2 Graphing Quadratic Equations in Two Variables § 8.3Graphing Quadratic Equations in Two Variables
3 Graphs of Quadratic Equations We spent a lot of time graphing linear equations in chapter 3.The graph of a quadratic equation is a parabola.The highest point or lowest point on the parabola is the vertex.Axis of symmetry is the line that runs through the vertex and through the middle of the parabola.
4 Graphs of Quadratic Equations ExamplexyGraph y = 2x2 – 4.(–2, 4)(2, 4)x y241–2(–1, – 2)(1, –2)–4–1–2(0, –4)–24
5 Intercepts of the Parabola Although we can simply plot points, it is helpful to know some information about the parabola we will be graphing prior to finding individual points.To find x-intercepts of the parabola, let y = 0 and solve for x.To find y-intercepts of the parabola, let x = 0 and solve for y.
6 Characteristics of the Parabola If the quadratic equation is written in standard form, y = ax2 + bx + c,1) the parabola opens up when a > 0 and opens down when a < 0.2) the x-coordinate of the vertex isTo find the corresponding y-coordinate, you substitute the x-coordinate into the equation and evaluate for y.
7 Graphs of Quadratic Equations ExampleGraph y = –2x2 + 4x + 5.xy(1, 7)Since a = –2 and b = 4, the graph opens down and the x-coordinate of the vertex is(0, 5)(2, 5)x y3–1(–1, –1)(3, –1)25175–1–1
8 The Graph of a Quadratic Function The vertical line which passes through the vertex is called the Axis of Symmetry or the AxisRecall that the equation of a vertical line is x =cFor some constant cx coordinate of the vertexThe y coordinate of the vertex isThe Axis of Symmetry is the x =
9 The Quadratic function Opens up when a>oopens down a < 0Vertexaxis of symmetry
10 Identify the Vertex and Axis of Symmetry of a Quadratic Function Vertex =(x, y). thusVertex =Axis of Symmetry: the line x =Vertex is minimum point if parabola opens upVertex is maximum point if parabola opens down
11 Identify the Vertex and Axis of Symmetry Vertex x =y =Vertex = (-1, -3)Axis of Symmetry is x == -1=-3= -1
12 8.5 Quadratic Solutions The number of real solutions is at most two. No solutionsOne solutionTwo solutions
13 Identifying Solutions Example f(x) = x2 - 4Solutions are -2 and 2.
14 Identifying Solutions Now you try this problem.f(x) = 2x - x2Solutions are 0 and 2.
15 Graphing Quadratic Equations The graph of a quadratic equation is a parabola.The roots or zeros are the x-intercepts.The vertex is the maximum or minimum point.All parabolas have an axis of symmetry.
16 Graphing Quadratic Equations One method of graphing uses a table with arbitraryx-values.Graph y = x2 - 4xRoots 0 and 4 , Vertex (2, -4) ,Axis of Symmetry x = 2xy1-32-434
17 Graphing Quadratic Equations Try this problem y = x2 - 2x - 8.RootsVertexAxis of Symmetryxy-2-1134
18 A quadratic equation is written in the Standard Form, 8.6 – Solving Quadratic Equations by FactoringA quadratic equation is written in the Standard Form,where a, b, and c are real numbers andExamples:(standard form)
19 If a and b are real numbers and if , then or . 8.6 – Solving Quadratic Equations by FactoringZero Factor Property:If a and b are real numbers and if ,then orExamples:
20 If a and b are real numbers and if , then or . 8.6 – Solving Quadratic Equations by FactoringZero Factor Property:If a and b are real numbers and if ,then orExamples:
21 Solving Quadratic Equations: 1) Write the equation in standard form. 8.6 – Solving Quadratic Equations by FactoringSolving Quadratic Equations:1) Write the equation in standard form.2) Factor the equation completely.3) Set each factor equal to 0.4) Solve each equation.5) Check the solutions (in original equation).
29 8.6 – Quadratic Equations and Problem Solving A cliff diver is 64 feet above the surface of the water. The formula for calculating the height (h) of the diver after t seconds is:How long does it take for the diver to hit the surface of the water?seconds
30 8.6 – Quadratic Equations and Problem Solving The square of a number minus twice the number is 63. Find the number.x is the number.
