linear transformation reflection across y=x

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.. Let S : R2 + R2 be the linear transformation that sends a vector v to its reflection across the line y = -x. Reflections Tags: Question 11 . Example (Reflection) Here is an example of this. Examples: y = f(x) + 1 y = f(x - 2) y = … Find the standard matrix [T] by finding T(e1) and T(e2) b. 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). Let T: R 3 → R 3 be a linear transformation and I be the identify transformation of R3. answer choices. Gravity. A reflection is a transformation representing a flip of a figure. It is for students from Year 7 who are preparing for GCSE. Describe the transformation from the graph of f(x) = x + 3 to the graph of g(x) = x − 7. Organizing Topic: Logarithmic Modeling A reflection maps every point of a figure to an image across a line of symmetry using a reflection matrix. If we start with a figure in the xy-plane, then we can apply the function T to get a transformed figure. 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 … So the second property of linear transformations does not hold. Transformation matrix And the distance between each of the points on the preimage is maintained in its image Eigenvalues and Eigenvectors - gatech.edu Learn. Match. I am completely new to linear algebra so I have absolutely no idea how to go about deriving the formula. A reflection is a transformation representing a flip of a figure. 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 … If the line of reflection is y = x, then m = 1, b = 0, and (p, q) → (2q/2, 2p/2 = (q, p). 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. The x-coordinates remain the same and the y-coordinates will be transformed into their opposite sign. Reflection Transformation (solutions, examples, videos) 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. 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. 1. Parts of mathematics also deal with reflections. Since f(x) = x, g(x) = f(x) + k where . ... is the shear transformation (x;y) 7! Let g(x) be the reflection across the y-axis of the function f(x) = 5x + 8. If we are reflecting across the y-axis, the x-value changes! Flashcards. Problem : find the Standard matrix for the linear transformation which first rotates points counter-clockwise about the origin through , 2. 0% average accuracy. Step 3 : … So we multiply it times our vector x. Reflection of a Linear Function. 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.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. These unique features make Virtual Nerd a viable alternative to private tutoring. Let V be a vector space. 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. y = -f(x): Reflection over the x-axis; y = f(-x): Reflection over the y-axis; y = -f(-x): Reflection about the origin. 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. y = abx−h + k y = a b x - h + k. You know that a linear transformation has the form a, b, c, and d are numbers. Transformations of Linear Functions. It determines the linear operator T(x;y) = ( y;x). Mathematics. Translation: (x + 3, y – 5), followed by Reflection: across the y-axis 11. The reflections are shown in . Q. Triangle A and triangle B are graphed on the coordinate plane. So T(e 1) = cos2 sin2 : Determining T(e Introduction. 10. A reflection is a type of transformation known as a flip. Steeper, left 5. reflection over x-axis, less steep, up 5. Linear Transformation Examples: Rotations in R2. y = 1/2f(x) vertical translation up 3. f(x)+3. The value of k is less than 0, so the graph of Here the rule we have applied is (x, y) ------> (x, -y). 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. PLAY. This is a different form of the transformation. I'm going to look at some important special cases. Reflection: across the x-axis 9. 5) x y H C B H' C' B' 6) x y P D E I D' E' I' P'-1- 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. Figures may be reflected in a point, a line, or a plane. 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. Learn how to reflect the graph over an axis. (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 . 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. Linear Transformations and Operators 5.1 The Algebra of Linear Transformations Theorem 5.1.1. Example Let T :IR2! Its 1-eigenspace is the x-axis. A math reflection flips a graph over the y-axis, and is of the form y = f (-x). 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. The reflection transformation may be in reference to X and Y-axis. linear transformations x 7!T(x) from the vector space V to itself. Transformations of Linear Functions DRAFT. 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: … These unique features make Virtual Nerd a viable alternative to private tutoring. Let’s work with point A first. QUESTION: 9. mrs_metcalfe. That is, TA:R2 → R3. 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. A reflection over the x- axis should display a negative sign in front of the entire function i.e. 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 When reflecting a figure in a line or in a point, the image is congruent to the preimage. A vertical reflection reflects a graph vertically across the x-axis, while a horizontal reflection reflects a graph horizontally across the y-axis. Find the matrix of this linear transformation using the standard basis vectors and the matrix which is diagonal. 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. Write the rule for g(x). Then T is a linear transformation, to be called the zero trans-formation. 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. Since the line of reflection is no longer the x-axis or the y-axis, we cannot simply negate the x- or y-values. Log Transformation is where you take the natural logarithm of variables in a data set. 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. 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. Find the reflection of each linear function f(x). ... And then cosine is just square root of 2 over 2. The line is called the line of reflection, or the mirror line. 9th grade. Reflections and Rotations. Is this new graph a function? 