Vertex Form

5 min

Narrative

The purpose of this Warm-up is to get students thinking about a new form of quadratic, which will be useful when students explore “vertex form” in a later activity. While students may notice and wonder many things about these functions, a comparison of the forms is the important discussion point.

When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language used to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.

Launch

Arrange students in groups of 2. Display the 2 sets of equations for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss, with their partner, the things they notice and wonder about.

Student Task

What do you notice? What do you wonder?

Set 1:

f(x)=x2+4xf(x)= x^2 +4x

g(x)=x(x+4)g(x)=x(x+4)

h(x)=(x+2)2−4h(x)=(x+2)^2 -4

Set 2:

p(x)=-x2+6x−5p(x)=\text-x^2+6x-5

q(x)=(5−x)(x−1)q(x)=(5-x)(x-1)

r(x)=-1(x−3)2+4r(x)=\text-1(x-3)^2+4

 

Sample Response

Students may notice:

  • Each set has three equations.
  • Each equation has the output in function notation on the left side of the equal sign, and an expression on the right side.
  • The expressions are in standard form, factored form, and one other form.
  • The equations in standard and factored form are equivalent.

Students may wonder:

  • Why are the equations in each set grouped together?
  • In each set, is the expression in the last equation written in a particular form? What is it called?
  • In each set, is the expression in the last equation equivalent to the other two?
Activity Synthesis (Teacher Notes)

Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary for all to see. If possible, record the relevant reasoning on or near the equations. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.

If the idea that all three functions in a set represent the same thing, and the new form of the third function in each set does not come up during the conversation, ask students to discuss this idea.

Standards
Building Toward
  • F-IF.8.a·Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
  • F-IF.8.a·Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
  • F-IF.8.a·Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
  • HSF-IF.C.8.a·Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

10 min

20 min