Curriculum  /  Math  /  11th Grade  /  Unit 2: Quadratics  /  Lesson 5

Quadratics

Lesson 5

Math

Unit 2

11th Grade

Lesson 5 of 11

Objective

Solve quadratic equations written in vertex form and describe graphical features from vertex form.

Common Core Standards

Core Standards

  • A.REI.A.1 — Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
  • A.REI.B.4.B — Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

Criteria for Success

  1. Describe the features that you can "see" easily in vertex form. 
  2. Convert vertex form to standard form to identify the y-intercept of a quadratic function. 
  3. Solve quadratic equations written in vertex form. 
  4. Identify the number and kind of roots of an equation in vertex form. Describe, with minimal calculation, which quadratic equations will result in non-real solutions. 
  5. Check solutions to problems using a graphing calculator. 
  6. Show that calculating $${\sqrt{-1}}$$ results in an "Error" on a graphing calculator. Describe why an error results. 

Tips for Teachers

Allow students to have a graphing calculator to check their solutions and confirm their thinking of what the graph will look like.

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Anchor Problems

25-30 minutes

Problem 1

How would you find the roots of the following quadratic function without doing the following? 

  • Converting to standard form and factoring
  • Using the quadratic formula
  • Graphing
  • Using a graphing calculator

$${f(x)=3(x+1)^2-12}$$

Guiding Questions

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Problem 2

John was solving the following quadratic equation:

$${f(x)={1\over2}(x+2)^2+6}$$

When he tried to find the decimal approximation for the roots, he got this message:

What went wrong?

Guiding Questions

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Target Task

5-10 minutes

Problem 1

Which of the following quadratic functions will result in non-real solutions?

$${{1\over2}(x-1)^2-5=0}$$

$${{1\over2}(x-1)^2+5=0}$$

$$-{{1\over2}(x-1)^2+5=0}$$

$$-{{1\over2}(x-1)^2-5=0}$$

Explain your resoning.

Problem 2

Solve the following quadratic function. Identify the number and kind of solutions.

$${g(x)={2\over3} \left({x-{1\over9}} \right)^2 -{8\over27}}$$

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

  • Include problems where students need to solve the vertex form and the roots are rational numbers, not integers. 
  • Include problems where students are given an equation showing a difference of two squares. Ask students to find the solutions without factoring. 
  • Include problems written in vertex form and ask students to identify all of the features of the quadratic function from the vertex form. Include a question similar to “How can you find the y-intercept without converting the equation to standard form?”
  • Include problems where students need to link the altering of an equation to form real solutions to the idea of transformations, and that real solutions result when a parabola crosses the x-axis. 
  • Include problems such as: “Below is a screenshot of a quadratic function with the vertex noted but no roots shown in the window. Calculate the roots of the function using the vertex. The quadratic function is not vertically or horizontally dilated.”

Next

Complete the square to convert an equation written in standard form to an equation written in vertex form.

Lesson 6
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Lesson Map

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Topic A: Features of Quadratic Functions

Topic B: Imaginary Solutions and Operating with Complex Numbers

Topic C: Applications, Systems, and Inverse with Quadratics

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