If students can identify most transformations but struggle with the vertical stretches, consider saying:

• “Explain your representation to me.”

• “What’s the same and what’s different about the vertex and points 1 unit to the left and right on the graph of compared to the corresponding points on the graph of ?”

The purpose of this discussion is for students to articulate that vertex form shows the transformations of a parabola from an original equation . Display this definition for students: The vertex form of a quadratic expression is .

Here are some questions for discussion:

• “What transformations of the graph of a parabola can you see from the vertex form of the equation, and how do they relate to , , and ?” ( is the vertical stretch and vertical reflection, is the horizontal translation, and is the vertical translation.)
• “How can you know the vertex of a parabola if you know the horizontal and vertical translation from the original function?” (Since the vertex of is , a horizontal translation by and a vertical translation by will move the vertex to the point .) 

Students complete the square to identify the transformations by rewriting an equation in vertex form. Students connect the vertex form of a parabola with the transformations needed to transform a parabola from the graph of . When the equation for a parabola is written in vertex form, the transformations can be seen directly from the equation without graphing. As students identify the transformations from the equation in vertex form, they compare the structures of equations of a transformed function and vertex form of the equation of a parabola (MP7).

In this activity, students are using algebraic manipulation to rewrite an equation before graphing, so technology is not an appropriate tool. Instead, invite students to rewrite the equation and identify the vertex before graphing. If desired, graphing technology may be used to check student work.

• “What strategies did you use to rewrite the equations?” (I started by factoring the fraction out of the terms with an , then looked for what to add to make a perfect square.)
• “Where could you see the transformations in the vertex form of the equation?” (The vertex gives the horizontal and vertical translation of the parabola from . The factor that is multiplied by , which is the fraction in these equations, gives the vertical stretch.)
• “How did rewriting the equation in vertex form help with graphing the parabola?” (In standard form, we can see the -intercept, but it’s hard to know what the other important features of the graph are. In vertex form, we can start by plotting the vertex, then looking at the vertical stretch to see if it is reflected and how tall or wide to make the parabola.)