (Key) Created Date:

  4. Chapter 3: Parent Functions and Transformations Flashcards

    f (x)=-|2x|+7. Parent Functions and Transformations: For the following functions, give the parent function or give the vertex and graph. Give the parent function: Absolute Value Functions. Give the vertex and graph. State the domain and range. f (x)=|x-1|-1.

  5. Homework 6 Transformations Practice

    Homework 6 Page 1 - Angles to Radians. Exponential Growth and Decay Practice HW. Appendix A.4 - Polynomial Expressions. Angles of Elevation and Depression Practice. Activity-Template -Risk-management-plan. 5.9 Graphing Sine, Cosine and Tangent Functions. Related documents. 5.8 Area of Triangles using Sine and Cosine. 5.7 Law of Cosines.

  6. PDF -5 -4 -3 -2 -1 1 2 3 4 5 -1 3 4 -1 3 4 -134 Parent Functions and

    Sample Problem 1: Identify the parent function and describe the transformations. Sample Problem 2: Given the parent function and a description of the transformation, write the equation of the transformed function ( ). Sample Problem 3: Use the graph of parent function to graph each function. Find the domain and the range of the new function. a.

  7. Functions Transformations

    Function Transformations. Transformation of functions means that the curve representing the graph either "moves to left/right/up/down" or "it expands or compresses" or "it reflects". For example, the graph of the function f (x) = x 2 + 3 is obtained by just moving the graph of g (x) = x 2 by 3 units up. Function transformations are very helpful ...

  8. PDF Homework #6: Parent Functions Day 1

    Homework #6: Parent Functions Day 1 a) Determine the parent function of each equation b) Determine the transformation c) Make a table and graph the transformation d) Determine domain and range of the transformation 1) 𝑓(𝑥)= 𝑥+5 +1 2) )𝑓(𝑥=−(𝑥−3)2−2

  9. Unit 3: Parent Functions & Transformations Flashcards

    x = -b/2a. quadratic standard form. y = ax^2+bx+c. 1st step in converting quadratic standard form to quadratic vertex form: group (ax^2 + bx) 2nd step in converting quadratic standard form to quadratic vertex form: check a; if a != 1, factor outside so it becomes a (x^2 + bx) 3rd step in converting quadratic standard form to quadratic vertex form:

  10. Parent Functions and Transformations

    The include the points (ordered pairs) of the original parent functions, and also the transformed or shifted points. The first two transformations are , the third is a , and the last are forms of. Absolute value transformations will be discussed more expensively in the ! Transformation.

  11. Function Transformations

    Here are some simple things we can do to move or scale it on the graph: We can move it up or down by adding a constant to the y-value: g (x) = x2 + C Note: to move the line down, we use a negative value for C. C > 0 moves it up C < 0 moves it down We can move it left or right by adding a constant to the x-value: g (x) = (x+C)2

  12. Parent Functions and Transformations (Algebra 2

    A chart is provided with all the parent functions that can be used throughout future units. This Parent Functions and Transformations Unit Bundle includes guided notes, homework assignments, three quizzes, a study guide and a unit test that cover the following topics: • Piecewise Functions. • Graphing Absolute Value Functions and ...

  13. 6A 2.7 Parent Functions Parent Graphs & Transformations

    View Notes - 6A (2.7) Parent Functions, Parent Graphs, & Transformations from MATH Algebra II at Stanton College Preparatory. Parent Functions, Parent Graphs, & Transformations (2-7) Homework 6 Read

  14. Wolf, Matthew / Unit 3: Parent Functions

    Unit 3: Parent Functions. 3.1 Completing the Square. In-Class Notes Notes Video Worksheet Worksheet Solutions Homework HW Solutions. Textbook HW Pg. 287 #73-75, 79-81. 3.2 Graphing Quadratic Functions. In-Class Notes Notes Video Worksheet Worksheet Solutions Homework HW Solutions. Textbook HW: Pg. 253 #20-22, 26-28.

  15. Transformations of Parent Functions Practice

    Some activities/pages use only the first four functions, some use all seven types. Transformations include: translations (up, right, left, and down), reflections across the x-axis, and vertical stretches and compressions. Students will likely need some familiarity with the parent functions described above to complete the activity, as most of ...

  16. Parent Function Transformations Practice Teaching Resources

    4.0. (1) $3.00. PDF. Easel Activity. Transformations of Parent Functions PracticeStudents will practice identifying transformations that have occurred from a graph or equation of a transformed function, and also practice writing the equation of a transformed function. This no prep activity contains fifteen pages of practice problems (9 pages of ...

  17. Parent Functions with Transformations HW practice sheet

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  18. 2-4 Transformations of Absolute Value Functions

    Problem 6 Problem 5 continued To find the y-intercept, set x = 0. y = 300 - 20 + 4 y = 10 The y-intercept is (0, 10) or 10. The parent function y = 0x 0 is translated 2 units to the right, vertically stretched by the factor 3, and translated 4 units up. Check Check by graphing the equation on a graphing calculator. PRACTICE and APPLICATION EXERCISES O N LI N

  19. Homework 6 Parent Functions Transformations

    Some of the worksheets for this concept are Work parent functions transformations no, Parent functions and transformations work answers, Name date, Transformations work name date, The parent functions, Transforming functions work, Function parent graph characteristics name function, To of parent functions with their graphs tables and.