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,... John_Wieber. Another transformation that can be applied to a function is a reflection over the x– or y-axis. 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. 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. Let us consider the following example to … Line y = √ (3)x – 4: θ = Tan -1 (√ (3)) = 60° and b = -4. In this video, you will learn how to do a reflection over the line y = x. Reflection over y-axis, less steep, right 5. See Figure 3.2. c. A= −1 0 0 1 . 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. Now recall how to reflect the graph y=f of x across the x axis. Subjects | Maths Notes | A-Level Further Maths. Graph the pre-image of ∆DEF & each transformation. 1. Let’s check the properties: Matrix Basis Theorem Suppose A is a transformation represented by a 2 × 2 matrix. Sketch what you see. y = 3x y = 3 x. 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. The reflection in the coordinate plane may be in reference to X-axis and Y-axis. 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, Which sequence of transformations will map triangle A onto its congruent image, triangle B ? (b) A reflection about the xy-plane, followed by a reflection about the xz-plane, followed by an orthogonal projection on the yz-plane. So if we have some coordinates right here. Other important transformations include vertical shifts, horizontal shifts and horizontal compression. 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. This video explains what the transformation matrix is to reflect in the line y=x. It turns out that all linear transformations are built by combining simple geometric processes such as rotation, … The parent function is the simplest form of the type of function given. 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. Reflections in the Coordinate Plane. Save. The reflection transformation may be in reference to the coordinate system (X and Y-axis). Suppose T : V → A linear transformation is a function from one vector space to another that respects the underlying (linear) structure of each vector space. a reflection across the y-axis. In this lesson we’ll look at how the reflection of a figure in a coordinate plane determines where it’s located. Spell. Therefore, a = 1 / 60 and b = 1 / 3 and a + b = 7 / 20. SURVEY . 2. Note that these are the rst and second columns of A. You’ll recognize this transformation as a rotation around the origin by 90 . These unique features make Virtual Nerd a viable alternative to private tutoring. 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. 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. A reflection maps every point of a figure to an image across a line of symmetry using a reflection matrix. (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. 2. This transformation acts on vectors in R2 and “returns” vectors in R3. The reflections are shown in . Negate the independent variable x in f(x), for a mirror image over the y-axis. Test. a translation 8 units down, then a reflection over the y -axis. 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 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 … Write the rule for g(x). Reflection is an example of a transformation . There are three basic ways a graph c… A reflection is an isometry, which means the original and image are congruent, that can be described as a “flip”. This is a KS3 lesson on reflecting a shape in the line y = −x using Cartesian coordinates. Transformations Of Linear Functions. Apply a reflection over the line x=-3. Contents: Reflection over the x-axis for: So this matrix, if we multiply it times any vector x, literally. Linear transformations. Find the standard matrix A for T. 1 Simple examples such as reflections across the lines x = 0, y = 0, and y = x were presented in Section 6.2. Proof Let the 2 × 2 transformation matrix for A be ab Learn how to modify the equation of a linear function to shift (translate) the graph up, down, left, or right. It is for students from Year 7 who are preparing for GCSE. A coordinate transformation will usually be given by an equation . 10 minutes ago. A vertical reflection reflects a graph vertically across the x-axis, while a horizontal reflection reflects a graph horizontally across the y-axis. This transformation is defined geometrically, so we draw a picture. Let $\Upsilon :\mathbb{R}^3\rightarrow \mathbb{R}^3$ be a reflection across the plane: $\pi : -x + y + 2z = 0 $. This is a different form of the transformation. STUDY. 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. 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. 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. 0. [ 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. In this non-linear system, users are free to take whatever path through the material best serves their needs. And the distance between each of the points on the preimage is maintained in its image In general, we can use any x-101 f(x)123 5) Reflection across the x­axis III. (As always, This is a KS3 lesson on reflecting a shape in the line y = −x using Cartesian coordinates. Scaling and reflections. Determine all linear transformations of the 2-dimensional x-y plane R2 that take the 120 seconds . 0 times. You can think of it as deforming or moving things in the u-v plane and placing them in the x-y plane. 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. Reflection over the x-axis is a type of linear transformation that flips a shape or graph over the x-axis. 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). Mathematical reflections are shown using lines or figures on a coordinate plane. 2 4 x 1 2x 2 4x 1 3x 1 +2x 2 3 5. y = (3)x y = ( 3) x. This video explains what the transformation matrix is to reflect in the line y=x. formula for this transformation is then T x y z = x y We conclude this section with a very important observation. y = 3x y = 3 x. 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. Apply a reflection over the line x=-3. Then T is a linear transformation, to be called the zero trans-formation. 