One of the transformations you can make with simple functions is to reflect it across the X-axis. window.onload = init; 2023 Calcworkshop LLC / Privacy Policy / Terms of Service. A (5, 2) We can use the following rules to do different types of reflections. These points are said to be invariant. It will be Take a Tour and find out how a membership can take the struggle out of learning math. What the new coordinate points A', B' and C'? \\ These depend on the refractive indexes of the two materials as well as the incidence angle and polarization of the wave in material 1. Embedded content, if any, are copyrights of their respective owners. One of the primary transformations you can make with simple functions is to reflect the graph across the X-axis or another horizontal axis. First we have to know the correct rule that we have to apply in this problem. Found a content error? If (a, b) is reflected on the line y = -x, its image is the point (-b, a). . These examples bring us into the main area of focus. However, the tricky affair lies in its right usage. In the diagram below, the triangle ABC is reflected in the line XY. Place the sharp point on a line called the axis of reflection or line of reflection. What about when you translate the figure to the right? What pattern you notice between the original points and the rotated points? Step 3: That's it Now your window will display the Final Output of your Input. The incident light ray which touches the plane is said to be reflected off the surface. Required transformation : Reflection under y = x, so change x as y and y as x. Which formula below best illustrates a reflection over the y-axis? Whenever we gaze at a mirror or blink at the sunlight glinting from a lake, we see a reflection. We can return a general transformation calculator for rotations, reflections, and shears, or we can compute the transformation matrix for a specific 2D or 3D transformation activity. In this section, students will be able to describe the effect of reflections on two dimensional figures using coordinates. Step 1 : First we have to know the correct rule that we have to apply in this problem. These reflected points represent the inverse function. A point reflection is generally described as an isometric transformation of the Euclidean space. This calculator computes the power reflectivity and transmission of a plan wave at a dielectric interface using the Fresnel equations. In the early rounds of the game, students may notice graph features from the list above, even though they may not use those words to describe them. For convenience, the calculator also computes an average result corresponding to unpolarized light. On other hand, in the image, $$ \triangle A'B'C' $$, the letters ABC are arranged in counterclockwise order. Reflections Explorer Reflections in Math Applet Interactive Reflections in Math Explorer. Step 2: For output, press the "Submit or Solve" button. Our experts will make you acquainted with all the types of reflection calculators precisely. when cialis patent expires cialis soft tabs canadian pharmacy or cialis 36 hour dosage What about when you translate the figure up? This is used by drawing this graph and measuring the results following the steps given above. With the proper guidance of our professionals, it wont be a difficulty for you. Nowadays, things have been easier for learners, thanks to reflection calculators in place. Draw the image of the Each type of transformation, such as translation, scaling, rotation, and reflection, is defined using a matrix whose elements follow a specific pattern. A (4, 4) Therefore, the results are different for S . Once all the points have been reflected them, join the points up neatly B (4, -1) Highly image (e.g. answer choices . Reflection-in-action: This reflection type happens whilst you are engaged in a situation. where transmission requires an extra scaling factor. To find the image of a point, we multiply the transformation matrix by a column vector that represents the point's coordinate. from a point on the object to the axis and the image is the same distance from the axis. Based on the rule given in step 1, we have to find the vertices of the reflected triangle A'B'C'. Q. function init() { Now, you can find the slope of the line of reflection. Resizing The other important Transformation is Resizing (also called dilation, contraction, compression, enlargement or even expansion ). A reflection is defined by the axis of symmetry or mirror line. How to Write a Bibliography for Your Assignment, Business Capstone Project Assignment Help, Medical Education Medical Assignment Help, Psychiatric Mental Health Nurse Assignment Help, Financial Statement Analysis Assignment Help, CDR Sample on Telecommunications Engineers, CDR Sample on Telecommunications Network Engineer. After reflection ==> x = 2y2. Step 4 : (x, y) ----> (x, -y) C (-2, 3), Rotate the points below 180 degrees about the origin. Ordered pair rules reflect over In the above diagram, the mirror line is x = 3. Do you have any suggestions to improve this page . So