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Math II Chapter 1 Vocabulary & Notes Vertex is the highest or lowest point. It is written as a coordinate pair; for example (2,3) Axis of Symmetry is the line that divides the parabola in half (left to right) and creates a mirror image. The Axis of Symmetry will be the X value of the Vertex. The Axis of Symmetry is always written as X = ____. For example, if the Vertex is (2,3), the Axis of Symmetry is X = 2 Zeros are the X-intercepts or where the parabola crosses the X axis. The zeros are written as a coordinate pair where the y value will always be zero X-intercept is where the parabola crosses the X-axis. Put zero in for Y value then solve for the X value. Y-intercept is where the parabola crosses the Y-axis. Put a zero in for the X value and solve to find the value for Y. Parabola is a U-shaped graph of a quadratic equation. The parent function is f(x)= X2. Quadratic equation is f(x)= ax2 + bx + c or f(x)= a(x – h)2 + k Domain is the collection/ set of X-values For a quadratic function, the domain is always “All Real Numbers” or . Range is the collection/ set of Y-values. For a quadratic function, the range will either be ≥ or ≤ that y value of the vertex. If a is positive in f(x)= ax2 + bx + c the vertex is the lowest point and the y-values are decreasing on the left side of the vertex and increasing on the right side of the vertex. If a is negative in f(x)= ax2 + bx + c the vertex is the highest point and the y-values are increasing on the left side of the vertex and decreasing on the right side of the vertex. Read graphs Left to Right (like reading a book) Extreme points or Extrema is the maximum or minimum point on a graph (e.g., the vertex) Interval notation: Open interval (a, b) is all numbers between a and b, but does not include a or b Interval notation: Closed interval [a, b] is all numbers between a and b and also includes a and b. Interval notation: Half-closed/ half-open interval. The interval includes the number on the side with the brackets, with the parenthesis does not include the number. [a, b) = numbers between a and b, including a but not including b (5, 10] = numbers between 5 and 10, including 10 but not 5 [0, ∞) = the set of numbers greater than and equal to zero To factor quadratic equations use the area model. Standard form is ax2 + bx + c = y If you are given an X coordinate of a vertex, plug that value into the equation to get your Y coordinate The X coordinate of the vertex = -b/ (2a) The Y coordinate of the vertex = f(-b/2a) (Plug the x value in and solve!) In Standard Form, when a < 0, the vertex is the highest point of the parabola (the parabola opens down). In Standard Form, when a > 0, the vertex is the lowest point of the parabola (the parabola opens up). Vertex form: f(x) = a(x-h)2 + k where your vertex is (h, k) Converting from Vertex Form to Standard Form: a(x-h)2 + k =0 a(x-h)(x-h) + k = 0 You have to FOIL the (x-h)2 terms When graphing a parabola, find the vertex and then pick two x values greater and two x values less than the x value of the vertex. Math II Chapter 2 Notes & Vocabulary The Quadratic Formula gives you the roots/ zeros/ x-intercepts/ solutions (remember these are all the same thing) of a quadratic formula. b b 2 4ac The Quadratic Formula is x 2a Roots/ Solutions are written as x = ____ (what you find when you solve the Quadtratic Formula) Zeros are written as a coordinate pair with the x-value as whatever you find when you solve the Quad Formula and the y-value as 0. (___, 0) The Discriminant is b2 -4ac The Discriminant tells you the number and the nature of and the number of the roots of the Quadratic Functions. When b2 -4ac < 0 no real roots When b2 -4ac = 0 one real root When b2 -4ac > 0 two real roots Finding the Vertex (written as a coordinate pair) of a parabola: The x-value of the coordinate pair = b ; The y-value is found by plugging in your x-value and solving. 2a Complex Numbers: Standard form is a + bi where a is a real number and bi is an imaginary term i 1 i 2 1 Simplifying negative square roots: For example: 80 1 80 1 * 16 * 5 4i 5 Adding & Subtracting Complex Numbers: Combine like terms (Combine real numbers and then combine imaginary numbers) For example (3 + 2i) – (1 + 6i) = 3 -1 + 2i -6i = 2 -4i Multiplying Complex Numbers: FOIL or Distribute (Depending on your problem) and then simplify! For example (3 + 2i)(1 + 6i) = 3 + 18i + 2i + 12i2 = 3 + 20i + (12*-1) = -9 + 20i Dividing Complex Numbers: Multiply by the Conjugate and then simplify. Remember the Conjugate is simply the complex number with the opposite sign (For example the conjugate of (2 + 7i) is (2 – 7i) Example of Dividing: 1 6i (1 6i ) (8 3i ) (8 3i 48i 18i 2 ) (8 51i 18) (10 51i) * 2 8 3i (8 3i ) (8 3i ) (64 24i 24i 9i ) (64 9) 73 Multiplying Conjugate Shortcut (8 – 3i)(8 + 3i) Square the first number, Square the second number, and add them together 82 = 64 32 = 9 64 + 9 = 73 The Quadratic Formula with imaginary numbers: When you work the Quadratic Formula and on the outside of the b2 4ac gives you a negative square root, simplify, putting the i . These are roots, but they are imaginary roots, not real roots. Solving Quadratic Inequalities Example: x2 -4x – 2 > -5 1. Set the equation equal to zero. X2 – 4x -2 +5 > -5 + 5 x2 – 4x + 3 > 0 X2 – 4x + 3 = 0 2. Calculate the zeros/ solutions of the function. (x – 3)(x – 1) X = 3 and X = 1 3. Roughly sketch the parabola 4. Find the y-values on the graph that correspond to the inequality. (Substitute y in for the equation) x2 – 4x + 3 > 0 y>0 Y values greater than 0 5. Decide which values of x will give you these y values. x < 1 and x >3 Sequence List of numbers Arithmetic Sequence List of numbers with a particular pattern (e.g., add 2 each time) Term A number in a sequence Common Difference (d) How much you add or subtract to each term in an arithmetic sequence Explicit form a n dn a0 ; This gives you a particular term in an arithmetic sequence by plugging that number (e.g. 20 to find the 20th term) for n Arithmetic Series Adding the terms of a arithmetic sequence together Gauss’s Formula for finding sum Sn n(a1 an ) , where a1 is your first term and an is your last term 2