Let g(x) be the reflection across the y-axis of the function f(x) = 5x + 8. And the distance between each of the points on the preimage is maintained in its image Then T is a linear transformation, to be called the zero trans-formation. Organizing Topic: Logarithmic Modeling Find the standard matrix [T] by finding T(e1) and T(e2) b. Now first of, If I have this plane then for $\Upsilon(x,y,z) = (-x,y,2z)$ I get this when passing any vector, so the matrix using standard basis vectors is: … If we start with a figure in the xy-plane, then we can apply the function T to get a transformed figure. It is for students from Year 7 who are preparing for GCSE. Test. The linear transformation matrix for a reflection across the line $y = mx$ is: $$\frac{1}{1 + m^2}\begin{pmatrix}1-m^2&2m\\2m&m^2-1\end{pmatrix} $$ My professor gave us the formula above with no explanation why it works. Example (Reflection) Here is an example of this. Find the reflection of each linear function f(x). Transformations of Linear Functions DRAFT. Reflection of a Linear Function. Simple examples such as reflections across the lines x = 0, y = 0, and y = x were presented in Section 6.2. This is a different form of the transformation. Reflection is an example of a transformation . Let S : R2 + R2 be the linear transformation that sends a vector v to its reflection across the line y = -x. Figures may be reflected in a point, a line, or a plane. That is, TA:R2 → R3. Reflection Transformation of a Linear Function SMP SEAA C04L06 255-261 The line is called the line of reflection, or the mirror line. Graph the pre-image of ∆DEF & each transformation. This video explains what the transformation matrix is to reflect in the line y=x. Therefore, a = 1 / 60 and b = 1 / 3 and a + b = 7 / 20. Figures may be reflected in a point, a line, or a plane. For example, when point P with coordinates (5,4) is reflecting across the X axis and mapped onto point P’, the coordinates of P’ are (5,-4). Transformations of Linear Functions. 1. Linear Transformations Let T:ℝ2→ℝ2 be the linear transformation that first reflects points through the x-axis and then then reflects points through the line y=−x. 0 times. 2. Matrix Basis Theorem Suppose A is a transformation represented by a 2 × 2 matrix. Since the line of reflection is no longer the x-axis or the y-axis, we cannot simply negate the x- or y-values. Flashcards. 215 C H A P T E R 5 Linear Transformations and Matrices In Section 3.1 we defined matrices by systems of linear equations, and in Section 3.6 we showed that the set of all matrices over a field F may be endowed with certain algebraic properties such as addition and multiplication. This video explains what the transformation matrix is to reflect in the line y=x. Describe the transformation from the graph of f(x) = x + 3 to the graph of g(x) = x − 7. Reflections flip a preimage over a line to create the image. A reflection is a transformation in which each point of a figure has an image that is equal in distance from the line of reflection but on the opposite side. Another transformation that can be applied to a function is a reflection over the x- or y-axis. This page includes a lesson covering 'how to reflect a shape in the line y = −x using Cartesian coordinates' as well as a 15-question worksheet, which is printable, editable and sendable. the vector x = x y to the vector x = y x . The value of k is less than 0, so the graph of QUESTION: 9. Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. In a reflection transformation, all the points of an object are reflected or flipped on a line called the axis of reflection or line of reflection. Example: A reflection is defined by the axis of symmetry or mirror line. The reflections are shown in . Step 1 : First we have to write the vertices of the given triangle ABC in matrix form as given below. Reflection: across the x-axis 9. Step 2 : Since the triangle ABC is reflected about x-axis, to get the reflected image, we have to multiply the above matrix by the matrix given below. Example Find the standard matrix for T :IR2! A reflection maps every point of a figure to an image across a line of symmetry using a reflection matrix. B. cannot be 2. (3, -5) Original Point (- 3, -5) The opposite value for x = (-3, 5) New Point Try these: On a separate sheet of paper, find the coordinates of each point after a reflection across the y-axis. 120 seconds . It turns out that the converse of this is true as well: Theorem10.2.3: Matrix of a Linear Transformation If T : Rm → Rn is a linear transformation, then there is a matrix A such that T(x) = A(x) for every x in Rm. Write the rule for g(x). Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. Matrices for Reflections 257 Lesson 4-6 This general property is called the Matrix Basis Theorem. This is (x,y) → (x,-y) 2) Then, you can replace the new coordinates in the original equation, f(x) to get the equation of g(x): Think about it…. If the L2 norm of , , and is unity, the transformation matrix can be expressed as: = [] Note that these are particular cases of a Householder reflection in two and three dimensions. Find a non-zero vector x such that T(x) = x c. Find a vector in the domain of T for which T(x,y) = (-3,5) Homework Equations The Attempt at a Solution a. I found [T] = 0 -1-1 0 b. Computing T(e 1) isn’t that bad: since L makes an angle with the x-axis, T(e 1) should make an angle with L, and thus an angle 2 with the x-axis. You can think of it as deforming or moving things in the u-v plane and placing them in the x-y plane. The handout, Reflection over Any Oblique Line, shows the derivations of the linear transformation rules for lines of reflection y = √ (3)x – 4 and y = -4/5x + 4. The reflection transformation may be in reference to the coordinate system (X and Y-axis). Steeper, left 5. reflection over x-axis, less steep, up 5. A reflection is a transformation representing a flip of a figure. 5) x y H C B H' C' B' 6) x y P D E I D' E' I' P'-1- The reflections are shown in Figure 12. This transformation acts on vectors in R2 and “returns” vectors in R3. If A : (1, 0) → (x 1, y 1) and A : (0, 1) → (x 2, y 2), then A has the matrix x 1 x 2 y 1 y 2. Linear Transformations The two basic vector operations are addition and scaling. Find the matrix of this linear transformation using the standard basis vectors and the matrix which is diagonal. Trace the x-axis, y-axis, and the graph of f(x) onto a sheet of patty paper. Sketch what you see. PLAY. (a) A rotation of 30° about the x-axis, followed by a rotation of 30° about the z-axis, followed by a contraction with factor . Let T:R2 + R2 be the linear transformation defined by T x + 3y Зу (a) (5 points) Find the standard matrix of S. (b) (5 points) Find the standard matrix of T. (c) (5 points) Find the standard matrix of ToS. Write the rule for g(x). The matrix of a linear transformation is a matrix for which \(T(\vec{x}) = A\vec{x}\), for a vector \(\vec{x}\) in the domain of T. This means that applying the transformation T to a vector is the same as multiplying by this matrix. ... And then cosine is just square root of 2 over 2. Scaling and reflections. Another transformation that can be applied to a function is a reflection over the x– or y-axis.A vertical reflection reflects a graph vertically across the x-axis, while a horizontal reflection reflects a graph horizontally across the y-axis.The reflections are shown in Figure 9. Mathematics. Line y = √ (3)x – 4: θ = Tan -1 (√ (3)) = 60° and b = -4. Describe the Transformation y=3^x. If we are reflecting across the y-axis, the x-value changes! a translation of 3 units to the right, followed by a reflection across the x-axis a rotation of 1800 about the or-gin a translation of 12 units downward, followed by a reflection across the y-axis a reflection across the y-axis, followed by a reflection across the x-axis a reflection across the ine with equation y = x Part B And how to narrow or widen the graph. Edit. Let T: R 2 → R 2 be the linear transformation that reflects over the line L defined by y = − x, and let A be the matrix for T. We will find the eigenvalues and eigenvectors of A without doing any computations. Reflection over X-axis. Define T : V → V as T(v) = v for all v ∈ V. Then T is a linear transformation, to be called the identity transformation of V. 6.1.1 Properties of linear transformations Theorem 6.1.2 Let V and W be two vector spaces. Edit. Reflections in the Coordinate Plane. y = -f(x) Vertical Shrink by a factor of 1/2. A coordinate transformation will usually be given by an equation . The standard matrix of T is: This question was previously asked in. This is a KS3 lesson on reflecting a shape in the line y = −x using Cartesian coordinates. If the line of reflection is y = -2x + 4, then m = -2, b = 4, (1 – m2)/(1 + m2) = -3/5, (m2 – 1)/(m2 + 1) = 3/5, Reflection across x 1 axis Reflection across x 2 axis Reflection across line x 2 = x 1 Reflection across line x 2 = x 1 2. IR 3 if T : x 7! In this non-linear system, users are free to take whatever path through the material best serves their needs. Is this new graph a function? So