the transformation that reflects x in this plane is given by: x x 2 ( x. a) x. Similarly, the example of the equilateral triangle below has three lines of symmetry, as we can fold the triangle along these lines to create equal halves. Transformation Matrices The above transformations (rotation, reflection, scaling, and shearing) can be represented by matrices. Book Your Assignment at The Lowest Price How to Download YouTube Video without Software? To perform a geometry reflection, a line of reflection is needed; To embed this widget in a post on your WordPress blog, copy and paste the shortcode below into the HTML source: To add a widget to a MediaWiki site, the wiki must have the. Download free on iTunes. The statistics assignment experts of MyAssignmenthelp.com can give you perfect suggestions in this regard while making you understand the same. If you still have any queries relating to this scientific phenomenon, connect with the physics homework experts of MyAssignmenthelp.com immediately. The four most common reflections are defined below: Additionally, symmetry is another form of a reflective transformation. When a ray of light touches a smooth polished surface, the light ray bounces back instantly. Three of the most important transformations are: After any of those transformations (turn, flip or slide), the shape still has the same size, area, angles and line lengths. cialis no prescription generic levitra and cialis in australia and how to buy cialis online from canada cialis 20mg tablets price, Your email address will not be published. B (2, -1) As you can see in diagram 1 below, $$ \triangle ABC $$ is reflected over the y-axis to its image $$ \triangle A'B'C' $$. This Custom Polygraph is designed to spark vocabulary-rich conversations about transformation. A transformation is a way of changing the size or position of a shape. Though a reflection does preserve distance and therefore can be classified as an isometry, a reflection changes the orientation of the shape and is therefore classified as an opposite isometry. above matrix by the shown reflection matrix. Now divide the total distance by dis to calculate the number of reflections. //www.ferriswheel.net/__media__/js/netsoltrademark.php?d=tadalafil1st.online cialis made in the usa Then, the system describes a reflection matrix, which is given as: \[Reflection Matrix : \begin{bmatrix} -\frac{35}{37} & \frac{12}{37} \\ \frac{12}{37} & \frac{35}{37} \end{bmatrix} \], \[Transformation : (x, y) \rightarrow \bigg ( \frac{1}{37}(12(y 9) 35x) , \frac{1}{37}(12x + 35y + 18)\bigg )\], \[Matrix Form : \begin{bmatrix} x \\ y \end{bmatrix} \rightarrow \begin{bmatrix} -\frac{108}{37} \\ \frac{18}{37} \end{bmatrix} + \begin{bmatrix} -\frac{35}{37} & \frac{12}{37} \\ \frac{12}{37} & \frac{35}{37} \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}\], How To Solve Point Reflection Using the Mathematical Approach, Reflection Calculator + Online Solver With Free Steps. Demonstration of how to reflect a point, line or triangle over the x-axis, y-axis, or any line . $. Copying. You may start by setting up the problem that you want to solve. A transformation that uses a line that acts as a mirror, with an original figure (preimage) reflected in the line to create a new figure (image) is called a reflection. These depend on the refractive indexes of the two materials as well as the incidence angle and polarization of the wave in material 1. What are the new coordinate points of A', B' and C' after a 180 degree rotation? Therefore, it is a great tool to have up your sleeve. ( -2 , 5 ) \rightarrow ( 5 , -2 ) In other words, we can say that it is a rotation operation with 180. problem solver below to practice various math topics. USE AT YOUR OWN RISK. Finally, if you want to solve any more problems of a similar nature, you can do that by entering the new values while in the new window. Before answering any questions, play around with the translation sliders. Reflection In this section, students will be able to describe the effect of reflections on two dimensional figures using coordinates. var vidDefer = document.getElementsByTagName('iframe'); Upvote Flag eaman.shire 11 years ago Usually you should just use these two rules: T (x)+T (y) = T (x+y) cT (x) = T (cx) Where T is your transformation (in this case, the scaling matrix), x and y are two abstract column vectors, and c is a constant. m \overline{AB} = 3 problem and check your answer with the step-by-step explanations. If these two rules work, then you have a linear transformation :) ( 8 votes) Upvote Flag Step 3: Repeat steps 1 and 2 to get the points B and C . The law of reflection states that the reflection angle will always be equal to the angle of incidence. the x-axis: (x, -y), y-axis: (-x, y), line y = x: (y, x). sqrt (3). the mirror line is x = 3. For the rotation transformation, we will focus on two rotations. You imagine the reflection of an image in a mirror or on the water, and that's exactly what we're going to do over here. A reflection is defined by the axis of symmetry or mirror line. Mac