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K12MATH011: Algebra II

Unit 3: Quadratic Equations and Functions   You have now been introduced to linear functions. Starting with unit 3, you begin to find solutions for functions that form curves rather than lines. Unit 3 covers solutions of quadratic functions (which have variables of power 2, but no higher), beginning with graphing. Then, you will consider solving by factoring. One issue that arises with quadratic (and higher powered) functions is the possibility of complex or imaginary solutions. That is, the solutions may have “i” in them, where i = -1. You are familiar with real numbers up to this point, but you may not have worked with so-called imaginary numbers. Although complex numbers (which have a real part added to an imaginary part, such as 3 – 2i) cannot be found among real numbers, the uses of complex numbers are many and varied. In fact, if you wear glasses, chances are imaginary numbers played a role in the shaping of your lenses. The remainder of the discussion of quadratic functions focuses on methods for finding solutions, real or complex. You will end the unit with an introduction to quadratic inequalities.

Unit 3 Time Advisory
Completing this unit should take approximately 14 hours and 15 minutes.

☐    Subunit 3.1: 1 hour and 15 minutes

☐    Subunit 3.2: 1 hour

☐    Subunit 3.3: 1 hour

☐    Subunit 3.4: 1 hour and 45 minutes

☐    Subunit 3.5: 1 hour and 15 minutes

☐    Subunit 3.6: 4 hours and 15 minutes

☐    Subunit 3.7: 1 hour and 15 minutes

☐    Subunit 3.8: 2 hours and 30 minutes

Unit3 Learning Outcomes
Upon successful completion of this unit, you will be able to: - Graph a quadratic function. - Rewrite a quadratic function in order to use transformations of basic graphs. - Solve a quadratic function by graphing. - Solve a quadratic function by factoring. - Solve a quadratic function by completing the square. - Solve a quadratic function by using the quadratic formula. - Determine the types of solutions of a quadratic function by using the discriminant. - Graph quadratic inequalities.

Standards Addressed (Common Core): - CCSS.Math.Content.HSA-SSE.A.1 - CCSS.Math.Content.HSA-SSE.A.2 - CCSS.Math.Content.HSA-SSE.B.3a - CCSS.Math.Content.HSA-SSE.B.3b - CCSS.Math.Content.HSA-APR.A.1 - CCSS.Math.Content.HSA-APR.B.3 - CCSS.Math.Content.HSA-REI.B.4 - CCSS.Math.Content.HSA-REI.D.11 - CCSS.Math.Content.HSF-IF.C.8 - CCSS.ELA-Literacy.RST.9-10.1 - CCSS.ELA-Literacy.RST.11-12.2 - CCSS.ELA-Literacy.RST.11-12.4 - CCSS.ELA-Literacy.RST.11-12.5 - CCSS.ELA-Literacy.WHST.11-12.1 - CCSS.ELA-Literacy.WHST.11-12.2

3.1 Graphing Quadratic Functions   Quadratic functions are usually the next type to which students are introduced. They are symmetrical across a vertical line through the function’s vertex (which is the highest or lowest point on the curve) and are easy to rewrite in translation form (which will be discussed later in this unit). Once in translation form, they are easy to graph.

3.2 Transformations of Quadratic Functions   - Explanation: YouTube: Khan Academy’s “Graphs of Quadratic Functions” Link: YouTube: Khan Academy’s “Graphs of Quadratic Functions” (YouTube)
 
Instructions: This video discusses in detail how to graph quadratic functions. Write a two-paragraph essay on how to find the number of solutions of a quadratic function just based on looking at its graph.
 
Completing this activity should take approximately 1 hour.
 
Standards Addressed (Common Core):

-   [CCSS.Math.Content.HSA-SSE.A.1](http://www.corestandards.org/Math/Content/HSA/SSE/A/1)
-   [CCSS.Math.Content.HSA-SSE.A.2](http://www.corestandards.org/Math/Content/HSA/SSE/A/2)
-   [CCSS.ELA-Literacy.RST.9-10.1](http://www.corestandards.org/ELA-Literacy/RST/9-10/1)
-   [CCSS.ELA-Literacy.RST.11-12.4](http://www.corestandards.org/ELA-Literacy/RST/11-12/4)
-   [CCSS.ELA-Literacy.RST.11-12.5](http://www.corestandards.org/ELA-Literacy/RST/11-12/5)
-   [CCSS.ELA-Literacy.WHST.11-12.2](http://www.corestandards.org/ELA-Literacy/WHST/11-12/2)


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Attribution-NonCommercial-ShareAlike 3.0 United States
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is attributed to Khan Academy.

3.3 Solving Quadratic Functions by Graphing   As with linear functions, quadratic equations are solved when the function intersects the x-axis, which is also the y = 0 line (take a look at a typical coordinate grid and you’ll see this fact). While not always practical, if the solutions have noninteger values, graphs of parabolas are interesting in that they cross the x at one point (if the vertex is the only thing touching the x-axis), at two points (if the vertex is above or below the x-axis), or at no points (if the curve is entirely above or below the x-axis).

3.4 Solving Quadratic Equations by Factoring   When we solved linear equations algebraically, we used equation properties to reduce the equation to “x = ….” That is we had x, by itself, on one side of the equation, and some value on the other. This is the value of x that caused y = 0, and the solution as a point was in the form (x, 0). It is not so simple with quadratic equations but requires only an additional step or two. First, set the equation equal to 0, factor, and then set each factor equal to 0 to solve. While this method is not as visual as graphing, you can more easily find a precise solution this way.

3.5 Complex Solutions   When the parabola never touches the x-axis, there is no way to find a real number solution. In such a case, the only solutions possible are complex numbers (that is, have an a + bi form, where i = -1). We will quickly discover that we cannot factor a quadratic equation when it has complex roots, but luckily, there is more than one way to solve a quadratic form.

