Q1_G10_ADV_ALGEB.docx.pdf

ADVANCED ALGEBRA - 1ST GRADING REVIEWER OUTLINE I. LINEAR EQUATIONS AND FUNCTIONS A. The Cartesian Coordinate System B. Relations and Functions 1. Relations a. Ways of Expressing Relations...

ADVANCED ALGEBRA - 1ST GRADING REVIEWER OUTLINE I. LINEAR EQUATIONS AND FUNCTIONS A. The Cartesian Coordinate System B. Relations and Functions 1. Relations a. Ways of Expressing Relations 2. Functions a. Concepts of Functions b. Some Special Types of Functions c. Functional Notation i. Operations Involving Functions ii. Evaluating Functions iii. Composition of Function iv. Inverse of a Function NOTE: Please attend the A.M.E peer tutoring session on September 18-19, 2024. RECALL MATH CHANTS Addition and Same signs, add and keep. Different signs, subtract. Follow the sign of the greater Subtraction of Integers number. Then, you’ll be exact. Examples: 5 + (−5) = 0 −8 − 6 = −14 5 + 10 = 5 (−7) + (−2) = −9 Multiplication and If you love to love, then you love. Division of Integers If you hate to hate, then you love. If you love to hate, then you hate. If you hate to love, then you hate. Examples: (−12)(−3) = 36 (−4)( 7) = −28 −25 /(−5) = 5 −16/2 = −8 SPECIAL PRODUCTS Product of Two (𝑎x + b)( 𝑐x + d) = acx² + (ad + bc)x + bd Binomials STEPS: EXAMPLE: (x+5)(x-2) 1. Multiply the first terms. (x)(x)=x² 2. Multiply the outer terms. (x)(-2) = -2x 3. Multiply the inner terms. TRANSCRIBED BY: THE ALLIANCE OF MATH ENTHUSIASTS (A.M.E) - JHS  ADVANCED ALGEBRA - 1ST GRADING REVIEWER (5)(x) = 5x 4. Multiply the last terms. (5)(-2) = -10 5. Then, combine similar terms if there are. −2x + 5x = 3x Product: (𝐱 + 𝟓) (𝐱 − 𝟐) = 𝐱² + 𝟑𝐱 − 𝟏𝟎 Square of a Binomial (a ± b)² = a² ± 2ab + b² STEPS: EXAMPLE: (x+4)² 1. Square the first term. (x)² = x² 2. Twice the product of the first and second terms. 4(𝑥)(2) = 8𝑥 3. Square the second term. (4)² = 16 Product: (𝒙 + 𝟒)² = 𝒙² + 𝟖𝒙 + 𝟏𝟔 Product of the Sum and (a + b)( a − b ) = a² − b² Difference of Two STEPS: Terms EXAMPLE: (x+3)(x-3) 1. Square the first term or multiply the first terms (x)(x) = x² 2. Then, get the negative square of the last term or multiply the last terms. (3)(-3) = -9 Product: ( 𝒙 + 𝟑)( 𝒙 − 𝟑 )= 𝒙² − 𝟗 FACTORING Factoring Quadratic acx²+ (ad + bc)x + bd = (𝑎x + b)(𝑐x + d) Trinomial where 𝐀 = 𝟏 STEPS: EXAMPLE: x²+5x+6 1. Get the factors of the first term. x² = (x)(x) 2. Get the factors of the last term that gives a sum similar to the numerical coefficient of the middle term. 6 = (2)(3), 2 +3 = 5 3. Write the factors as the sum or difference of two terms FACTORS: (𝒙 + 𝟑 )(𝒙 + 2) NOTE: If the last term is positive, both factors share the middle term's sign. If negative, the factors have opposite signs, with the larger factor matching the middle term's sign. Factoring Perfect a² ± 2ab + b²= (a ± b)² Square Trinomial STEPS: EXAMPLE: x²-6x+9 1. Get the square root of the first term. √x² = x 2. Get the square root of the last term. √9 = 3 3. For the operation, follow the sign of the middle term. FACTORS: (𝒙 − 𝟑 )(𝒙 − 3) TRANSCRIBED BY: THE ALLIANCE OF MATH ENTHUSIASTS (A.M.E) - JHS  ADVANCED ALGEBRA - 1ST GRADING REVIEWER Factoring the a² − b² =(a + b) (a − b) Difference of Two STEPS: Squares EXAMPLE: x² - 4 1. Get the square root of the first term. √x² = x 2. Get the square root of the last term. √4 = 2 FACTORS: (𝒙 + 2 )(𝒙 − 2) THE CARTESIAN COORDINATE SYSTEM Cartesian Coordinate 2 Coplanar perpendicular number lines system Nicole Oresme 1320 – Discussed locating points using coordinates 1382 Came up with fractional exponents and contributed to infinite series Preceded Descartes in inventing coordinate geometry René Descartes 1596 – Discovered the Cartesian coordinate system while lying sick in bed 1650 Coplanar Lines Lines that lie on the same plane Perpendicular Lines Two lines intersect at a right angle Number Line A straight line with numbers placed at equal intervals or segments along its length Origin The point of intersection of the x and y axes and written in ordered pair as (0,0) x – axis The horizontal number line y – axis The vertical number line Coordinates The numerical descriptive