Quadratic functions and Equations.pdf
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This document discusses quadratic functions and equations. A quadratic function is defined as f(x) = ax^2 + bx + c, where a ≠ 0. The graph of a quadratic function is a parabola. A quadratic equation is an equation of degree 2 in the form ax^2 + bx + c = 0, where a, b, c are real numbers and a ≠ 0. There are three methods for solving quadratic equations: factorizing the equation, completing the square, and using the quadratic formula. Factorizing and completing the square are demonstrated through examples.
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The document then lists topics that will be covered in the exam, grouped under the categories of Algebra, Geometry, Calculus, Trigonometry, Statistics, Science and Technology. Specific topics include functions, quadratic equations
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This document discusses linear functions and systems of linear equations. It begins by defining the standard form of a linear function as y = mx + b, where m is the slope and b is the y-intercept. It then discusses various ways to determine the equation of a linear function given different inputs like slope and a point, or two points. The document also discusses how to graph linear functions and systems of linear equations. It describes three possible solutions for a system of two equations with two unknowns: a unique solution, no solution, or infinitely many solutions. Finally, it covers two methods for solving systems of linear equations: elimination and substitution.
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The document discusses graphing quadratic functions. It begins with reviewing key concepts like the vertex and axis of symmetry and how the a, b, and c coefficients affect the graph. Examples are provided for determining the width, direction opened, and vertical shift based on these coefficients. The remainder of the document provides step-by-step examples of graphing quadratic functions by finding the axis of symmetry, vertex, y-intercept, and other points to plot the parabolic curve.
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This document provides an overview of quadratic equations, including definitions, methods for solving quadratic equations such as factoring, completing the square, and using the quadratic formula, and applications of quadratic equations. Key topics covered include defining linear and quadratic equations, solving quadratics by factoring when possible and using completing the square or the quadratic formula when not factorable, deriving the quadratic formula, interpreting the discriminant, and modeling real-world situations with quadratic equations.
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- 1. Quadratic functions and equations • Quadratic functions: A function f given as f(x) = a 𝑥2 + b x + c, a ≠0, is called a quadratic function. The graph of quadratic function is a parabola. e.g. 1) f(x) = 𝑥2 − 4x + 5. 2) f(x) = 3 𝑥2 − 9 x + 13 are quadratic functions.
- 2. Quadratic equations An equation of degree 2 is called quadratic equations. General from: a x2 + b x + c = 0 where a, b, c, ∈ R and a ≠ 0. The values of x satisfying the quadratic equation are called roots of the equation. Examples: 1) 3x2 + x + 5 = 0, 2) -x2 + 7x + 5 = 0, 3) x2 + x = 0 are the quadratic equations.
- 3. • There are three methods of solving the quadratic equations, 1) Factorizing of Quadratic Equation 2) Method of Completing the Square 3) Formula Method of Finding Roots 1) Factorizing of Quadratic Equation Ex. Solve the following Quadratic Equations 1) x2 + 5x + 6 = 0 Sol.: x2 + 5x + 6 = 0 x2 + 2x + 3x + 6 = 0 x(x + 2) + 3(x + 2) = 0 (x + 2)(x + 3) = 0 Thus the two obtained factors of the quadratic equation are (x + 2) and (x + 3) i.e. x =-2 & x = -3. 2) x2 - 2x - 24 = 0 Sol.: x =-4 & x = 6.
- 4. 2) Method of Completing the Square Ex.1) Solve x2 + 4x + 4 =0 Sol: x2 + 4x + 4 =0 (x+2)2 = 0 (x+2)(x+2) =0 i.e. x= -2, -2 Ex.2) Solve x2 + 8x + 16 =0
- 5. THANK YOU