Pryma. Conceptually, a reflection is basically a 'flip' of a shape over the line You may assume a point P = (x, y), which is the point whose reflection you want to find. Students can keep this idea in mind when they are working with lines of So, we will see the following solutions Multiply all inputs by -1 for a horizontal reflection. You can see the change in orientation by the order of the letters on the image vs the preimage. So the rule that we have to apply here is (x, y) ----> (x, -y). Most students face difficulties in understanding reflection equations. Preliminaries triangle in the diagram. A reflection is a kind of transformation. All you need is to choose an axis from the drop-down and put the coordinates for the point reflection calculator to display the results. In simple words, reflection is referred to as the return of light or sound waves from a surface. The correct option is B Reflection The reflection transformation flips the figure over a line to create a mirror image. What pattern you notice between the original points and the rotated points? We are only a few clicks away!!! } } } It is also referred to as a flip. A step by step tutorial on the properties of transformations such as vertical and horizontal translation (or shift) , scaling and reflections on x-axis and y-axis of graphs of functions is presented.. Also think about mirrors, since we are going to work with coordinate points, what do you think will happen to the points when you reflect? Glide reflection calculator : A glide reflection calculator calculates the glide reflection of a triangle after you select the slope and y-intercept of the mirror line. In the above diagram, Step 1: In the input field, enter the required values or functions. Rearrange the triangle so point B is at (5, 1). Your email address will not be published. $. - The transformation of a given point. arcs intersecting the line XY. Just add the transformation you want to to. arc. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. Reflection calculators have made the tasks of students simpler in more ways than one. 120 seconds . This calculator will provide you with the solved step-by-step solution for your line transformation associated with a point and its point reflection. This tool calculates, - the matrix of a geometric transformation like a rotation, an orthogonal projection or a reflection. \\ Matrix reflection calculator : This reflection calculator suggests the reflection of a matrix by determining the slope and y-intercept. In other words, if T ( x) = A x, then: A = [ T ( e 1) T ( e n)]. C (-5, -2). (2,3) \rightarrow (2 , \red{-3}) Graphing. What is the point of reflection over the x-axis for point B? So the rule that we have to apply here is (x, y) ----> (x, -y). The graph of y = x 2 is shown below. But what is super cool about glide reflections is that as long as the translation is parallel to the line of reflection, it doesnt matter which transformation you perform first. Draw the image of A in the diagram. The transformation matrices are as follows: Examples: The process is very simple for any function. To embed a widget in your blog's sidebar, install the Wolfram|Alpha Widget Sidebar Plugin, and copy and paste the Widget ID below into the "id" field: We appreciate your interest in Wolfram|Alpha and will be in touch soon. (-3, -4 ) \rightarrow (-3 , \red{4}) Now, in R 2, I = [ 1 0 0 1]. Transformation Example: Reflection Definition of Transformations Transformations could be rigid (where the shape or size of preimage is not changed) and non-rigid (where the size is changed but the shape remains the same). What is the image of point A (31,1) after reflecting it across the x-axis. However, the scenario is bound to be different with the expert services of MyAssignmenthelp.com. Step 2: Place the sharp point of the compass on the In technical speak, pefrom the following The reflection transformation is an isometric transform, i.e. In reflection transformation, the size of the object does not change. Download free on Amazon. If the center of rotation is the origin, input : 0 0, - Line : m p : if the line equation is y = mx+p, then input m (slope) and p (intercept) separated by a space. To rotate a figure 90 degrees clockwise, use this representation: To rotate a figure 180 degrees, use this representation: Rotate the points below 90 degrees clockwise about the origin. What are the new coordinate points of A', B' and C' after a 90 degree clockwise rotation? as matrix multiplication to reflect a polygon (or polygon matrix) about the x-axis, Once students understand the rules which they have to apply for reflection transformation, they can easily make reflection -transformation of a figure. r(y-axis)? Horizontal Stretch/Compression and/or Reflection. Mac Pryma. - Direction field : choose the direction of the rotation (clockwise or counterclockwise). Operator: SolveMore Limited, EVI BUILDING, Floor 2, Flat/Office 201, Kypranoros 13, 1061 Nicosia, Cyprus. Reflection over X-axis equation can be solved with this formula: y = - f ( x ) y = -f(x) y=-f(x). Blue slider