T1 INTRODUCTION TO TRANSFORMATIONS
Homework
1. Let g(x) = 2x – 1.
a. Find g (8) and g (-8).
b. Find the image of 5.
c. Find the preimage of 7.
d. Is g(x) a one-to-one function?
Answers
15, and -17
9
X=4
Yes
2. Mapping G maps each point (x, y) to the point (2x, y – 1).
a. Find A’ and B’, the images of A (3,0) and B (1,4). A’ = (6, -1) , B’ (2, 3)
b. Let P (x 1, y 1) and Q(x 2, y 2). Decide whether G maps M, the midpoint of PQ, to M’ the
midpoint of P’Q’.
Yes
c. Decide whether PQ = P’Q’. No
d. Is G a one-to-one function? Yes
3. Let the point (x, y) be mapped to the point (x’, y’) by
x’ = 2x and y’ = 3y
a. Find the images of (1, - 1), (1/2, 1/3) and (- 3,4). (1, 1) ⇒ (2, 3); (1/2, 1/3) ⇒ (1, 1)
and (- 3,4) ⇒ (-6, 12)
b. Find the preimage of (2, - 3) and (4, 6). (2, - 3) does not have a preimage and the
preimage of (4, 6) is (2, y)
c. Is this mapping a transformation? Why or Why not? No (2, 1) ⇒ (4, 6) and (2, 2) ⇒
(4, 6).
4. Show the image of the parabola y = x 2 under the mapping defined by
x’ = y and y’ = x 2 + y
a. Is part of a straight line? Identify the line. x’ = y = x 2 and y’ = x 2 + x 2 = 2x 2.
So (x, y) maps to (x 2, 2x 2) which is on the line y’ = 2x’ for x’ ≥ 0.
b. Is this mapping a transformation? Why or Why not?
No (2, 4) and (-2, 4) have the same image and (-1, 1) has no preimage. Not 1-1 nor onto.
5. a. Find the line into which all points of the plane map, under the mapping
x’ = 2x – y and y’ = - 6x + 3y.
y’ = -3x’
b. Is the mapping a transformation? No, not 1-1. (1,3) and (2, 5) have the same image.
An Isometry is a transformation that preserves distance.
6. ABCD is a trapezoid with bases AB and CD. Describe a way of mapping each point of CD to a
point of AB so that the mapping is one-to-one. Is your mapping an isometry?
Extend the two sides until them meet. Draw a segment from the point of intersection thru both
bases. This will give a 1-1 mapping. It is not an isometry.
7. Prove that an isometry maps a triangle ABC into a congruent triangle A’B’C’.
Since it is an isometry mAB = mA’B’, mAC = mA’C’ and mCB = mC’B’ and the two triangles
are congruent by SSS.
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8. Prove that an isometry preserves angle measure.
Directly from 7.
9. Prove that the identity map is an isometry.
A = A’ and B = B’ so mAB = mA’B’ and thus an isometry.
10. Prove that an isometry maps parallel lines to parallel lines.
Use the theorem that proved parallel lines are everywhere equidistant apart.
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