MATH 9 Reviewer for 1st Periodical Examination PDF

— MATHEMATICS 9 — Reviewer for 1st Periodical Test || by Ethan Serquiña of 9 - Bohr (Thank you Ethan for letting me steal borrow it) Topics for the First Quarter: I. Quadratic Equations (Module 1): ➔ Illustrating Quadratic Equations...

— MATHEMATICS 9 — Reviewer for 1st Periodical Test || by Ethan Serquiña of 9 - Bohr (Thank you Ethan for letting me steal borrow it) Topics for the First Quarter: I. Quadratic Equations (Module 1): ➔ Illustrating Quadratic Equations ➔ Solving Quadratic Equations through the following: ➔ Extracting the Square Roots ➔ Factoring ➔ Completing the Square ➔ Quadratic Formula (includes the derivation) II. Discriminants and the Sum and Product of the Roots (Module 2): ➔ Discriminant Formula ➔ Determine the Nature of the Roots of a Quadratic Equation using the Discriminant ➔ Formula for the Sum and Product of Roots of Q.E. (includes the derivation) ➔ Getting the Sum and Product of the Roots of a Q.E. ➔ Transforming the Sum and Product of Roots into a Q.E. ➔ Case 1: If the Sum and Product of the Roots are Given ➔ Case 2: If the Roots are Given III. Word Problems Involving Quadratic Equations (Module 4) (mostly examples) IV. Solving Problems Involving Rational Algebraic Equations (Module 4) V. Quadratic Inequalities (Module 5): ➔ Introduction to Quadratic Inequalities ➔ Solving Quadratic Inequalities and Word Problems VI. Quadratic Functions (Modules 6 to 9): ➔ Review of Relations, Functions, Domain, and Range ➔ Introduction to Quadratic Functions ➔ Representing Quadratic Functions using an Equation, Table of Values, and Graph ➔ The Vertex Form and the Formulas for h and k ➔ Transforming Quadratic Functions to the Vertex Form and Vice Versa ➔ Graph of a Quadratic Function ➔ Zeros of a Quadratic Function (and its Sum and Product) ➔ Transforming the Zeros and a Given Point or Vertex to Quadratic Functions ➔ Quadratic Function Transformations I. Quadratic Equations The characteristics of a quadratic equation in one variable are the following: 1. A polynomial that has a degree of 2 (highest exponent value = 2) 2. Can be written in the standard form: ax2 + bx + c = 0 3. Has three terms: ax2 = quadratic term, bx = linear term, c = constant 4. In the standard form, a, b, and c are real numbers, and a ≠ 0 and non-negative. The solutions of a quadratic equation are called the roots. In solving quadratic equations, we can use 4 methods: * Remember! Polynomials CANNOT have a variable at the denominator and a negative exponent! Get it? Now it’s your turn! You may answer these questions for reviewing. Check your answers using a Math AI solver. *Extra Definitions: Factors are two or more expressions that produce the given expression when multiplied together. A perfect square trinomial is a trinomial where the first and last terms are perfect squares while the middle term is positive or negative TWICE the product of the square roots of the first and last tems. II. Discriminant The discriminant is used to determine the nature of the roots of a quadratic equation. The discriminant (D) is represented by the formula: D = b2 - 4ac The following are the nature of the roots of a quadratic equation: Here are some examples: Get it? Now it’s your turn! You may answer these questions for reviewing. III. Sum and Product of the Roots Here are some examples showing the three situations (checking process is optional): 1. If you are asked to find the product and roots given a quadratic equation: 2. If you are asked to find the quadratic equation given the sum and product of roots.  3. If you are asked to find the quadratic equation given only the roots. Get it? Now it’s your turn! You may answer these questions for reviewing. IV. Word Problems Involving Quadratic Equations In solving word problems involving quadratic equations, you can do the following process: Here are some examples of word problems involving quadratic equations:  Get it? Now it’s your turn! You may answer these questions for reviewing. V. Solving Problems Involving Rational Quadratic Equations Please note that in these types of problems, you may encounter work problems frequently. Work problems are represented by the equation: In solving word problems involving rational quadratic equations, you can do the same process in solving word problems involving quadratic equations. Here are some examples:  Get it? Now it’s your turn! You may answer these questions for reviewing. VI. Quadratic Inequalities A quadratic inequality is an inequality that contains a polynomial of 2nd degree and can be written in any of the following forms: 1. ax2 + bx + c > 0 2. ax2 + bx + c < 0 3. ax2 + bx + c ≥ 0 4. ax2 + bx + c ≤ 0 You can use this process to solve quadratic inequalities: Can’t understand it? Here are