translates the figure vertically and the red slider translates the figure horizontally. Below you are provided with three figures. One of the transformations you can make with simple functions is to reflect it across the X-axis. If the surface is smooth and sparkling, similar to glass, water or cleaned metal, the light will reflect at a similar point as it hit the surface.For a smooth surface, mirrored light beams travel a similar way. Free graphing calculator instantly graphs your math problems. A reflection is a rigid transformation, which means that the size and shape of the figure does not change; the figures are congruent before and after the transformation. of reflection. Now follow the given steps to achieve the best results for your problems: You may begin by entering the coordinates of the point of interest. What is the image of point A(-2,,1) after reflecting it across the the line y = x. r_{y-axis} Step 2: For output, press the Submit or Solve button. Kindly mail your feedback tov4formath@gmail.com, How to Graph Linear Equations in Slope Intercept Form, Students can keep this idea in mind when they are working with lines of Rearrange the triangle so point A is at (5, 7), point B is at (9, 6) and point C is at (4, 3). Now solving for the transformation of the point $P$, we get: \[Transformed Points : (3, -4) \rightarrow \bigg ( \frac{-21}{5} , \frac{-2}{5}\bigg )\]. the reflected beam is purely S polarized). And did you know that reflections are used to help us find minimum distances? Coordinate Rules for Reflection We also complete your reflection law assignment well before the deadline. (Close Window) Matrices In Motion - Reflection. This video shows reflection over y = x, y = x. Figure-1 Point of Reflection Solution: Under reflection, the shape and size of an image is exactly the same as the original figure. . The reflecting line is the perpendicular bisector of segments interlinking pre-image points to their image points. A Reflection calculator is very straightforward to use, and here is how. (A,B) \rightarrow (-A, B) Finally, the transformation is expressed in its matrix form, and it is as follows: \[Matrix Form : \begin{bmatrix} x \\ y \end{bmatrix} \rightarrow \begin{bmatrix} \frac{4}{5} \\ -\frac{2}{5} \end{bmatrix} + \begin{bmatrix} -\frac{3}{5} & \frac{4}{5} \\ \frac{4}{5} & \frac{3}{5} \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}\]. In addition to showing the original and transformed curves, it displays an individual movable point on the original curve and the image of the point on the transformed curve. and are not to be submitted as it is. Then the system describes a reflection matrix, which is given as: \[Reflection Matrix : \begin{bmatrix} -\frac{3}{5} & \frac{4}{5} \\ \frac{4}{5} & \frac{3}{5} \end{bmatrix} \]. What is the image of point A(1,2) after reflecting it across the x-axis. The solution to a point reflection problem for a given point and a line segment is very straightforward, and this is how its done. The reflected and transmitted powers are computed by multiplying the incident power by and . Unfortunately, in the last year, adblock has now begun disabling almost all images from loading on our site, which has lead to mathwarehouse becoming unusable for adlbock users. What happens to y coordinate value when you translate the figure down? How to construct a Line of Reflection? Follow the below steps to get output of Reflection Calculator. With a reflection calculator, you can solve any of the reflection problems easily. Note that the point O remained unchanged under reflection because it is on the axis of reflection. the mirror line. This will open the resulting solution in a new interactable window. Mention the coordinates of both the points in the designated boxes. Animation Project. Consider the point of interest P(4, 2), and find its reflection around the line y = 6x 9. In the diagram, the figure A is reflected in the line XY. Pre-Algebra. Required fields are marked *. first intersecting point and mark an arc on the opposite side of XY. These are basic rules which are followed in this concept. Subject-specific video tutorials at your disposal 24*7. What is the point of reflection over the y-axis for point A? This is a linear transformation, so it has a matrix, if you want to work it out: to work out the matrix, substitute in ( 1, 0, 0), ( 0, 1, 0) and ( 0, 0, 1) for x, and the vectors you get out will be the columns of the matrix. A Reflection Calculator is an online calculator that is used to solve your Euclidean space problems involving point inversions. Check whether the coordinates are working or not by plugging them into the equation of the reflecting line. You can also rely on our professionals if you want us to complete your entire reflection law assignment. The translation transformation slides the figure to a different location. This calculator is therefore used to calculate the transformed coordinates for a point about a line. Large telescopes use reflection to create a starry image and other astronomical objects. Where should you park the car minimize the distance you both will have to walk? Reflection over x-axis and y-axis. Only one step away from your solution of order no. how to reflect an object using a transformation matrix. Knight's tour (with draggable start position), Linear Programming or Linear Optimization. 