Properties and Definitions of Transformations
Transformations are fundamental concepts in mathematics, pivotal in various fields from geometry to data analysis. They encapsulate the manipulation of objects in a mathematical space, altering their position, size, or orientation. Defined by specific rules, transformations encompass translations, rotations, reflections, and more intricate operations. Each transformation possesses unique properties governing its behavior, such as preserving distances, angles, or orientations. Understanding these properties and definitions not only illuminates the mechanics of transformations but also underpins their applications in diverse mathematical contexts, making them indispensable tools for modeling and problem-solving.
Questions
- Describe a sequence of transformations that transform the graph of f(x) into the graph of g(x)? #f(x)=sqrtx# and #g(x)=-3(sqrt(x+1))-4#
- What transformation transforms (p, q) to (q, p)?
- Triangle RST with vertices R(2, 5), S(1, 4), and T(3, 1) is Translated 3 units right. What are the coordinate of S', R' &T'?
- Image point B'(4,-8) was transformed using the translation (x-2, y+ 3). What were the coordinates of B?
- What is the image of point (3, 5) if the rotation is -180°?
- Using the graph of f(x)= 1/x as a starting point, describe the transformations to get to #g(x) = 1/x-4#?
- Using the graph of #f(x)=x^2# as a guide, describe the transformations, and then graph the function #g(x)=-2x^2#?
- If the following graph is of a function #f(x)#, how will the graph of (i) #f(x+3)#, (ii) #f(x)+3#, (iii) #-f(x)# and (iv) #1-f(x-3)# appear?
- What are the rules of transformation - specifically, of dilation, rotation, reflection and translation?
- Which single transformation that would have the same result as the two transformations (a) rotation by #180^@# about origin and (b) reflection in #y#-axis?
- Describe a sequence of transformations that transform the graph of f(x) into the graph of g(x)? #f(x)=x^2# and #g(x)=(x-4)^2+4#
- What are the coordinates of the image of point #A(2,-7)# under the translation #(x,y)-> (x-3,y+5)#?
- What transformation is represented by the rule (x, y)→(x, −y) ?
- What are the coordinates of the point (−4, 2)(−4, 2) after a translation 2 units left and 2 units up?
- Describe the transformations applied to #y=x²# to obtain the graph of #y=-(x+3)²-2#?
- The vertices of #triangleEFG# are E(0,2), F(-3,-4), and G(2,-5). If this shape is translated to the right 2 units, and down 3 units what are the new vertices of #triangleE'F'G'#?
- Point (w, z) is transformed by the rule (w+5, z) ?
- What is a transformation? And what are the four types of transformations?
- What is a manifold in topology?
- What are the different coordinate transformation conjectures?