Mathematics 10 - 2nd Quarter PDF

MATHEMATICS 10 - 2ND QUARTER Prepared By: Kyle Benavente POLYNOMIAL FUNCTIONS Example: a function that involves only f(x) = 4x3 + 2x2 - 5x + 7 = 0 non-negative integer powers or only FACTOR FORM: positive integ...

MATHEMATICS 10 - 2ND QUARTER Prepared By: Kyle Benavente POLYNOMIAL FUNCTIONS Example: a function that involves only f(x) = 4x3 + 2x2 - 5x + 7 = 0 non-negative integer powers or only FACTOR FORM: positive integer exponents of a variable EX 1: (X - 2) (X + 1) (X - 5) = 0 in an equation. Degree: 3 Roots: 2, -1, 5 f(x) = X2 - 5x + 6, and y = X2 - 5x + 6 is an example of Polynomial Function. EX 2: (x + 1)3 (x - 2)2 (x + 1) = 0 Degree: 6 Roots: -1 of multiplicity 3 X2 - 5x + 6 is an example of Polynomial 2 of multiplicity 2 Expression. -1 X2 - 5x + 6 = 0 is an example of The polynomial function is in Polynomial Equation. factor form where the degree is the sum of all of the exponent, and the roots should be the - Polynomial Equation is a opposite sign as the one in the polynomial that has been set equal to equation. zero in an equation. FINDING POLYNOMIAL EQUATION STANDARD FORM: GIVEN THE ROOTS: EX: X2 - 5x + 6 = 0 Degree: 2 Example: -1, 3, 2 Roots: 2 (x + 1) (x - 3) (x - 2) V The polynomial function is in X2 + 1x - 3x -3 standard form when the terms are arranged from the highest (X2 - 2x - 3) (x - 2) exponent down to the lowest. Which the highest exponent is X3 - 2x2 - 3x - 2x2 + 4x + 6 = 0 determined to be the degree. = X3 - 4x2 + 1x + 6 = 0 X-intercepts Y-intercepts: X values when y = 0 Y = (3) (-1) (-4) Y-intercepts Y = 6 | (0,6) Y values when x = 0 GRAPH: Odd Positive (Falls to the Example: left, Rises to the right). Y = x3 - 4x2 + x + 6 3 | 1 -4 1 6 3 -3 -6. 2 | 1 -1 -2 0 2 2. -1 | 1 -1 0 1. 1 0 roots/x-intercepts = 3, 2, -1 (3,0) (2,0) (-1,0) Example 2: Y = x5 + 3x4 - 1x3 - 7x2 + 4 Y = (x + 2)2 (x - 1) (x + 2)2 Degree: 5 Odd - X-intercepts: (-2,0) (1,0) (2,0) Y-intercepts: Y = (4) (1) (4) Y = 16 | (0,16) GRAPH: Odd Negative (Rises to the left, Falls to the right). FACTORS: Y = (x - 3) (x - 2) (x + 1) Y = (0 - 3) (0 - 2) (0 + 1) = ( -3) (-2) (1) Y=6 (0,6) GRAPHING POLYNOMIAL FUNCTION Example 1: Y = (2x + 3) (x - 1) (x - 4) Y = 2x3 - 7x2 - 7x + 2 = 0 Degree: 3 Odd + X-intercepts: (-3,0) (1,0) (4,0) 2 Example 3: CIRCLE Y = x4 - 7x2 + 6x Is a shape consisting of all points in Y = x (x + 3) (x - 1) (x - 2) a plane that are at a given distance Degree: 4 Even + from a given point, the centre. X-intercepts: (0,0) (-3,0) (1,0) (2,0) PARTS OF A CIRCLE Y-intercepts: Center - Is the fixed point from Y = (0) (3) (-1) (-2) which all points on the boundary Y = 0 | (0,0) of the circle are equidistant. Radius - Distance from the GRAPH: Even Positive (Rises to the center of a circle to the outside. left, Rises to the right). Diameter - Distance across the circle going through the centre. Circumference - Distance once around the circle. Arc - A part of the circumference. 1. Major arc - Is greater than half the circumference. 2. Minor arc - Is less than half the circumference. Area - The space inside the shape. Chord - Line segment going from one point of the circumference to Example 4: another but does not go through Y = -x4 + 2x3 + 13x2 - 14x - 24 the center. Y = - (x + 3) (x + 1) (x - 2) (x - 4) Secant - Line that goes through Degree: -4 Even - the circle at two points. X-intercepts: Tangent - Straight line that (-3,0) (-1,0) (2,0) (4,0) touches the circle at a single Y-intercepts: point only. Y = (3) (1) (2) (4) Sector - Section of the circle Y = -24 | (0,-24) created by two radii. Segment - Section of the circle GRAPH: Even Negative (Falls to the created by a chord. left, Falls to the right).  Central Angle - The angle formed when two radii of a circle meet at the center of the circle. The vertex of a central angle is always at the center of the circle. EXTERNAL SECANT SEGMENT An external secant segment is the part of a secant segment that is outside a circle. Inscribed Angle - The angle formed in the interior of a circle when two chords intersect on the THEOREMS ON SECANT circle. It can also be defined as SEGMENTS, TANGENT SEGMENTS, the angle subtended at a point on AND EXTERNAL SECANT the circle by two given points on SEGMENTS the circle. 1. If two secant segments are drawn to a circle from an exterior point, then the product of the lengths of one secant segment and its external secant segment is equal to the product of the lengths of the other secant segment and its external secant segment. THEOREM ON TWO INTERSECTING CHORDS 2. If a tangent segment and a If two chords of a circle intersect, secant segment are drawn to a then the product of the measures circle from an exterior point, then of the segments of one chord is the square of the length of the equal to the product of the tangent segment is equal to the measures of the segments of the product of the lengths of the other chord. secant segment and its external secant segment.  MIDPOINT FORMULA Example 1: A (6, 8) AND B (12,10) x1 y1 x2 y2 DISTANCE FORMULA Example 1: M (2, -3) and N (10, -3) x1 y1 x2 y2 Example 2: P (-5, -1) and Q (8, 6) x1 y1 x2 y2 Example 2: A (0, 0) and B (6, 8) x1 y1 x2 y 2