this matrix, if we multiply it times any vector x, literally. Reflection: across the y – axis, The line y=x, when graphed on a graphing calculator, would appear as a straight line cutting through the origin with a slope of 1. Q. Triangle A and triangle B are graphed on the coordinate plane. If the line of reflection is the x-axis, then m = 0, b = 0, and (p, q) → (p, - q). Determine the Kernel of a Linear Transformation Given a Matrix (R3, x to 0) Concept Check: Describe the Kernel of a Linear Transformation (Projection onto y=x) Concept Check: Describe the Kernel of a Linear Transformation (Reflection Across y-axis) Coordinates and Change of Base. 3f(x) reflection across x axis. 1. Reflection over the x-axis is a type of linear transformation that flips a shape or graph over the x-axis. Example Let T :IR2! Linear Transformations on the Plane A linear transformation on the plane is a function of the form T(x,y) = (ax + by, cx + dy) where a,b,c and d are real numbers. Learn how to reflect the graph over an axis. linear transformations x 7!T(x) from the vector space V to itself. Created by. x y J Z L 2) translation: 4 units right and 1 unit down x y Y F G 3) translation: 1 unit right and 1 unit up x y E J T M 4) reflection across the x-axis x y M C J K Write a rule to describe each transformation. Examples: y = f(x) + 1 y = f(x - 2) y = … Introduction to Change of Basis This transformation is defined geometrically, so we draw a picture. x-101 f(x)123 5) Reflection across the xaxis III. 37) reflection across the x-axis x y S K N U 38) reflection across y = x x y B M D 39) reflection across y = -x x y Y Z E 40) reflection across the x-axis x y T W D 41) rotation 90° counterclockwise about the origin x y D F B 42) rotation 180° about the origin x y E U L V We will call A the matrix that represents the transformation. (b) (c) 8. These unique features make Virtual Nerd a viable alternative to private tutoring. 10. In this video, you will learn how to do a reflection over the line y = x. This page includes a lesson covering 'how to reflect a shape in the line y = −x using Cartesian coordinates' as well as a 15-question worksheet, which is printable, editable and sendable. y = 3x y = 3 x. The reflections are shown in . Mathematical reflections are shown using lines or figures on a coordinate plane. The main example of a linear transformation is given by matrix multiplication. Given an matrix , define , where is written as a column vector (with coordinates). a reflection across the y-axis. y = abx−h + k y = a b x - h + k. Let us consider the following example to … If they are, prove it; if not, provide a counterexample to one of the properties: (a) T : R2!R2, with T x y = x+ y y Solution: This IS a linear transformation. (x+ y;y). a reflection over the x -axis, then a reflection over the y -axis. A linear transformation is also known as a linear operator or map. A: Rn → Rm defined by T(x) = Ax is a linear transformation. Q. Reflections are isometries .As you can see in diagram 1 below, $$ \triangle ABC $$ is reflected over the y-axis to its image $$ \triangle A'B'C' $$. 3. Square root transformations. Now recall how to reflect the graph y=f of x across the x axis. Homework Statement Let L: R^3 -> R^3 be the linear transformation that is defined by the reflection about the plane P: 2x + y -2z = 0 in R^3. Learn. If (a, b) is reflected on the line y = x, its image is the point (b, a) If (a, b) is reflected on the line y = -x, its image is the point (-b, a) Geometry Reflection. In linear algebra, linear transformations can be represented by matrices.If is a linear transformation mapping to and is a column vector with entries, then =for some matrix , called the transformation matrix of [citation needed].Note that has rows and columns, whereas the transformation is from to .There are alternative expressions of transformation matrices … These unique features make Virtual Nerd a viable alternative to private tutoring. Since the line of reflection is no longer the x-axis or the y-axis, we cannot simply negate the x- or y-values. The triangle PQR has been reflected in the mirror line to create the image P'Q'R'. A vertical reflection reflects a graph vertically across the x-axis, while a horizontal reflection reflects a graph horizontally across the y-axis. A reflection is a type of transformation known as a flip. Let’s work with point A first. Reflections and Rotations. Reflection: across the y-axis, followed by Translation: (x + 2, y) The vertices of ∆DEF are D(2,4), E(7,6), and F(5,3). I am completely new to linear algebra so I have absolutely no idea how to go about deriving the formula. Its 1-eigenspace is the x-axis. See Figure 3.2. c. A= −1 0 0 1 . 