Unit 4 Polnomial/Rational Functions Quadratic Functions (Chap 0.3) William (Bill) Finch Lesson Goals When ou have completed this lesson ou will: Graph and analze the graphs of quadratic functions. Solve quadratic equations b factoring, completing the square, and the quadratic formula. Mathematics Department Denton High School Quadratics 2 / 24 Famil of Quadratic Functions Parabolas The Standard Form of a Quadratic Function is All parabolas: f () = a 2 + b + c where a, b, and c are real numbers and a 0. are smmetric wrt a vertical line called the ais of smmetr Ais The parent quadratic function is f () = 2. Graph called parabola Domain (, ) have a point called a verte that is either a maimum or a minimum Verte Range [0, ) Verte (0, 0) Ais of Smmetr = 0 Quadratics 3 / 24 Quadratics 4 / 24
Parabolas In a quadratic equation f () = a 2 + b + c, the leading coefficient is associated with the direction the parabola opens. Maimum Verte of a Parabola Some applications involve finding the minimum or maimum value of a quadratic function. The verte of the graph of f () = a 2 + b + c is ( b2a (, f b )) 2a Minimum If a > 0, minimum at = b 2a a > 0 a < 0 If a < 0, maimum at = b 2a Quadratics 5 / 24 Quadratics 6 / 24 Eample 1 Use the ais of smmetr, -intercept, and verte to graph f. f () = 2 + 5 + 6 Eample 2 Determine the following for g() = 5 2 + 3. Does g have a minimum or maimum? Find the minimum or maimum value. State the domain and range of g. Quadratics 7 / 24 Quadratics 8 / 24
Verte Form of a Quadratic Function Verte Form: f () = a( h) 2 + k, a 0 Verte Form of a Quadratic Function Recall the lesson on transformations of functions. f () = a( h) 2 + k, a 0 opens up if a > 0 and down if a < 0 verte at (h, k) ais of smmetr = h a produces a vertical stretch or shrink a < 0 produces a reflection wrt -ais h produces a horizontal shift k produces a vertical shift Quadratics 9 / 24 Quadratics 10 / 24 Completing the Square To rewrite a quadratic function into verte form: Eample 3 Rewrite in verte form: f () = 2 + 10 5 1. Write the function in the form f () = ( 2 + b) + c. 2. Complete the square b adding zero to the right hand side. 3. Factor the perfect square trinomial. Quadratics 11 / 24 Quadratics 12 / 24
Eample 4 Rewrite in verte form: g() = 2 2 8 + 3 Eample 5 Rewrite in verte form: h() = 4 2 + 20 + 1 Quadratics 13 / 24 Quadratics 14 / 24 3 Methods of Solving Quadratic Equations Factoring Eample 6 Solve b factoring: 3 2 + 2 = 0 Completing the Square Quadratic Formula Quadratics 15 / 24 Quadratics 16 / 24
Eample 7 Solve b factoring: 4 2 = 0 Completing the Square To solve a quadratic equation: 1. Write one side of the equation in the form 2 + b. 2. Complete the square b adding (b/2) 2 to both sides of the equation. 3. Factor the perfect square trinomial. 4. Take the square root of both sides (don t forget plus or minus ) and finish solving for. Quadratics 17 / 24 Quadratics 18 / 24 Eample 8 Solve b completing the square: 2 + 6 3 = 0 Eample 9 Solve b completing the square: 2 3 + 8 = 0 Quadratics 19 / 24 Quadratics 20 / 24
Quadratic Formula For a quadratic equation in standard form Eample 10 Solve with the quadratic formula: 2 2 3 = 0 a 2 + b + c = 0 The quadratic formula gives the two solutions = b ± b 2 4ac 2a Quadratics 21 / 24 Quadratics 22 / 24 Eample 11 Solve with the quadratic formula: 2 6 + 4 = 0 What You Learned You can now: Graph and analze the graphs of quadratic functions. Solve quadratic equations b factoring, completing the square, and the quadratic formula. Do problems Chap 0.3 #9, 13, 15, 21, 23-39 odd, 43-49 odd Quadratics 23 / 24 Quadratics 24 / 24