31 8.6 – Quadratic Equations and Problem Solving The length of a rectangular garden is 5 feet more than its width. The area of the garden is 176 square feet. What are the length and the width of the garden?The width is w.The length is w+5.feetfeet
32 8.6 – Quadratic Equations and Problem Solving Find two consecutive odd numbers whose product is 23 more than their sum?Consecutive odd numbers:
33 8.6 – Quadratic Equations and Problem Solving The length of one leg of a right triangle is 7 meters less than the length of the other leg. The length of the hypotenuse is 13 meters. What are the lengths of the legs?metersmeters
34 Solving Quadratic Equations by the Square Root Property § 8.7Solving Quadratic Equations by the Square Root Property
35 Square Root PropertyWe previously have used factoring to solve quadratic equations.This chapter will introduce additional methods for solving quadratic equations.Square Root PropertyIf b is a real number and a2 = b, then
40 Solving Quadratic Equations by Completing the Square § 8.8Solving Quadratic Equations by Completing the Square
41 Completing the SquareIn all four of the previous examples, the constant in the square on the right side, is half the coefficient of the x term on the left.Also, the constant on the left is the square of the constant on the right.So, to find the constant term of a perfect square trinomial, we need to take the square of half the coefficient of the x term in the trinomial (as long as the coefficient of the x2 term is 1, as in our previous examples).
42 Completing the Square Example x2 – 10x x2 + 16x x2 – 7x What constant term should be added to the following expressions to create a perfect square trinomial?x2 – 10xadd 52 = 25x2 + 16xadd 82 = 64x2 – 7xadd
43 Completing the Square Example We now look at a method for solving quadratics that involves a technique called completing the square.It involves creating a trinomial that is a perfect square, setting the factored trinomial equal to a constant, then using the square root property from the previous section.
44 Completing the SquareSolving a Quadratic Equation by Completing a SquareIf the coefficient of x2 is NOT 1, divide both sides of the equation by the coefficient.Isolate all variable terms on one side of the equation.Complete the square (half the coefficient of the x term squared, added to both sides of the equation).Factor the resulting trinomial.Use the square root property.
45 Solving Equations Example Solve by completing the square. y2 + 6y = 8 y = 4 or 2
46 Solving Equations Example (y + ½)2 = Solve by completing the square. y2 + y – 7 = 0y2 + y = 7y2 + y + ¼ = 7 + ¼(y + ½)2 =
47 Solving Equations Example Solve by completing the square. 2x2 + 14x – 1 = 02x2 + 14x = 1x2 + 7x = ½x2 + 7x = ½ =(x + )2 =
48 Solving Quadratic Equations by the Quadratic Formula § 8.9Solving Quadratic Equations by the Quadratic Formula
49 The Quadratic FormulaAnother technique for solving quadratic equations is to use the quadratic formula.The formula is derived from completing the square of a general quadratic equation.
50 The Quadratic FormulaA quadratic equation written in standard form, ax2 + bx + c = 0, has the solutions.
51 The Quadratic Formula Example Solve 11n2 – 9n = 1 by the quadratic formula.11n2 – 9n – 1 = 0, soa = 11, b = -9, c = -1
52 The Quadratic Formula Example Solve x2 + x – = 0 by the quadratic formula.x2 + 8x – 20 = (multiply both sides by 8)a = 1, b = 8, c = 20
53 The Quadratic Formula Example Solve x(x + 6) = 30 by the quadratic formula.x2 + 6x + 30 = 0a = 1, b = 6, c = 30So there is no real solution.
54 The DiscriminantThe expression under the radical sign in the formula (b2 – 4ac) is called the discriminant.The discriminant will take on a value that is positive, 0, or negative.The value of the discriminant indicates two distinct real solutions, one real solution, or no real solutions, respectively.
55 The Discriminant Example Use the discriminant to determine the number and type of solutions for the following equation.5 – 4x + 12x2 = 0a = 12, b = –4, and c = 5b2 – 4ac = (–4)2 – 4(12)(5)= 16 – 240= –224There are no real solutions.
56 Solving Quadratic Equations Steps in Solving Quadratic EquationsIf the equation is in the form (ax+b)2 = c, use the square root property to solve.If not solved in step 1, write the equation in standard form.Try to solve by factoring.If you haven’t solved it yet, use the quadratic formula.
57 Solving Equations Example Solve 12x = 4x2 + 4. 0 = 4x2 – 12x + 4 Let a = 1, b = -3, c = 1
58 Solving EquationsExampleSolve the following quadratic equation.
59 The Quadratic FormulaSolve for x by completing the square.
60 Yes, you can remember this formula Pop goes the WeaselGilligan’s IslandThis one I can’t explain
63 The DiscriminantThe number in the square root of the quadratic formula.
64 The Discriminant The Discriminant can be negative, positive or zero If the Discriminant is positive,there are 2 real answers.If the square root is not a prefect square( for example ),then there will be 2 irrational roots( for example ).
65 The Discriminant The Discriminant can be negative, positive or zero If the Discriminant is positive,there are 2 real answers.If the Discriminant is zero,there is 1 real answer.If the Discriminant is negative,there are 2 complex answers.complex answer have i.