Draw the x- and y-axes and the line / with equation y = x. Plot severalpoints and their

Explain why each of the correspondences pictured below is not a one-toone mapping from set A to set B.

a. Hf'.x> \x\, find the images of -3. 6. and b. Is / a one-to-one function? Explain.

a. If mapping M:(x. y) - (2x, 2y). find the images of P and Q in the diagram. b. Is M a transformation? c. Does M appear to be an isometry? d. Decide whether M maps the midpoint of /"(?,?) P0.3) 2(4.1) PQ to the midpoint of P'Q'

a. If g{x) = 2x - 1. find g(8) and g{-%). b. Find the image of 5.c. Find the preimage of 7.

Use the transformation T:(.x. y) > l.x + 1. y + 2) in this exercise. a. Plot the following points and their images on the chalkboard: A(0. 0). 5(3. 4). C(5, 1). and D(-\. -3). b. Find AB, A'B' . CD. and CD' . c. Does this transformation appear to be an isometry? d. What is the preimage of (0. 0)? of (4. 5)?

What is the image of point AT?

Is the distance between TV and P on the globe the same as the corresponding distance on the polar map

Does the polar map preserve or distort distances?

Explain how Corollary 1 follows from Theorem 14-1.

Explain how Corollary 2 follows from Theorem 14-1.

O is a point equidistant from parallel lines /, and /2 . A mapping M maps each point P of /, to the point P' where PO intersects /: . a. Is the mapping a one-to-one mapping of /, onto l2l b. Does this mapping preserve or distort distance? c. If /, and /: were not parallel, would the mapping preserve distance? Illustrate vour answer with a sketch.

AXYZ is isosceles with XY = XZ. Describe a way of mapping each point of XY to a point of XZ so that the mapping is an isometry.

ABCD is a trapezoid. Describe a way of mapping each point of DC to a point of AB so that the mapping is one-to-one. Is your mapping an isometry

The red and blue squares are congruent and have the same center O. A mapping maps each point P of the red square to the point P' where OP intersects the blue square. a. Is this mapping one-to-one? b. Copy the diagram and locate a point X that is its own image. c. Locate two points R and 5 on the red square and their images R' and S' on the blue square that have the property that^S + R'S'. d. Does this mapping preserve distance? e. Describe a mapping from the red square onto the blue square that does preserve distance.

The transformation T: (.v. y) (x + y. y) preserves areas of figures even though it does not preserve distances. Illustrate this by drawing a square with vertices A(2. 3).fl(4. 3). C(4, 5), and D(.2. 5) and its image .A 'B' CD'. Find the area and perimeter of each figure.

Describe the image of the globe's equator.

Is the image of the Arctic Circle congruent to the image of the equator?

Are distances near the equator distorted more than or less than distances near the Arctic Circle?

Does the North Pole (point N) have an image?

Consider the mapping S: (x, y) *- (x, 0). a. Plot the points P(4, 5), Q(-3, 2), and /?(-3, - 1) and their images. b. Does S appear to be an isometry? Explain. c. Is S a transformation? Explain.

Mapping M maps points A and B to the same image point. Explain why the mapping M does not preserve distance.

Fold a piece of paper. Cut a design connecting the top and bottom point of the fold, as shown. Unfold the shape. Consider a mapping M of the points in the gray region to the corresponding points in the red region. a. Does M appear to be an isometry? b. If a point P is on line k, what is the image of PI c. If a point Q is not on line k, and M(Q) = Q' , what is the relationship between line k and QQ'l

a. Plot the points A(6, 1), B(3, 4), and C(l, -3) and their images A', B' , and C under the transformation R:(x, y) > ( x, y). b. Prove that R is an isometry. (Hint: Let P(xx , y,) and Q(x2 , y2 ) be any two points. Find P' and Q' , and use the distance formula to show that PQ = P'Q'.)

A rolling ball that does not have much spin will bounce off a wall so that the two angles that the path forms with the wall are congruent. Thus, to roll the ball from B off the wall shown and into hole //, you need to aim the ball so that L\ = Z.2. a. Let //' be the image of// by reflection in the wall. BH' intersects the wall at P. Why is L. 1 = Z.3? Why is L 3 = L 2? Why is L 1 = L 2? You can conclude that if you aim for //', the ball will roll to H. b. Show that the distance traveled by the ball equals the distance BH'

In the two-wall shot illustrated at the right, a reflection in one wall maps H to //', and a reflection in a second wall (extended) maps //' to //". To roll the ball from B to //, you aim for //". Show that the total distance traveled by the ball equals the distance BH"

Show how to score a hole in one on the fifth hole of the golf course shown by rolling the ball off one wall.

Repeat Exercise 26 but roll the ball off two walls.

Repeat Exercise 26 but roll the ball off three walls.

A ball rolls at a 45 angle away from one side of a billiard table that has a coordinate grid on it. If the ball starts at the point (0, 1) it will eventually return to its starting point. Would this happen if the ball started from other points on the y-axis between (0, 0) and (0, 4)?

The line with equation y =-2x + 3 is reflected in the y-axis. Find an equation of the image line

The line with equation y = x + 5 is reflected in the x-axis. Find an equation of the image line.

In each exercise Rk :A -* A'. Find an equation of line k.

In each exercise Rk :A -* A'. Find an equation of line k.

In each exercise Rk :A -* A'. Find an equation of line k.

In each exercise Rk :A -* A'. Find an equation of line k.

In each exercise Rk :A -* A'. Find an equation of line k.

In each exercise Rk :A -* A'. Find an equation of line k.

Draw the x- and y-axes and the line / with equation y = x. Plot several points and their images under /?,. What is the image of (a, b)l

Draw the x- and y-axes and the vertical line 7 with equation x = 5. Find the images under Rt of the following points. a. (4, 3) b. (0, -2) c. (-3, 1) d. (x, y)

Repeat Exercise 39 letting j be the horizontal line with equation y = 6.