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Ch. 1 Quadratics 1.1 Quadratics in Standard Form A. OBJ: to graph quadratics in standard form B.Facts/Formulas: 1. A quadratic function is a function that can be written in the standard form y = ax2+bx+c where a β 0. The graph of a quadratic equation is a parabola. 2. The lowest or highest point on a parabola is the vertex. 3. The axis of symmetry divides the parabola into mirror images and passes through the vertex. 4. For y = ax2+bx+c, the vertexβs y-coordinate is the minimum value of the function if a > 0 and the maximum value if a < 0. 5. a>0 up and a< 0 down 6. |a| > 1 narrow and |a| < 1 wide 7. axis of symmetry = βπ ππ 8. y β intercept = (0, c) 1.2 Quadratics in Intercept and Vertex Form A.OBJ: to graph quads in intercept and vertex form B. FACTS/FORMULAS: 1. The vertex form of a quadratic equation is given by y = a(x - h)2 +k. 2. The intercept form of a quadratic equation is given by y = a(x - p)(x - q). C. Examples: 1. 1.3 / 1.4 Solving Quadratics A.OBJ: to solve quads by factoring B.Facts/Formulas: 1. A monomial is an expression that is a number, a variable, or the product of a number and one or more variables. 2. A binomial is the sum of two monomials. 3. A trinomial is the sum of three monomials. 4. A quadratic equation in one variable can be written as ax2 + bx + c = 0 where a β 0. 5. A solution of a quadratic equation is called the root or solution of the equation. 6. Because a functionβs value is zero when x = p and x = q, the numbers, p and q are also called the zeros or xintercepts of the function. 7. Zero Product Property: If A*B = 0, then A=0 or B=0. C. Examples: 1.5 Solving by β A. OBJ: to solve quads by using square roots B.Facts/Formulas: 1. A number r is a square root of a number s if r2= s. 2. The expression βπ is called a radical. The symbol β is a radical sign and the number s beneath the radical sign is the radicand of the expression. 3. To rationalize a denominator βπ of a fraction, multiply the numerator and denominator by βπ 4. To rationalize a denominator a + βπ of a fraction, multiply the numerator and denominator by a - βπ , and to rationalize a denominator a - βπ of a fraction, multiply the numerator and denominator by a + βπ 5. The expressions a + βπ and a - βπ are called conjugates. 1.6 Complex #βs A. OBJ: to perform operations on complex numbers. B. Facts/Formulas: 1. The imaginary unit i is defined as i = ββπ 2. A complex number written in standard form is a number a ± bi, where a and b are real numbers. If b β 0, then a ± bi is an imaginary number. 3. Two complex numbers of the form a + bi and a - bi are called complex conjugates. 4. Every complex number corresponds to a point in the complex plane. 5. The complex plane has a horizontal axis called the real axis and a vertical axis called the imaginary axis. 6. The absolute value of a complex number z = a ± bi, denoted |z| is a nonnegative real number defined as z = βππ + ππ . 1.7 Completing the Square A. OBJ: to solve quads by completing the square. B.Facts/Formulas: 1. To complete the square for ax2+bx=c, add (b/2)2 to both sides. 2. Remember that a = 1! 3. Factor and solve C. Ex: 1.8 Quadratic Formula A. OBJ: to solve using the quadratic formula. B.Facts/Formulas: 1. The quadratic formula: Let a, b, and c be real numbers where a β 0. 2. The solutions of the quadratic equation ax2 ± bx ± c = 0 βπ±βπ2 β4ππ are π₯ = 2π 3. In the quadratic formula, the expression b2 - 4ac is called the discriminant of the quadratic equation. a. IF the discriminant > 0, then there are 2 real solutions b. IF the discriminant < 0, then there are 2 imaginary solutions c. IF the discriminant = 0, then there is only 1 real solution. 1.9 Quadratic Inequalities A. OBJ: to solve and graph quadratic inequalities B.Facts/Formulas: 1. A quadratic inequality in two variables can be written in one of the following forms: y < ax2 ±bx ± c y οΎο ax2 ± bx±c y β₯ ax2 ± bx ± c y β€ο ax2 ± bx ± c 2. A quadratic inequality in one variable can be written in one of the following forms: ax2 ± bx ± c < 0 ax2 ± bx ± c οΎο 0 ax2 ± bx ± c β₯ 0 ax2 ± bx ± c β€ο 0 C. Ex: