ReflectionStudents had a difficult time in this section due to the many ways you can factor a quadratic. The students found factoring square roots the easiest but completing the square the hardest. Even using the quadratic formula proved to be difficult since they haven't had much experience with formulas. In addition, the students were able to read quadratic graphs and identify key features of the graph with ease.
|
Essential Questions and Standards
1. How can you determine what is a quadratic function in the real world? MAFS.912.A-SSE.2.3, MAFS.912.F-IF.2.4 2. How do you solve quadratic equations using the quadratic formula? MAFS.912.A-REI.2.4 3. How do you factor quadratic equations using the distributive property? MAFS.912.A-SSE.2.3, MAFS.912.F-IF.3.8, MAFS.912.A-REI.2.4 4. How do you factor quadratic equations using the zero product property? MAFS.912.A-SSE.2.3, MAFS.912.F-IF.3.8, MAFS.912.A-REI.2.4 5.How do you factor quadratic equations by factoring out a greatest common factor? MAFS.912.A-SSE.2.3, MAFS.912.F-IF.3.8, MAFS.912.A-REI.2.4 6. How do you factor a perfect square trinomial? MAFS.912.A-REI.2.4 7. How do you solve a quadratic equation by taking the square root? MAFS.912.A-REI.2.4 8.How do you solve quadratic equations by completing the square? MAFS.912.A-REI.2.4, MAFS.912.A-SSE.1.2 9.How do you solve real world situations using quadratics? MAFS.912.F-IF.2.4 MAFS.912.A-SSE.1.2 - Use the structure of an expression to identify ways to rewrite it. MAFS.912.A-SSE.2.3.a.b - Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a.Factor a quadratic expression to reveal the zeros of the function it defines. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. MAFS.912.F-IF.3.8.a - Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function a. Use the process of factoring and completing the square in quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. MAFS.912.A-REI.2.4 - Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p) ^2) = q.that has the same solutions. Derive the quadratic formula from this form. Solve quadratic equations by inspection (e.g. x^2 = 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. MAFS.912.F-IF.2.4 - For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. |
Key Features of a GraphQuadratic equations are graphed on a coordinate as a parabola. A parabola is a U-shaped graph. Quadratic equations are also known as nonlinear functions since their rate of change is not constant. In order to find the key features of a quadratic graph, you need to understand the parts of a graph. The parts are: x - intercepts, y-intercepts, vertex, independent variable, and dependent variable. To solve a quadratic equation using a graph:
a. look for the 𝒙 −intercepts of the graph. b. the solutions are the values where the graph intercepts the 𝒙 − axis. Below is a tutorial. www.virtualnerd.com/algebra-2/quadratics/equation-definition.php Zero Product PropertyOnce a quadratic equation is factored, we can use the zero product property to solve the equation.
a. The zero product property states that if the product of two factors is zero, then one (or both) of the factors must be zero. ob. If 𝑎𝑏 = 0, then either 𝑎 = 0, 𝑏 = 0, or 𝑎 = 𝑏 = 0. To solve a quadratic equation by factoring: Step 1: Set the equation equal to zero. Step 2: Factor the quadratic. Step 3: Set each factor equal to zero and solve. Step 4: Write the solution set. Below is a tutorial. www.virtualnerd.com/algebra-1/polynomials-and-factoring/zero-product-property-definition.php Taking Square RootsWhen quadratic equations are in the form 𝑎𝑥^2 + 𝑐 = 0, solve by taking the square root.
Step 1: Get the variable on the left and the constant on the right. Step 2: Then take the square root of both sides of the equation. (Don’t forget the negative root!) Below is a tutorial: www.virtualnerd.com/algebra-1/quadratic-equations-functions/square-root-method-two-solutions.php |
Quadratic FormulaTo find the solutions of a quadratic equation, you can use the quadratic formula. The formula uses the a,b,and c terms for the quadratic in the formula.
-b+ √(b^2-4ac) 2a Below is a tutorial. www.virtualnerd.com/algebra-1/quadratic-equations-functions/quadratic-formula-definition.php Factoring using the GCFMany quadratic equations will not be in standard form:
a. The equation won’t always equal zero. b. There may be a greatest common factor (GCF) within all of the terms. Solve by Completing the SquareSolve 2𝑥^2 + 12𝑥 + 2 = 3 by completing the square.
Step 1: Write the equation in standard form. 𝟐𝒙^𝟐 + 𝟏𝟐𝒙 − 𝟏 = 𝟎 Step 2: Move the constant term to the right side of the equation. 𝟐𝒙^𝟐 + 𝟏𝟐𝒙 = 𝟏 Step 3: If the coefficient of the 𝑥^2 term does not equal 1, then factor out the coefficient. 𝟐(𝒙^𝟐 + 𝟔𝒙 + _____) = 𝟏 Step 4: Divide the coefficient of the middle term by two and square the result. Use the addition property of equality to make the trinomial a perfect square. 𝟐 (𝒙𝟐 + 𝟔𝒙 + 𝟗) = 𝟏 + 𝟏𝟖 Step 5: Factor and solve the perfect square trinomial. 𝟐 (𝒙𝟐 + 𝟔𝒙 + 𝟗) = 𝟏𝟗 𝟐 (𝒙 +𝟑)^ 𝟐 =𝟏𝟗 𝟐 √(𝒙+𝟑)^ 𝟐 =± √𝟗.𝟓 𝒙 + 𝟑 = ± √𝟗. 𝟓 𝒙 = −𝟑 ± √𝟗. 𝟓 𝒙 = −𝟑 ± √𝟗. 𝟓 𝒙 ≈ −𝟑 ± 𝟑. 𝟎𝟖 𝒙 ≈ 𝟎. 𝟎𝟖 or 𝒙 ≈ −𝟔. 𝟎𝟖 Below is a tutorial: www.virtualnerd.com/algebra-1/quadratic-equations-functions/completing-the-square-solution-example.php |
Factoring using the Distributive PropertyTo factor using the distributive property, follow the steps above. You can check your answer to every factor by
using the distributive property. The product of the factors should always result in the original trinomial. Perfect Square Trinomials and Difference of Perfect SquaresPerfect Square Trinomials
a. 𝑥^2 + 6𝑥 + 9 is an example of perfect square trinomial. We see this when we factor. b. A perfect square trinomial is created when you square a binomial. A quadratic expression can be factored as a perfect square trinomial if it can be re-written in the form 𝑎^2 + 2𝑎𝑏 + 𝑏^2. c. If 𝑎^2 + 2𝑎𝑏 + 𝑏^2 is a perfect square trinomial, then = (𝑎 + 𝑏)^2. d.If 𝑎^2 − 2𝑎𝑏 + 𝑏^2 is a perfect square trinomial, then = (𝑎 − 𝑏)^2. When we have a binomial in the form 𝑎^2 − 𝑏^2, it is called the difference of two squares. We can factor this as (𝑎 + 𝑏) (𝑎 − 𝑏) . Below is a tutorial. www.virtualnerd.com/algebra-1/quadratic-equations-functions/completing-the-square-solution-example.php Quadratics in the Real-WorldUse the image above to help find solutions to specific problems. When talking about time, you will use the positive solution since time cannot be negative. You will use the quadratic formula often to find solutions especially if there are decimals.
|