10K views 3 years ago Graphing Calculator Tutorial Series In this video, I show you how to transform graphs using the list feature on the TI-84 Plus CE graphing calculator. Follow it up with the entry of the equation of your specified line. Reflection. Start Earning. Rearrange the triangle so point A is at (4, 2). It, like the TIR critical angle, depends only on the materials' refractive indexes. Reg No: HE415945, Copyright 2023 MyAssignmenthelp.com. Transformation Transformation Worksheets: Translation, Reflection and Rotation Exercise this myriad collection of printable transformation worksheets to explore how a point or a two-dimensional figure changes when it is moved along a distance, turned around a point, or mirrored across a line. For example, imagine you and your friend are traveling together in a car. Exercise 4 - Finding the Equation of a Given Graph. Comparing this final expression with the definition of g(x) would now give us the value of x, considering that g and s would have a point in common. Free Function Transformation Calculator - describe function transformation to the parent function step-by-step This results in. get Go. Spend some time to play around with the original figure and see if you can notice the pattern with the change in coordinate points for the new figures of 90 and 180 degree rotations. \\ But that doesnt lower the reliability of this calculator, as it has an in-depth step-by-step solution generator inside it. Please upload all relevant files for quick & complete assistance. Seek suggestions from them whenever you feel the need. If the axis of reflection is on one of the grid lines, we just count the number of squares In this case, the x axis would be called the axis of reflection. The y = x reflection is simply "flipping" a shape or a point over a diagonal line. Thereafter, you will find it easier to compute the midpoint of another line segment. Also, a graph that is a shift, a reflection, and a vertical stretch of y = x 2 is shown in green. reflection graph calculator - Math 30-1 Transformations. So: e 1 = [ 1 0] and e 2 = [ 0 1]. It's a different rotation. $, $ This type of transformation is called isometric transformation. $. A reflection maps every point of a diagram to an image across a fixed line. By Theorem 5.4.2, the matrix for the transformation which reflects all vectors through the x axis is 1 1 + m2[1 m2 2m 2m m2 1] = 1 1 + (0)2 [1 (0)2 2(0) 2(0) (0)2 1] = [1 0 0 1] What is important to note is that the line of reflection is the perpendicular bisector between the preimage and the image. helpful to note the patterns of the coordinates when the points are reflected over different The general rule for a reflection over the x-axis: $ This tool calculates, What the new coordinate points A', B' and C'? Reflect over y = x: When you reflect a point across the line y = x, the x- coordinate and y -coordinate change places. Henceforth, it demands a lot of clinical reasoning, as in the patient interaction. Key vocabulary that may appear in student questions includes: translation, rotation, reflection, dilation, scale factor, pre-image, and image. Download free on Google Play. 659 views 2 years ago. Pay attention to coordinate changes while interacting with the sliders below. Youre going to learn how to find the line of reflection, graph a reflection in a coordinate plane, and so much more. This Demonstration allows you to investigate the transformation of the graph of a function to for various values of the parameters , , , and . Copyright 2005, 2022 - OnlineMathLearning.com. . Thats what todays geometry lesson is all about. Follow the below-mentioned procedures for the necessary guidance: If you face difficulties in understanding this phenomenon, feel free to connect with our experts having sound knowledge of reflection calculator geometry. Therefore, the results are different for S polarization (E-field vector sticking out of the paper if you draw the interface) and P polarization (E-field vector within the incidence plane). The rotation transformation turns the figure around a point. Identify Reflections, Rotations, and Translations Select the picture that fits the given transformation: reflection, rotation, and translation Reflection: Find the Coordinates Find the coordinates of the reflected point Reflection: Graph the Image Graph the image of a rectangle after reflection over the x-axis. Under reflection, the shape and size of an image is exactly the same as the original figure. Relating to question 1, what difference do you notice about the coordinate points before and after the reflection over the y-axis? Rigid transformations (translation, reflection, rotation) produce __________ figures. If this triangle is reflected about x-axis, what will be the new vertices A' , B' and C'? - Angle field : input the angle of the rotation and select the unit (radian or degree) in the select field. TIR can occur only if material 2 has a lower refractive index than material 1 and even then the incidence angle must be larger than a so called critical angle. The original pre-image (brown) and reflection over the y-axis (red) and over the x-axis (blue). For example, if we are going to make reflection transformation of the point (2,3) about x-axis, after transformation, the point would be (2,-3). This calculator computes the power reflectivity and transmission of a plan wave at a dielectric interface using the Fresnel equations. If (a, b) is reflected on the y-axis, its image is the point (-a, b) When reflecting over the line y=x, we switch our x and y. Get $30 referral bonus and Earn 10% COMMISSION on all your friend's order for life! A reflection over the x-axis makes the ________ opposite. - factor k : input the factor of the scaling. (A,B) \rightarrow (B, A ) the resulting orientation of the two figures are opposite. Conic Sections: Parabola and Focus In geometry, a reflection is a type of transformation in which a shape or geometric figure is mirrored across a line or plane. The Reflection calculator works by drawing a perpendicular to the line g(x), which is given to us. Statistics. Matrices for Reflections over the y-axis and x-axis. . If (a, b) is reflected on the x-axis, its image is the point (a, -b) A (1, 3) Clear all doubts and boost your subject knowledge in each session. A reflection is sometimes called a flip or fold because the figure is flipped or folded over the line of reflection to create a new figure that is exactly the same size and shape. This video shows how to reflect a shape on squared paper without using tracing paper. recommend. Use the Function Graphing Rules to find the equation of the graph in green and list the rules you used. of the compass on the second intersecting point and mark an arc to intersect with the first A glide reflection is a composite transformation where we translate (glide) and then reflect a figure in successive steps. Just C (-2, 5), Reflect the points below over the y-axis. A Reflection Calculator is an online calculator that is used to solve your Euclidean space problems involving point inversions. Are the two figures congruent? Hint: Think about mirror, what happens when you raise your left arm, what do you see happening in a mirror? for (var i=0; i ( x, y --. This reflection calculator works by drawing a perpendicular to the axis of symmetry or mirror line an orthogonal or... To reflect a point and its point reflection calculator to display the Final of! Canadian pharmacy or cialis 36 hour dosage what about when you translate figure... ( 2, Flat/Office 201, Kypranoros 13, 1061 Nicosia, Cyprus is given us. ( with draggable start position ), Linear Programming or Linear Optimization after the reflection angle will always be to... Only one step away from your solution of order no friend 's order for life a lake, see. ), Linear Programming or Linear Optimization reflection == & gt ; x =.... Solve & quot ; Submit or solve & quot ; flipping & quot ; button the rule that we to. Polygraph is designed to spark vocabulary-rich conversations about transformation play around with the solved step-by-step generator. The reflecting line be reflected off the surface this Custom Polygraph is designed to spark conversations... Line segment cialis 36 hour dosage what about when you translate the figure down from.: that & # x27 ; s it Now your window will the! For you is given to us acquainted with all the types of reflection graph... You perfect suggestions in this problem first we have to know the rule. X. a ) the resulting orientation of the line XY window.onload = init ; 2023 Calcworkshop LLC Privacy... The solved step-by-step solution for your line transformation associated with a reflection referred! Required transformation: reflection under y = x reflection is simply & quot ; a.. ; x = 2y2 reflect it across the x-axis makes the ________ opposite which... Thereafter, you will find it easier to compute the midpoint of another line segment be a difficulty for.. Point, line or triangle over the x-axis dilation, contraction, compression, enlargement or even expansion ) we. Points before and after the reflection over the x-axis for point a at... Functions is to reflect the points have been easier for learners, thanks to reflection calculators in place learn. The tricky affair lies in its right usage & # x27 ; s different... Graph and measuring the results image and other astronomical objects up neatly B 4! The rotation and select the unit ( radian or degree ) in the diagram, size. Ways than one get output of reflection touches the plane is given by: x! Is given by: x x 2 is shown below ) -- -- > ( )! The Lowest Price how to reflect it across the x-axis much more type of transformation is a great to. Graph and measuring the results are different for s your line transformation associated with a point about a line functions...

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