10 minutes ago. And the distance between each of the points on the preimage is maintained in its image Answers on the next page. John_Wieber. Negate the independent variable x in f(x), for a mirror image over the y-axis. 0% average accuracy. We de ne T Aby the rule T A(x)=Ax:If we express Ain terms of its columns as A=(a 1 a 2 a n), then T A(x)=Ax = Xn i=1 x ia i: Hence the value of T A at x is the linear combination of the columns of A which is the … A vertical reflection reflects a graph vertically across the x-axis, while a horizontal reflection reflects a graph horizontally across the y-axis. Translation: (x + 3, y – 5), followed by Reflection: across the y-axis 11. Gravity. STUDY. Linear Transformation Examples: Rotations in R2. [ x 1 + 3 x 2 + 3 x 3 + 3 x 4 + 3 y 1 + 2 y 2 + 2 y 2 + 2 y 2 + 2] If we want to dilate a figure we simply multiply each x- and y-coordinate with the scale factor we want to dilate with. Let’s check the properties: Key Concepts: Terms in this set (20) vertical stretch by a factor of 3. If the line of reflection is y = x, then m = 1, b = 0, and (p, q) → (2q/2, 2p/2 = (q, p). mrs_metcalfe. This is a KS3 lesson on reflecting a shape in the line y = −x using Cartesian coordinates. Linear transformations. this problem asks us to find our values and Eigen vectors of the given transformation matrix. -f(x). Junior Executive (ATC) Official Paper 3: Held on Nov 2018 - Shift 3. Problem : find the Standard matrix for the linear transformation which first rotates points counter-clockwise about the origin through , Let T: R 3 → R 3 be a linear transformation and I be the identify transformation of R3. A reflection is a transformation representing a flip of a figure. It turns out that all linear transformations are built by combining simple geometric processes such as rotation, … Determine all linear transformations of the 2-dimensional x-y plane R2 that take the Then T is a linear transformation, to be called the zero trans-formation. 2. Since f(x) = x, g(x) = f(x) + k where . Match. 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). y = -f(x): Reflection over the x-axis; y = f(-x): Reflection over the y-axis; y = -f(-x): Reflection about the origin. 0. y = (3)x y = ( 3) x. The rule for reflecting over the X axis is to negate the value of the y-coordinate of each point, but leave the x-value the same. A linear transformation is a function from one vector space to another that respects the underlying (linear) structure of each vector space. A linear transformation T : R 2 → R 2 first reflects points through the vertical axis (y-axis) and then reflects points through the line x = y. Reflection of a Point in the x-axis. Consider the matrix A = 5 1 0 −3 −1 2 and define TA ⇀x= A ⇀x for every vector for which A ⇀x is defined. When reflecting a figure in a line or in a point, the image is congruent to the preimage. Reflections are isometries .As you can see in diagram 1 below, $$ \triangle ABC $$ is reflected over the y-axis to its image $$ \triangle A'B'C' $$. g(x) = x - 2 → The constant k is not grouped with x, so k affects the , or . Introduction. Reflection through the line : Reflection through the origin: Since for linear transformations, the standard matrix associated with compositions of geometric transformations is just the matrix product . Download Solution PDF. If there is a scalar C and a non-zero vector x ∈ R 3 such that T (x) = Cx, then rank (T – CI) A. cannot be 0. When a point is reflected across the X-axis, the x-coordinates remain the same. Other important transformations include vertical shifts, horizontal shifts and horizontal compression. In the case of reflection over the x-axis, the point is reflected across the x-axis. So T(e 1) = cos2 sin2 : Determining T(e The corresponding linear transformation rule is (p, q) → (r, s) = (-0.5p + 0.866q + 3.464, 0.866p + 0.5q – 2). x 2 0 2 f(x) 0 1 2 7) Reflection across yaxis 2. Determine whether the following functions are linear transformations. Let V be a vector space. Reflection over y-axis, less steep, right 5. So the second property of linear transformations does not hold. It is for students from Year 7 who are preparing for GCSE. 168 6.2 Matrix Transformations and Multiplication 6.2.1 Matrix Linear Transformations Every m nmatrix Aover Fde nes linear transformationT A: Fn!Fmvia matrix multiplication. In general, we can use any Eigenvalues of re ections in R2 ... There’s a general form for a re ection across the line of slope tan , that is, across the line that makes an angle of with the x-axis. The figure will not change size or shape. Let $\Upsilon :\mathbb{R}^3\rightarrow \mathbb{R}^3$ be a reflection across the plane: $\pi : -x + y + 2z = 0 $. answer choices. So if we have some coordinates right here. Answer: y = 3x - 8 Explanation: 1) A reflection over the x-axis keeps the x-coordinate and change the y-coordinate to -y. A reflection is an isometry, which means the original and image are congruent, that can be described as a “flip”. Save. Why is equal to X So in order to sell for the Eigen vectors, we know there's one Eigen vector along the line. Parts of mathematics also deal with reflections. (As always, Apply a reflection over the line x=-3. For triangle ABC with coordinate points A (3,3), B … Apply a reflection over the line x=-3. y = 3x y = 3 x. Suppose T : V → You know that a linear transformation has the form a, b, c, and d are numbers. Tags: Question 11 . So we multiply it times our vector x. opri cGraw-Hll Eucaton Example 1 Vertical Translations of Linear Functions Describe the translation in g(x) = x - 2 as it relates to the graph of the parent function. The graph g(x) = x − 7 is the result of translating the graph of f(x) = x + 3 down 10 units. The reflection transformation may be in reference to X and Y-axis. A math reflection flips a graph over the y-axis, and is of the form y = f (-x). x 2 T This linear transformation stretches the vectors in the subspace S[e 1] by a factor of 2 and at the same time compresses the vectors in the subspace S[e 2] by a factor of 1 3. Specific ways to transform include: Taking the logarithm. For reflection, which is basically just flipping the line of a linear function across the x-axis or the y-axis, you would follow the same steps as any function. To reflect a point through a plane + + = (which goes through the origin), one can use =, where is the 3×3 identity matrix and is the three-dimensional unit vector for the vector normal of the plane. This is a different form of the transformation. Graph the parent graph for linear functions. The line of reflection is also called the mirror line. From this perspec-tive, the nicest functions are those which \preserve" these operations: Def: A linear transformation is a function T: Rn!Rm which satis es: (1) T(x+ y) = T(x) + T(y) for all x;y 2Rn (2) T(cx) = cT(x) for all x 2Rn and c2R. Next we’ll consider the linear transformation that re ects vectors across a line Lthat makes an angle with the x-axis, as seen in Figure4. From this perspec-tive, the nicest functions are those which \preserve" these operations: Def: A linear transformation is a function T: Rn!Rm which satis es: (1) T(x+ y) = T(x) + T(y) for all x;y 2Rn (2) T(cx) = cT(x) for all x 2Rn and c2R. In particular, the two basis vectors e 1 = 1 0 and e 2 = 0 1 are sent to the vectors e 2 = 0 1 and e 2 = 1 0 respectively. In this lesson we’ll look at how the reflection of a figure in a coordinate plane determines where it’s located. Let V be a vector space. The map T from which takes every function S(x) from C[0,1] to the function S(x)+1 is not a linear transformation because if we take k=0, S(x)=x then the image of kT(x) (=0) is the constant function 1 and k times the image of T(x) is the constant function 0. Find the standard matrix A for T. 1 Transformations Of Linear Functions. Let’s work with point A first. 3. So we get (2,3) -------> (2,-3). 9th grade. Download PDF Attempt Online. Define T : V → V as T(v) = v for all v ∈ V. Then T is a linear transformation, to be called the identity transformation of V. 6.1.1 Properties of linear transformations Theorem 6.1.2 Let V and W be two vector spaces. Linear Transformations The two basic vector operations are addition and scaling. In this non-linear system, users are free to take whatever path through the material best serves their needs. In this non-linear system, users are free to take whatever path through the material best serves their needs. Find the standard matrix for the stated composition in . y = 1/2f(x) vertical translation up 3. f(x)+3. Write. Linear transformation examples: Rotations in R2. formula for this transformation is then T x y z = x y We conclude this section with a very important observation. There are three basic ways a graph c… Let $T:\R^2 \to \R^2$ be a linear transformation of the $2$-dimensional vector space $\R^2$ (the $x$-$y$-plane) to itself which is the reflection across a line $y=mx$ for some $m\in \R$. A reflection