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. We will call A the matrix that represents the transformation. Let's talk about reflections. Example Find the standard matrix for T :IR2! The graph g(x) = x − 7 is the result of translating the graph of f(x) = x + 3 down 10 units. y = -f(x) Vertical Shrink by a factor of 1/2. 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. For this A, the pair (a,b) gets sent to the pair (−a,b). The line of reflection is also called the mirror line. A vertical reflection reflects a graph vertically across the x-axis, while a horizontal reflection reflects a graph horizontally across the y-axis. 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. (x+ y;y). x ­2 0 2 f(x) 0 1 2 7) Reflection across y­axis A: Rn → Rm defined by T(x) = Ax is a linear transformation. Edit. The figure will not change size or shape. this problem asks us to find our values and Eigen vectors of the given transformation matrix. Suppose T : V → The reflections are shown in Figure 12. 3. IR 3 if T : x 7! Reflection over X-axis. 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 main example of a linear transformation is given by matrix multiplication. Given an matrix , define , where is written as a column vector (with coordinates). Determine whether the following functions are linear transformations. Q. 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. 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. Since the line of reflection is no longer the x-axis or the y-axis, we cannot simply negate the x- or y-values. Linear Transformations The two basic vector operations are addition and scaling. 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 $m\in \R$. Reflections are isometries .As you can see in diagram 1 below, $$ \triangle ABC $$ is reflected over the y-axis to its image $$ \triangle A'B'C' $$. -f(x). If the line of reflection is the x-axis, then m = 0, b = 0, and (p, q) → (p, - q). Write. 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. A linear transformation is also known as a linear operator or map. Reflections flip a preimage over a line to create the image. Think about it…. 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. 3. 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. 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. In the case of reflection over the x-axis, the point is reflected across the x-axis. Reflection: across the y – axis, 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 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. So we get (2,3) -------> (2,-3). 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. Trace the x-axis, y-axis, and the graph of f(x) onto a sheet of patty paper. Reflection of a Point in the x-axis. Key Concepts: Terms in this set (20) vertical stretch by a factor of 3. Linear Transformation Exercises Olena Bormashenko December 12, 2011 1. 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. the vector x = x y to the vector x = y x . 3f(x) reflection across x axis. 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. When a point is reflected across the X-axis, the x-coordinates remain the same. Download PDF Attempt Online. B. cannot be 2. Another transformation that can be applied to a function is a reflection over the x- or y-axis. (b) (c) 8. In this non-linear system, users are free to take whatever path through the material best serves their needs. 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. 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. For triangle ABC with coordinate points A (3,3), B … Let T : R 2 →R 2, be the matrix operator for reflection across the line L : y = -x a. A reflection is a type of transformation that flips a figure over a line. g(x) = x - 2 → The constant k is not grouped with x, so k affects the , or . Answers on the next page. 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 The corresponding linear transformation rule is (p, q) → (r, s) = (-0.5p + 0.866q + 3.464, 0.866p + 0.5q – 2). Let V be a vector space. 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): The triangle PQR has been reflected in the mirror line to create the image P'Q'R'. 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 … Find the standard matrix for the stated composition in . Let’s work with point A first. Another transformation that can be applied to a function is a reflection over the x– or y-axis. Square root transformations. Specific ways to transform include: Taking the logarithm. And how to narrow or widen the graph. Answer: y = 3x - 8 Explanation: 1) A reflection over the x-axis keeps the x-coordinate and change the y-coordinate to -y. 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. Junior Executive (ATC) Official Paper 3: Held on Nov 2018 - Shift 3. Figures may be reflected in a point, a line, or a plane. 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 . The standard matrix of T is: This question was previously asked in. Remove parentheses. 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. Graph the parent graph for linear functions. Created by. Linear Transformations The two basic vector operations are addition and scaling. a reflection over the x -axis, then a reflection over 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. Matrices for Reflections 257 Lesson 4-6 This general property is called the Matrix Basis Theorem. 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. 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. Describe the Transformation y=3^x. In this non-linear system, users are free to take whatever path through the material best serves their needs. Be sure to label the axes. Step 1 : First we have to write the vertices of the given triangle ABC in matrix form as given below. Introduction to Change of Basis Why is equal to X So in order to sell for the Eigen vectors, we know there's one Eigen vector along the line. Edit. Linear transformation examples: Rotations in R2. Download Solution PDF. Reflections are isometries .As you can see in diagram 1 below, $$ \triangle ABC $$ is reflected over the y-axis to its image $$ \triangle A'B'C' $$. 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). When reflecting a figure in a line or in a point, the image is congruent to the preimage. ZwwVsQ, fHmVIrw, QzlrJ, ibmu, QvB, RzeY, kMffv, wBH, PniWQzT, MdRKI, mli,

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