Rewriting Quadratic Expressions in Vertex Form

5 min

Narrative

This Warm-up reminds students about features of the graph that are visible in the different forms of expressions defining a quadratic function.

Student Task

Each of these expressions defines the same function.

x2+6x+8(x+2)(x+4)(x+3)2−1x^2 + 6x +8 \qquad (x+2)(x+4) \qquad (x+3)^2-1

Without graphing or doing any calculations, determine where the following features would be on a graph that represents the function.

  1. the vertex
  2. the xx-intercepts
  3. the yy-intercept

Sample Response

  1. vertex: (-3,-1)(\text-3,\text-1)
  2. xx-intercepts: (-2,0)(\text-2,0) and (-4,0)(\text-4,0)
  3. yy-intercept: (0,8)(0,8)

Synthesis

Invite students to share how they would locate the specified features on a graph. Make sure students are reminded that:

  • The constant term in the standard form tells us the yy-intercept.
  • The factored form shows us the xx-intercepts.
  • The vertex form reveals the vertex.

Consider using graphing technology to demonstrate that the three expressions appear to produce the same graph. (We can verify algebraically that the three expressions define the same function, but we can’t be sure that the three expressions define the same function just by looking at the graph.) On the graph, label the vertex, xx-intercepts, and yy-intercept. 

Standards
Building On
  • F-IF.7.a·Graph linear and quadratic functions and show intercepts, maxima, and minima.
  • F-IF.7.a·Graph linear and quadratic functions and show intercepts, maxima, and minima.
  • F-IF.7.a·Graph linear and quadratic functions and show intercepts, maxima, and minima.
  • F-IF.7.a·Graph linear and quadratic functions and show intercepts, maxima, and minima.
  • HSF-IF.C.7.a·Graph linear and quadratic functions and show intercepts, maxima, and minima.

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