13. a) The graph of y x' is transformed to 1 obtain the graph of y -2(x- 1)* + 2. 4 List the parameters and describe the corresponding transformations.
b) Sketch a graph of y = -2(x- 1)]* + 2.
Answers
Transformations of the parabola
Translations
We can translate the parabola vertically to produce a new parabola that is similar to the basic parabola. The function y=x2+b has a graph which simply looks like the standard parabola with the vertex shifted b units along the y-axis. Thus the vertex is located at (0,b). If b is positive, then the parabola moves upwards and, if b is negative, it moves downwards.
Similarly, we can translate the parabola horizontally. The function y=(x−a)2 has a graph which looks like the standard parabola with the vertex shifted a units along the x-axis. The vertex is then located at (a,0). Notice that, if a is positive, we shift to the right and, if a is negative, we shift to the left.
2 graphs. 1. Two parabolas which are both translations parallel to the y axis of y = x squared. 2. Two parabolas which are both translations parallel to the x axis of y = x squared.
Detailed description of diagram
These two transformations can be combined to produce a parabola which is congruent to the basic parabola, but with vertex at (a,b).
For example, the parabola y=(x−3)2+4 has its vertex at (3,4) and its axis of symmetry has the equation x=3.
In the module Algebra review Open Algebra review in new window, we revised the very important technique of completing the square. This method can now be applied to quadratics of the form y=x2+qx+r, which are congruent to the basic parabola, in order to find their vertex and sketch them quickly.