3. A dilation (by a factor c about the origin) is a transformation of the plane

Question

3. A dilation (by a factor c about the origin) is a transformation of the plane described by D(x, y) = (cx, cy ). (a) Apply D to the graph of y = 02, and find the equation of the transformed graph. (Hint: let u= cr, and write v = cr? in terms of u.) (b) For what value of c does D map the graph of y = ar2 onto the graph of Y = br? (a, b > 0)? (c) Per the 2010 eighth grade Common Core Standards, students learn that “a two-dimensional figure such as a graph] is similar to another if the second can be obtained from the first by a sequence of rotations, reflections,

Explanation / Answer

(a) Applying the dilation to the curve, we get

D (x, y) = (cx, cy) => cy = (cx)2, for the graph y = x2.

Putting u = cx, and v = cy = (cx)2 = u2. Thus, the transformed graph is: v = u2

(b) From above, we see that by the dilation of factor c, the curve y = x2, transforms to y = c x2,

Thus, for c = a this curve will transform to y = a x2, Now, if we transform back to y = x2, we nee a dilation factor c' (=1/a), and transforming it to y = b x2, we need a further dilation factor c" (=b). Thus, we can apply both the transformation in one step as c (= c'c" = b/a). Hence, c = b/a, will transform y = a x2, to y = b x2.

(c) All the parabolas can be expressed as ay = b (dx+e)2, where a, b , d and e are real numbers. We have seen that the factors a, b can be dealt by vertical dilation (c = b/a on y = x2) . The factor e can be handled by shifting it to the left or right depending on its sign. The facor d is incorporated by horizontal dilation.

Thus, we can transform a parabola y = x2, by shifts and dilations, as described above to ay = b (dx+e)2 form. So, by these transformations we can conclude that all parabolas are similar, because they can be achieved by simple transformations from one to the other.

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