Using a Transformation In Exercises 11-14, sketch the image S in the uv -plane of the region R in the xy -plane using the given transformations.
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Chapter 14 Solutions
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- Transforming a square Let S = {(u, v): 0 ≤ u ≤ 1, 0 ≤ v ≤ 1} be a unit square in the uv-plane. Find the image of S in the xy-plane under the following transformation. T: x = 2uν, y = u2 - ν2arrow_forwardWhich transformation turns the pre-image ABCD into the image A'B'C'D'? O ( x,y) → (-x, y) O (x,y) (y, -x) O (X,y) → (y,x) O (x.y) → (x, -y)arrow_forwardParabolic coordinates Let T be the transformation x = u2 - ν2, y = 2uν.a. Show that the lines u = a in the uν-plane map to parabolas inthe xy-plane that open in the negative x-direction with verticeson the positive x-axis.b. Show that the lines ν = b in the uv-plane map to parabolas inthe xy-plane that open in the positive x-direction with vertices on the negative x-axis.c. Evaluate J(u, ν).d. Use a change of variables to find the area of the regionbounded by x = 4 - y2/16 and x = y2/4 - 1.e. Use a change of variables to find the area of the curvedrectangle above the x-axis bounded by x = 4 - y2/16,x = 9 - y2/36, x = y2/4 - 1, and x = y2/64 - 16.f. Describe the effect of the transformation x = 2uν,y = u2 - ν2 on horizontal and vertical lines in the uν-plane.arrow_forward
- Consider the transformation given by (a) Plot and label the square described by the points in the x-space E:x= (1,0), F :x (1, 1), G: x = (0, 1) and H : x = (0,0). (b) Plot the image of the square under the given transformation. (c) Plot the point P: y = (-3.5,-3.5) in the y--space. (d) Calculate the corresponding point in the x--space and plot it. (e) Calculate the angle between the lines HE and HG in the y-space. () Use the determinant to find the area of EFGH in the y-space...arrow_forwardB) IR2 --> IR3 (x, y) --> (x, y2, x+y) Is it linear transformation? investigate.arrow_forwardEvery linear transformation x = au + bv, y = cu + dv with ad – bc + 0 maps lines of the uv-plane onto lines of the xy-plane. Find the image (a) of a vertical line u = uo; (b) of a horizontal line v = vo.arrow_forward
- (3) Consider the linear map T : R² → R² which sends (1,0) → (-3,5) and (0, 1) → (4, – 1). (b) Where does this transformation send the area between the vector (4, 2) and the x-axis? Explain algebraically and draw a picture.arrow_forwardDefine Transformation? Write down the properties of Linear transformation and rotational transformation.arrow_forwardNAME: 1 1. Determine the coordinates of the image, plot the image and determine if it is an isometric transformation or not. PLOT THE IMAGE Transformation a) Pre-Image Points COORDINATES Image Points PRE-IMAGE A'( A (1,4) в (2, 1) C (6,-4) B'L C'L Isometry? Yes or No Coordinate Rule (x,y) E (x - 5, y + 3) Transformation Type: b) Pre-Image Points Image Points A'L A (1,4) в (2,-1) C (6,-4) B' L C'L Isometry? Yes or No Coordinate Rule (x,y) E (-x, y) Transformation Type: C) Pre-Image Points Image Points A'L A (1,-4) B (2,-1) C(6,-4) B' L %23 C'L Isometry? Yes or No Coordinate Rule (x,y) E (y, -x) Transformation Type: d) Pre-Image Points Image Points A (-5,-1) в (4,2) C (0,-1) A'L B' U. C'L Isometry? Yes or No Coordinate Rule (х,у) в (х, Зу) Transformation Type: el Pre-Image Points Image Points A' ( A (-2-1) в (-1,2) с (3,-1) B B' L. C'L Coordinate Rule Isometry? Yes or No (х,у) в (2х, 2у) Transformation Type: 2 Image Points A'L B'L f) Pre-Image Points A (1,-3) в (2,0) C (6,-3) CL…arrow_forward
- Consider the change of coordinates x=u2−v2, y=2uv. The Jacobian of this transformation is 4u2+4v2. (b) Let R be the image of the unit square [0, 1] × [0, 1] under this transformation. What is R? Describe it as a set and also plot a graph.arrow_forward(b) - 2+ (c) Sketch the image of {r+iy: z+y> 0} under the transformation w = T(z) = 0 W z+1 In part (b), find an equation of the image of the line x + y = 1 in the region.arrow_forwardName: PLOT THE IMAGE COORDINATES Transformation a) Pre-Image Points PRE-IMAGE Image Points A'L A (1,-4) в (2,-1) C (6,4) B'L Coordinate Rule (x,y) > (x – 5, y + 3) Isometry? Yes or No Transformation Type:arrow_forward
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