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Quadratic functions Rachel Collins and Ava LaRue Properties of Quadratics STANDARD FORM: y=ax2 + bx + c VERTEX FORM: y=a(x - h)2 + k ALWAYS GRAPHS A PARABOLA: -parent function= x2 Graphing Quadratics Standard Form: find vertex using x= –b/2a plug that X value into a T table and add 2 numbers below and above that value under the X column. To find the Y value, plug in your X values to the original equation. Then plot your points. For example: y= x2 + 4x + 6 –4/ 2(1) = -2 X y= x2 + 4x + 6 0 (0)2 + 4(0) + 6 = 6 -1 (-1)2 + 4(-1) + 6 = 3 -2 (-2)2 + 4(-2) + 6 = 2 (-3)2 + 4(-3) + 6 = 3 -3 -4 (-4)2 + 4(-4) + 6 = 6 Graphing Quadratics Vertex Form: Graph the function using transformations and using a T table. Vertex=(h,k) For example: y = 2(x - 1)2 + 3 Vertex:(1,5) -use the 2 numbers below and above the X value and plug them into T table to find the points Solving Quadratic Equations FactoringSteps: 1. Set y equal to 0 2. Check for a GCF. If yes, factor it out of each number 3. Make sure it is in the form 0=ax2+bx+c 4. Find the factors of c and look for the pair that add up to b 5. You should have 2 answers for the equation a. These answers are known as zeros, or x-intercepts 6. Example: 0=x2+3x-4 4 ,-1= (x-4)(x+1) x=4 and -1 Solving Quadratic Equations Quadratic FormulaThe Quadratic formula is The formula will always work when solving a quadratic equation, but you should try and factor first If the equation can not factor, then you must use the quadratic formula Example: y= x2 + 4x + 6 Plug the numbers into the formula and simplify Graphing Quadratic Inequalities First, make sure 0 is on one side of the < or > sign After, make a t table by finding the vertex Next graph the equation and shade in the graph using a test point (0,0) y< -x2 -2x +3 Vertex:(-1,4) (use -b/2a) -use the 2 numbers below and above the X value and plug them into T table to find the points Quadratic inequalities Make sure zero is on the other side of the inequality Then factor to get two numbers equal to x Set each number equal to x, solve and make a number line x2 -4x +1< 6 x2 -4x -5< 0 (x-5)(x+1) x=5 x=1 1<x<5 -1 5 Complex Numbers Complex number form: a+bi Addition: add like terms,(-2+4i)+(3-11i) = 1-7i Subtraction: distribute “-” sign, (-2+4i)-(3-11i)= -2+4i-3+11i= -5+15i Multiplication: multiply coefficients and add “i”s, (1+3i)(3i)= 3i+9i2 = 3i+9(-1)= 3i-9 Division: divide coefficients and subtract “i”s, 9i/3i = 3