Presentation on theme: "Solving Quadratic Equation by Graphing and Factoring"— Presentation transcript:
1Solving Quadratic Equation by Graphing and Factoring Section 6.2& 6.3CCSS: A.REI.4b
2Mathematical Practices: 1. Make sense of problems and persevere in solving them.2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated reasoning.
3CCSS: A.REI.4bSOLVE quadratic equations by inspection (e.g., for x2 = 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 and write them as a ± bi for real numbers a and b.
4Essential Question:How do I determine the domain, range, maximum, minimum, roots, and y-intercept of a quadratic function from its graph & how do I solve quadratic functions by factoring?
5Quadratic Equation y = ax2 + bx + c ax2__ is the quadratic term. bx--- is the linear term.c-- is the constant term.The highest exponent is two; therefore, the degree is two.
6Identifying Terms Example f(x)=5x2-7x+1 Quadratic term 5x2 Linear term xConstant term 1
7Identifying Terms Example f(x) = 4x2 - 3 Quadratic term 4x2 Linear termConstant term
8Identifying Terms Now you try this problem. f(x) = 5x2 - 2x + 3 quadratic termlinear termconstant term5x2-2x3
9Quadratic Solutions The number of real solutions is at most two. No solutionsOne solutionTwo solutions
10Solving EquationsWhen we talk about solving these equations, we want to find the value of x when y = 0. These values, where the graph crosses the x-axis, are called the x-intercepts.These values are also referred to as solutions, zeros, or roots.
11Identifying Solutions Example f(x) = x2 - 4Solutions are -2 and 2.
12Identifying Solutions Now you try this problem.f(x) = 2x - x2Solutions are 0 and 2.
13Graphing 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.
14Graphing 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
15Graphing Quadratic Equations Try this problem y = x2 - 2x - 8.RootsVertexAxis of Symmetryxy-2-1134
16Graphing Quadratic Equations The graphing calculator is also a helpful tool for graphing quadratic equations.
17Roots or Zeros of the Quadratic Equation The Roots or Zeros of the Quadratic Equation are the points where the graph hits the x axis.The zeros of the functions are the input that make the equation equal zero.Roots are 4,-3
18To solve a Quadratic Equation Make one side zero.Then factor then set each factor to zero
29How to write a quadratic equation with roots Given r1,r2 the equation is (x - r1)(x - r2)=0Then foil the factors,x2 - (r1 + r2)x+(r1· r2)=0
30How to write a quadratic equation with roots Given r1,r2 the equation is (x - r1)(x - r2)=0Then foil the factors,x2 - (r1 + r2)x+(r1· r2)=0Roots are -2, 5Equation x2 - (-2+5)x+(-2)(5)=0x2 - 3x -10 = 0
31How to write a quadratic equation with roots Roots are ¼, 8Equation x2 -(¼+8)x+(¼)(8)=0x2 -(33/4)x + 2 = 0Must get rid of the fraction, multiply by the common dominator. 44x2 - 33x + 8 = 0