Lesson 4.2: Graphing Parabolas Using Transformations y = x2 y = ax2 and y = –ax2

Recall… The graph of y = x2 x y –3 –2 –1 1 2 3 9 4

So… to graph, Remember y = x2 Apply the transformation to find the new vertex. Then, from the vertex, to find the next points you go: Over one, up one (12 = 1) Over two, up four (22 = 4) Over three, up nine (32 = 9) Etc…

y = ax2 Let’s try another transformation Please do not share your ideas out loud. One graph will correctly represent the equation, and one will incorrectly represent the equation. The dotted line in each graph represents y = x2 Look carefully at the position of the y-values for the corresponding x-values

y = 2x2 YES NO

y = 3x2 YES NO

YES NO

YES NO

y = 1.5x2 YES NO

Think you’ve got it??? Try these 

Which graph represents y = 5x2? b y = x2 a

Which graph represents y = 5x2?

Which graph represents ? b a y = x2

Which graph represents ? b

SUMMARY y = ax2, is called the “Congruent Curve” Each regular y-value from y = x2 is multiplied by “a” If a>1 or a<–1, the graph has a vertical stretch (narrows) If –1<a<1, the graph has a vertical compression (widens) If a>0 (+ve), the graph opens up If a<0 (–ve), the graph is reflected in the x-axis and opens down

Example 1: y = –2x2 The vertex is at (0, 0) a=-2: vertical stretch and reflection in the x-axis Opens down Each normal y-value is multiplied by -2 x y=x2 -2 4 -1 1 2 y=-2x2 -8 -2

y = –2x2

y = –2x2

y = –2x2

y = –2x2

Example 2: y = x2 The vertex is at (0, 0) a= : vertical compression, opens up Each normal y-value is multiplied by x y=x2 -3 9 -2 4 -1 1 2 3 y=2/3x2 6 8/3 2/3

y = x2

y = x2

y = x2

y = x2

HOMEWORK y = –¾x2 y = 2x2 y = 4x2 y = –½ x2 y = –3x2 y = ¼x2 Graph each of the following using transformations. State the direction of opening y = –¾x2 y = 4x2 y = –3x2 y = 2x2 y = –½ x2 y = ¼x2