Graphing the Standard Form (Part 2)

5 min

Narrative

In this Warm-up, students practice applying the distributive property to write equivalent quadratic expressions in standard and factored forms. They also notice that when a quadratic expression has no constant term (that is, is in the form of x2+bxx^2 + bx), its factors are a variable and a sum (or a difference). Their awareness of this structure (MP7) prepares them to think, later in the lesson, about how the linear term in a quadratic expression affects the behavior of the graph.

Launch

Arrange students in groups of 2. Give them a moment of quiet think time, and then time to briefly share their responses with their partner.

Student Task

  1. Complete each row with an equivalent expression in standard form or factored form.
    standard form factored form
    x2x^2
    x(x+9)x(x+9)
    x2−18xx^2-18x
    x(6−x)x(6-x)
    -x2+10x\text-x^2+10x
    -x(x+2.75)\text-x(x+2.75)
  2. What do the quadratic expressions in each column have in common (besides the fact that everything in the left column is in standard form and everything in the other column is in factored form)? Be prepared to share your observations.

Sample Response

  1. standard form factored form
    x2x^2 x⋅xx \boldcdot x
    x2+9xx^2 + 9x x(x+9)x(x+9)
    x2−18xx^2-18x x(x−18)x (x - 18)
    6x−x26x - x^2 or -x2+6x\text-x^2 + 6x x(6−x)x(6-x)
    -x2+10x\text-x^2+10x -x(x−10)\text-x (x-10) or x(-x+10)x(\text-x + 10) or x(10−x)x (10-x)
    -x2−2.75x\text-x^2 - 2.75x -x(x+2.75)\text-x(x+2.75)
  2. Sample response: In the left column, all the expressions in standard form have a squared term and a linear term but no constant term, and the squared term has a coefficient of either 1 or -1. In the right column, one of the factors is just a variable whose coefficient is 1 or -1, and the other factor is either a sum or a difference. 
Activity Synthesis (Teacher Notes)

Consider displaying the completed table for all to see. If needed, discuss only the last few expressions that can be written in a few different but equivalent ways.

Then, focus the discussion on the second question. Ask students:

  • “What do the quadratic expressions in standard form have in common?” (They have a squared term and a linear term. They do not have a constant term.)
  • “Can you tell where the xx-intercepts or the vertex will be?” (Not easily.)
  • “What do the expressions in factored form have in common?” (They are all in the form of xx or -x\text-x times a sum or a difference.)
  • “From these expressions, what can we predict, if anything, about the features of the graphs?” (From the standard form, we can predict the yy-intercept of the graph. From factored form, we can predict the xx-intercepts of the graph, which can be used to find the vertex of the graph.)
Standards
Addressing
  • A-SSE.3·Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
  • A-SSE.3·Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
  • A-SSE.3·Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
  • HSA-SSE.B.3·Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

20 min

10 min