3.6 Solving Quadratic Functions by Using Square Roots and Completing the Square   Completing the square is a process using equation properties to reach a perfect square trinomial (three-term polynomial) on the side of the equation. But what is interesting is that completing the square is often used to rewrite conic sections (circles, ellipses, parabolas, and hyperbolas) in transformation form, which makes them easier to graph. Pay close attention to the process of completing the square. It is not difficult but requires taking time when first exposed to it. Once mastered, it becomes a critical element in the process to graph many types of functions.

  • Activity: Carl Stitz and Jeff Zeager’s Precalculus: “Chapter 2, Section 3: Quadratic Functions” Link: Carl Stitz and Jeff Zeager’s Precalculus: “Chapter 2, Section 3: Quadratic Functions” (PDF)
     
    Instructions: These pages contain information on solving quadratic functions, including completing the square. Read pages 188–192, and then try the problems at the end of the chapter. Answers are on the pages that follow. Then write a three-page essay comparing and contrasting solving by graphing, solving by factoring, solving by using the quadratic formula, and solving by completing the square. What are the strengths and weaknesses of each method? When is the best time to use each method?
     
    Completing this activity should take approximately 3 hours.
     
    Standards Addressed (Common Core):

    Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. It is attributed to Carl Stitz and Jeff Zeager.

  • Explanation: YouTube: Khan Academy’s “Completing the Square” Link: YouTube: Khan Academy’s “Completing the Square” (YouTube)
     
    Instructions: This video discusses finding roots of a quadratic by completing the square. I realize this may be material you have seen before, but it does not hurt to reinforce the idea. Note that it is possible to have roots that are not real numbers. In this case, the values under the square root will be negative.
     
    Watching the video and writing notes should take approximately 30 minutes.
     
    Standards Addressed (Common Core):

     
    Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. It is attributed to Khan Academy. 

  • Web Media: GeoGebra: “Solving Quadratic Equations by Completing the Square” Link: GeoGebra: “Solving Quadratic Equations by Completing the Square” (HTML)
     
    Instructions: This page shows step-by-step instructions on solving quadratic equations by completing the square. This application is very unique; not only does it guide you through the steps, but it allows for interaction by using sliders on the third and fourth steps to find the right square term and the correct h term [for (x ± h)2 ] to complete the square. Note: The sliders, especially on the step to complete the square, are a little sensitive, so make sure to move them slowly. They will stop when the correct value is reached. What I especially like about this application is that it finishes with the final solution set, in set notation. We often forget that we are dealing with number sets in mathematics (domains, ranges, solutions, etc.), and it is refreshing to see the fact that the proper way to express a solution is in set form.
     
    To use this application, write the problem out on paper and work step-by-step. Use the check boxes to see the steps done by the teacher. Make sure you understand the steps before proceeding. It may help to write notes below or in the margin of your paper by each step. Compare your steps to the ones the teacher provides. Are they similar? Did you combine one or more steps? Click on the “Give me a new problem” button to try more problems. Try as many examples as you need to feel comfortable with this process.
     
    Completing this activity should take approximately 45 minutes.
     
    Standards Addressed (Common Core):

     
    Terms of Use: This resource is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License

3.7 Solving Quadratic Functions by Using the Quadratic Formula   The quadratic formula is perhaps the second most well-known formula in mathematics, after the Pythagorean Theorem. The purpose of the formula is to allow the user to find solutions to any quadratic equation simply and easily. The quadratic formula can be derived by completing the square on the standard form of a quadratic equation (y = ax2 + bx + c). There is a bonus: It does not matter whether there are two solutions, one solution, or no solutions. In fact, we can tell a lot about the nature of a quadratic equation’s solutions by just looking at the simplified value under the square root (b2 – 4ac). If positive, there are two real roots. If zero, there is only one. If negative, the two roots are complex.

  • Explanation: Khan Academy’s “How to Use the Quadratic Formula” Link: Khan Academy’s “How to Use the Quadratic Formula” (YouTube)
     
    Instructions: This detailed video discusses the process of solving quadratic functions by using the quadratic formula.
     
    Watching the video and writing notes should take approximately 30 minutes.
     
    Standards Addressed (Common Core):

     
    Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. It is attributed to Khan Academy. 

  • Web Media: GeoGebra: “Using the Quadratic Formula to Solve a Quadratic Equation” Link: GeoGebra: “Using the Quadratic Formula to Solve a Quadratic Equation” (HTML)
     
    Instructions: This page allows you to see how to solve quadratic equations by using the quadratic formula. It begins with two steps explaining that the equation must be set = 0 first, by adding opposites to both sides. Then you identify a, b, and c in the simplified equation, plug them into the formula, and use order of operations to simplify the solutions.
     
    To use this activity, write the problem out on paper and work step-by-step. Use the check boxes to see the steps done by the teacher. Make sure you understand the steps before proceeding. It may help to write notes below or in the margin of your paper by each step. Compare your steps to the ones the teacher provides. Are they similar? Did you combine one or more steps? Click on the “Give me a new problem” button to try more examples until you feel comfortable with this type of problem.
     
    Practicing with this application and writing notes should take approximately 45 minutes.
     
    Standards Addressed (Common Core):

     
    Terms of Use: This resource is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License

3.8 An Introduction to Quadratic Inequalities   As with linear forms, there are quadratic inequalities as well. And as with lines, graphing quadratic inequalities remains the quickest way to find a solution set. The reason is very similar. As with lines, it can be harder to find a solution by graphing an equation, but solutions of quadratic inequalities are half-planes, either above or below the curve. In this case, it is harder to find the set notation than to find the graph of the solutions.