some examples for visualization:  In graphing, a hollow point means that point is NOT included, and a closed point means that point IS included. Parentheses means NOT INCLUDED, while brackets mean INCLUDED. In finding the solution sets for quadratic inequalities, you can represent it by set notation or interval notation. Get it? Now it’s your turn! You may answer these questions for reviewing. VII. Word Problems Involving Quadratic Inequalities In solving word problems involving quadratic inequalities, you can do the following process: Here are some examples of word problems involving quadratic equations:  Get it? Now it’s your turn! You may answer these questions for reviewing. VIII. Quadratic Functions, Part 1 ! Review of Past Lessons: A relation shows the relationship between sets of values. There are four types of relations: 1. One-to-one Correspondence -> Each domain element (x) has a UNIQUE range element (y). 2. Many-to-one Correspondence -> Two or more domain elements (x) has the SAME range element (y). 3. One-to-many Correspondence -> Two or more range elements (y) has the SAME domain element (x). 4. Many-to-many Correspondence -> Two or more domain elements (x) is related to TWO OR MORE range elements (y). A function pairs each element in one set (x) UNIQUELY with one element from another set (y). In other words, each input has ONE output. The one-to-one and many-to-one correspondences are a function. The one-to-many and many-to-many correspondences are not a function. The domain is the set of x-coordinates (abscissa or inputs) of a function. The domain of a quadratic equation is the set of all real numbers. The range is the set of y-coordinates (ordinate or outputs) of a function. The range of a quadratic function depends on where its graph opens. ! The Quadratic Function Shenanigans: A quadratic function can be written in the standard form:  In the standard form, a, b, and c are real numbers, and a ≠ 0 and non-negative. The graph of a quadratic function is a parabola. ! Graphing Time In graphing a quadratic function, we can use a table of values. If you are given a quadratic function, you can compute for the values of the vertex h and k, which are represented by: In making a table of values given a quadratic function, determine the values of a, b, and c first. Then, compute for the value of h and k. You do not have to do these steps if the vertex is already given! In table of values, it is recommended to have 5 values. You may follow the formulas shown below: x y D1 D2 h+2 y2 =Value of y when (h + 2) is substituted to the Da = (y2) - (y1) (Da) - (Db) given quadratic function h+1 y1 = Value of y when (h + 1) is substituted to the Db = (y1) - (k) (Db) - (Dc) given quadratic function Value of h Value of k Dc = (k) - (y3) (Dc) - (Dd) h-1 y3 = Value of y when (h - 1) is substituted to the Dd = (y3) - (y4) given quadratic function h-2 y4 = Value of y when (h - 2) is substituted to the given quadratic function Take note: the values in the D1 column should be symmetrical, while the values in D2 should be consistent. After this, plot the points of x and y on the cartesian plane, and connect the lines. VIII. Quadratic Functions, Part 2 A quadratic function can also be written in the vertex form, which is represented as: y = a(x - h)2 + k or f(x) = a(x - h)2 + k, where the vertex is (h, k) ! How is it derived? In transforming quadratic functions to the vertex form, we use completing the square. However, if a > 1, we have to make a = 1 by dividing all terms by a OR in most cases, we use common monomial factoring, but we dont factor out constant c. However, in the situation above, we can also use the In transforming the vertex form to quadratic function, we simply expand. Here are some examples: VIII. Quadratic Functions, Part 3 In a quadratic function f(x), k is a zero of f(x) if f(x) = 0. A quadratic function has at most TWO distinct zeros. The zeros of a quadratic function are simply the roots. The axis of symmetry is the x-coordinate, or h in vertex (h, k), which divides the parabola into two. The y-coordinate is k in vertex (h, k).  The sum and product of zeros are basically the same as the sum and product of roots of a quadratic equation: Sum of Zeros = -b/2a Product of Zeros = c/a Quadratic Function Derived from Zeros: y = a[x2 - (-b/2a)x + (c/a)] VIII. Quadratic Functions, Part 4 The graph of a basic quadratic function is represented as f(x) = x2, where the vertex is at (0, 0) or the origin. If a < 0, the graph opens downwards. ! When put together, we get the vertex form: f(x) = a(x - h)2 + k! IX. Word Problems Involving Quadratic Functions Examples: Get it? Now it’s your turn! You may answer these questions for reviewing. You have reached the end of this reviewer! Good luck on the periodical exams!