Transformation properties #2 - Questions and Answers

Question 1 a) To find: The images of the following points under a 90° rotation counterclockwise about the origin: I. (2, 3) II. (-1, 2) III. (m,n) interms of m and n b) To show: That under a half-turn with the origin as center, the image of a point (a, b) has coordinates (-a, - b). c) To find: The image of P(a, b)under the rotation clockwise by90° about the origin. Answer a) Given: The image will have a 90° rotation counterclockwise a about the origin. Approach: A rotation is a transformation of the plane determined by holding one point the center fixed and rotating the plane about this point by a certain angle clockwise (negative) or counterclockwise (positive). Calculation: I. (2, 3) [Image of coordinate plane showing point (2,3) and its rotation (-3,2) with respect to the origin] II. (-1, 2) [Image of coordinate plane showing point (-1,2) and its rotation (-2,-1) with respect to the origin] III. (m, n) interms of m and n [Image of a coordinate plane showing the point (m,n) rotated 90 degrees to (-n, m)] b) Given: Under a half-turn with the origin as center the image of a point (a, b) has coordinates (-a, - b). Approach: A rotation is a transformation of the plane determined by holding one point-the center- fixed and rotating the plane about this point by a certain angle clockwise (negative) or counterclockwise (positive). Calculation: While taking half-turn rotation with center at the origin for (x, y) the image of it will fall into xy-plane with coordinates (-x,-y), thus the coordinates of the image of a point (a, b) will be (-a, - b) as shown in the graph below. [Image of a coordinate plane showing the point (a,b) and its rotation (-a,-b) with respect to the origin] c) Given: A point P(a, b). Approach: A rotation is a transformation of the plane determined by holding one point-the center- fixed and rotating the plane about this point by a certain angle clockwise (negative) or counterclockwise (positive). Calculation: The rotation of 180° about a point is a half-turn is a rotation thus it has all the properties of rotations. The image of P(-b, a) counterclockwise while turn of 180° will give the desired image as P(b, - a), i.e., clockwise. The graphical representation is as under: As given in question to take the reference from point (a) for turn of 90°, [Image of a coordinate plane showing the point (a,b) and its rotation (-b,a) with respect to the origin] Now, rotate the resulting figure to form the image of P(a, b), under 90°clockwise rotation as under: [Image of a coordinate plane showing the point (a,b) and its rotation (b,-a) with respect to the origin] Question 2 Write an equation for a sinusoidal graph with the following properties: A = -3 period = 2π/3 phase shift = π/4 Answer Step 1 The general equation of sinusoidal equation is y = Asin (kx - ϕ) where A is the amplitude, C is the phase shift. Equation the given period that is 2π/3 to 2π/k to obtain the value of k. 2π/3 = 2π/k k=3 Step 2 Equate the given phase shift to ϕ/k and Substitute 3 for k in ϕ/k = π/4 to obtain the value of ϕ ϕ/3 = π/4 ϕ = 3π/4 Substitute the obtained values of A, k, and ϕ in y = Asin (kx - ϕ) and simplify to obtain the equation of sinusoidal equation. y = Asin (kx - ϕ) y = -3sin (3x - (3π/4)) y = -3sin (3x + 3π/4) Question 3 If f(θ) = csc θ and f(a) = 2, find the exact value of: a) f(-a) b) f(a) + f(a + 2π) + f(a + 4π) Answer Step 1 Part (a) Now, f(-a) = csc (-a) = -csc (a) =-f(a) =-2 Step 2 Part (b) Now, f(a) + f(a + 2π) + f(a + 4π) = csc (a) + csc (a + 2π) + csc (a + 4π) = csc (a) + csc (a) + csc (a) + csc (a) = 3csc (a) = 3f(a) = 3(2) = 6 Question 4 Evaluate Definite Integral Using Integral Properties: If ∫(from 1 to 7) f(x)dx = 2.5 and ∫(from 1 to 7) g(x)dx = 4, find ∫(from 1 to 7) [4f(x) - 2g(x)]dz Answer Step 1 ∫(from 1 to 7) [4f(x) - 2g(x)]dx = ∫(from 1 to 7) 4f(x)dx - ∫(from 1 to 7) 2g(x)dx = 4 ∫(from 1 to 7) f(x)dx - 2 ∫(from 1 to 7) g(x)dx = 4(2.5) - 2(4) = 10-8 = 2 Question 5 A sequence of transformations is described below. A dilation centered at point C with a scale factor of 1/3. A reflection over the line x = 2 Select the statement that describes the resulting image after thee given transformations. The angle measures are preserved, but the image is a reduction The angle measure are preserved, but the image is an enlargement The angle measures are nto preserved, but the image is a reduction The angle measure are not preserved, but the image is an enlargement Answer Step 1 The dilation alters the image's size. The image will be smaller because the scale factor is less than one. While the rotation, just change the direction, while preserving the angles. Step 2 the correct option is : The angle measures are preserved but the image is a reduction