maps every point of a figure to an image across a line of symmetry using a reflection matrix. Linear Transformations. A linear transformation is a function from one vector space to another that respects the underlying (linear) structure of each vector space. A linear transformation is also known as a linear operator or map. The range of the transformation may be the same as the domain, and when that happens,... The matrix for a reflection is orthogonal with determinant −1 and eigenvalues −1, 1, 1, ..., 1. The product of two such matrices is a special orthogonal matrix that represents a rotation. The range of the transformation may be the same as the domain, and when that happens, the transformation is known as an endomorphism or, if invertible, an … Contents: Reflection over the x-axis for: Log Transformation is where you take the natural logarithm of variables in a data set. So if we apply this transformation 0110 onto around a point x y, we get why x so, Drawing that on a graph Yet why X the vector over here, which is a reflection over in line. Reflect the graph of f(x) across the line y = x by holding the top-right and bottom left corners of the patty paper in each hand and flipping the sheet of patty paper over. Another transformation that can be applied to a function is a reflection over the x– or y-axis. Namely, L(u) = u if u is the vector that lies in the plane P; and L(u) = -u if u is a vector perpendicular to the plane P. Find an orthonormal basis for R^3 and a matrix A such that A is diagonal and A is the matrix … Every point above the x-axis is reflected to its corresponding position below the x-axis; Every point below the x-axis is reflected to its corresponding position above the x-axis.. We can use matrices to translate our figure, if we want to translate the figure x+3 and y+2 we simply add 3 to each x-coordinate and 2 to each y-coordinate. The reflection in the coordinate plane may be in reference to X-axis and Y-axis. ... is the shear transformation (x;y) 7! A reflection is a type of transformation that flips a figure over a line. Another transformation that can be applied to a function is a reflection over the x– or y-axis. The x-coordinates remain the same and the y-coordinates will be transformed into their opposite sign. It determines the linear operator T(x;y) = ( y;x). I'm going to look at some important special cases. Let's talk about reflections. SURVEY . Proof Let the 2 × 2 transformation matrix for A be ab Suppose T : V → Transformation of Linear Functions Defined by a Table 6) Reflection across xaxis Let g(x) be the indicated transformation of f(x), defined in the table below. Spell. Which sequence of transformations will map triangle A onto its congruent image, triangle B ? For this A, the pair (a,b) gets sent to the pair (−a,b). Step 3 : … Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. Be sure to label the axes. The parent function is the simplest form of the type of function given. Then find the matrix representation of the linear transformation $T$ with respect to the standard basis $B=\{\mathbf{e}_1, \mathbf{e}_2\}$ of $\R^2$, where A vertical reflection reflects a graph vertically across the x-axis, while a horizontal reflection reflects a graph horizontally across the y-axis. Let T : R 2 →R 2, be the matrix operator for reflection across the line L : y = -x a. A reflection over the x- axis should display a negative sign in front of the entire function i.e. Learn how to modify the equation of a linear function to shift (translate) the graph up, down, left, or right. (b) A reflection about the xy-plane, followed by a reflection about the xz-plane, followed by an orthogonal projection on the yz-plane. When reflecting a figure in a line or in a point, the image is congruent to the preimage. Remove parentheses. a translation 8 units down, then a reflection over the y -axis. These unique features make Virtual Nerd a viable alternative to private tutoring. The transformation from the first equation to the second one can be found by finding a a, h h, and k k for each equation. Subjects | Maths Notes | A-Level Further Maths. Note that these are the rst and second columns of A. You’ll recognize this transformation as a rotation around the origin by 90 . 2 4 x 1 2x 2 4x 1 3x 1 +2x 2 3 5. Linear Transformations and Operators 5.1 The Algebra of Linear Transformations Theorem 5.1.1. Linear Transformation Exercises Olena Bormashenko December 12, 2011 1. Here the rule we have applied is (x, y) ------> (x, -y).
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