reference of a point from the two axes x – coordinate / The first coordinate which corresponds to a real number on the x-axis abscissa y – coordinate / The second coordinate which corresponds to a real number on the y-axis ordinate Ordered Pair A set of two well-ordered real numbers called coordinates. (x,y) Quadrants Four regions formed by the perpendicular number lines Quadrant 1 - (+,+) both the abscissa and the ordinate are positive Quadrant 2 - (-,+) the abscissa is negative, and the ordinate is positive Quadrant 3 - (-,-) both the abscissa and the ordinate are negative Quadrant 4 - (+,-) the abscissa is positive, and the ordinate is negative TRANSCRIBED BY: THE ALLIANCE OF MATH ENTHUSIASTS (A.M.E) - JHS  ADVANCED ALGEBRA - 1ST GRADING REVIEWER RELATIONS Relation It involves the association of an individual or object with another individual or object. It is the correspondence between two quantities. It is composed of an independent and dependent variable whose value freely changes and does not depend on quantity A set of ordered pairs that represent a relationship NOTE: A relation is formed when numbers come in pairs Independent Variable A variable that can freely change its value without being influenced by other variables in the study. (Cause) Dependent Variable A variable whose value is influenced by changes in the independent variable. (Effect) Domain It is a collection of the first values in the ordered pair (Set of all input (x) values) Range It is a collection of the second values in the ordered pair (Set of all output (y) values) TYPES OF RELATIONS One to One If every element of X is paired with a unique element of Y One to Many If an element of X is paired with more than one element of Y Many to One If an element of Y is paired with more than one element of X FUNCTIONS TRANSCRIBED BY: THE ALLIANCE OF MATH ENTHUSIASTS (A.M.E) - JHS  ADVANCED ALGEBRA - 1ST GRADING REVIEWER Function The term "function" was formally introduced by the German mathematician Gottfried Wilhelm Leibniz. A special type of relation where each input only has one output. For every x-value, there is exactly one y-value. Way of connecting input values to their corresponding output values. Also called a mapping EXAMPLE: F = { (1,1) , (2,8) , (3,27) , (4,64) … } NOTE: no two ordered pairs have the same abscissa. NOTE: All functions are relations, but not all relations are functions. Vertical line test A. If drawn passing through the graph and intersects it at exactly one point, then it is a function B. If drawn passing through the graph and intersects more than one point, then it is a mere relation. Function notation A way of representing functions in a more concise and organized manner It is typically written in the form f(x), where: - f is the name of the function. - x is the independent variable or the input to the function. - f(x) represents the output of the function when the input is x. It names functions and its variable It replaces "y =" in the equation SPECIAL TYPES OF FUNCTIONS Polynomial Function A function that involves only non-negative powers or only positive integer exponents like the quadratic equation, cubic equation, etc. Constant Function A function that stays constant. It is called the simplest function and is horizontal when graphed. A function that has the same output with different input values. Linear Function A function that is a straight line. f(x) = mx + b m = slope b = y intercept Absolute Value A function that has the variable inside the absolute value Function bars. It is also called “Modulus Function” and it produces non-negative outputs regardless of the input.f(x) = ∣x∣ TRANSCRIBED BY: THE ALLIANCE OF MATH ENTHUSIASTS (A.M.E) - JHS  ADVANCED ALGEBRA - 1ST GRADING REVIEWER Quadratic Function A polynomial function with one or more variables in which the highest exponent is 2. The graph is a curved parabola (intersects its axis of symmetry at a point called the vertex of parabola). f(x) = ax² + bx + c VERTEX Of QUADRATIC EQUATION U Shape where the function has a maximum or minimum value NOTE: If a>0, parabola opens upward, if a