Completing the square and vertex form (AASL 2.6)
Keywords
[table][br][tr][br][td]English[/td][br][td]Japanese[/td][br][td]Korean[/td][br][td]Chinese Simplified[/td][br][/tr][br][tr][br][td]Completing the Square[/td][br][td]平方完成[/td][br][td]완전제곱식[/td][br][td]配方法[/td][br][/tr][br][tr][br][td]Vertex Form[/td][br][td]頂点形式[/td][br][td]정점 형식[/td][br][td]顶点式[/td][br][/tr][br][tr][br][td]Quadratic Equation[/td][br][td]二次方程式[/td][br][td]이차방정식[/td][br][td]二次方程[/td][br][/tr][br][tr][br][td]Axis of Symmetry[/td][br][td]対称軸[/td][br][td]대칭축[/td][br][td]对称轴[/td][br][/tr][br][tr][br][td]General Form to Vertex Form Conversion[/td][br][td]一般形から頂点形への変換[/td][br][td]일반형에서 정점형으로의 변환[/td][br][td]一般形式转顶点形式[/td][br][/tr][br][tr][br][td]Parabola[/td][br][td]放物線[/td][br][td]포물선[/td][br][td]抛物线[/td][br][/tr][br][tr][br][td]Coefficient[/td][br][td]係数[/td][br][td]계수[/td][br][td]系数[/td][br][/tr][br][tr][br][td]Graph of Quadratic Function[/td][br][td]二次関数のグラフ[/td][br][td]이차함수의 그래프[/td][br][td]二次函数图像[/td][br][/tr][br][tr][br][td]Standard Form[/td][br][td]標準形[/td][br][td]표준형[/td][br][td]标准形式[/td][br][/tr][br][tr][br][td]Line of Symmetry[/td][br][td]対称線[/td][br][td]대칭선[/td][br][td]对称线[/td][br][/tr][br][/table][br]
Inquiry questions
[table][br][tr][br] [td][b]Factual Inquiry Questions[/b][br] [list][br] [*]What is the process of completing the square for a quadratic equation?[br] [*]How can completing the square be used to convert a quadratic equation into vertex form?[br] [/list][br] [/td][br] [td][b]Conceptual Inquiry Questions[/b][br] [list][br] [*]Why is completing the square a valuable method for solving quadratic equations, especially compared to factoring or using the quadratic formula?[br] [*]How does completing the square provide insight into the graph of a quadratic function, particularly its vertex and axis of symmetry?[br] [/list][br] [/td][br] [td][b]Debatable Inquiry Questions[/b][br] [list][br] [*]Is completing the square more intuitive and beneficial for understanding the properties of quadratic functions than other methods of solving quadratics?[br] [*]Can the technique of completing the square be considered foundational for more advanced topics in algebra and calculus? How so?[br] [*]How might the teaching of completing the square evolve with the integration of technology in mathematics education, especially with tools that can automatically solve and graph quadratic equations?[br] [/list][br] [/td][br][/tr][br][/table][br]
Part 1- Exploring the vertex form, [math]y=a(x-h)^2+k[/math] [br]The Vertex Voyage[br][br]Welcome aboard the Vertex Voyage, where we chart the path of parabolas and uncover the mysteries of their vertices! Let's navigate through the world of quadratic functions with these engaging challenges.
What is the effect on the parabola of changing the h and k values?
Expansion Expedition: Take the vertex form and expand it to get the general form [math]y=ax^2+bx+c[/math]. Do this for a parabola with a vertex at (3, -7) and a = 2.
Coefficient Cruise: Set sail on the coefficient sea by altering '[math]a[/math]' in the vertex form. What happens to the parabola when '[math]a[/math]' is greater than 1? Less than 1? Negative? Sketch your findings in your book.
Part 2 - Completing the square
We've seen that vertex form is useful for finding the vertex but what do we do if we have the equation in general form.[br][br]This part explores moving from general form [math]y=ax^2+bx+c[/math] to [math]y=a(x-h)^2+k[/math]
Standard Form Shift: Notice the equation [math]y=x^2-2x+3[/math]. It's in general form! [br]Can you rewrite it in vertex form by completing the square? [br]Check your answer with applet below. [br][br]Explain the general process from converting from general form to vertex form.
Standard Form Shift: Notice the equation [math]y=x^2-10x+23[/math]. It's in standard form! Can you rewrite it in vertex form by completing the square? Show your work and check it by comparing the vertex.[br][br]
Treasure of the Axis: The line of symmetry is the treasure map's "X marks the spot." If our vertex is at [math](5,-2)[/math], where do you predict the line of symmetry is? Verify your prediction algebraically.[br][br]
Expansion Expedition: Take the vertex form and expand it to get the standard form. Do this for a parabola with a vertex at[math](3,-7)[/math]and [math]a=2[/math]. Check your work by comparing the expanded form to the standard form.[br]
[br]Chart your findings and share them with your fellow math explorers. Keep a log of your discoveries, and remember, in the world of quadratics, every solution brings a new perspective. Happy voyaging!
Part 2 - Checking your understanding
Completing the square can be useful for putting the equation into vertex form. It can also be used for solving quadratic equations. Check out this video.
Question 1: Complete the square for the quadratic equation [math]x^2+6x+5[/math].
Question 2: What is the vertex of the quadratic equation [math]y=x^2+6x+5[/math]?
Question 3: Complete the square for the quadratic equation x^2 - 4x + 4.
Question 4: What is the vertex of the quadratic equation [math]y=x^2-4x+4[/math]?
Question 5: Complete the square for the quadratic equation [math]2x^2+8x+8[/math].
Question 6: What is the vertex of the quadratic equation [math]y=2x^2+8x+8[/math]?
Question 7: Complete the square for the quadratic equation 3x^2 - 6x + 9.A) 3(x - 1)^2 + 6B) 3(x + 1)^2 - 6C) 3(x - 1)^2D) 3(x - 1)^2 + 9Correct Answer: A) 3(x - 1)^2 + 6
Question 8: What is the vertex of the quadratic equation [math]y=3x^2-6x+9[/math]?[br]
We are now ready to tackle some exam-style questions.[br][br]Try Question 4, 5 ,17, 19, 20, 24, 25, 28 from the quadratic pack. [br][br]These all involve completing the square (vertex form).
[MAA 2.2] QUADRATICS
[MAA 2.2] QUADRATICS_solutions
Lesson Plan- Completing the Square and Understanding Vertex Form
Completing the square and vertex form- Intuition pump (thought experiments and analogies)
