Here are two sets of equations for quadratic functions that you saw earlier. In each set, the expressions that define the output are equivalent.

The expressions that define and are written in vertex form.

1. Show that the expressions defining and are equivalent.

2. Here are graphs representing the quadratic functions. Why do you think expressions such as those defining and are said to be written in vertex form?

   

   

1. Using graphing technology, graph . Then, add different numbers to before it is squared (for example, , ), and observe how the graph changes. Record your observations.

2. Graph . Then, experiment with each of the following changes to the function, and see how they affect the graph and the vertex:

   1. Adding different constant terms to (for example: , ).
   2. Multiplying by different coefficients (for example: , ).
3. Without graphing, predict the coordinates of the vertex of the graphs of these quadratic functions, and predict whether the graph opens upward or opens downward. Ignore the last row until the next question. 

   equations	coordinates of vertex	graph opens upward or downward?

4. Use graphing technology to check your predictions. If they are incorrect, revise them. Then complete the last row of the table.