TMAT 103 Chapter 4 Equations and Their Graphs

TMAT 103 §4.1 Functions

§4.1 – Functions Relations –Set of ordered pairs (x, y) –Independent variable x Domain –Dependent variable y Range

§4.1 – Functions Ex: Find the domain and range of the relation y + x = 2 Ex: Find the domain and range of the relation

§4.1 – Functions Function –Relation where no 2 ordered pairs have the same first element Ex: –Is {(1, 2), (5, 11), (4, 2), (1, 7)} a function? Ex: –Is {(1, 1), (5, 11), (4, 1), (-21, 7)} a function?

§4.1 – Functions Ex: Is y + x = 2 a function? Ex: Is x = y 2 a function?

§4.1 – Functions Functional notation –Isolate y and replace it with f(x) EquationFunction Notation y = 3 + xf(x) = 3 + x y – x 2 = 0f(x) = x 2 y = x 3 + x – 7f(x) = x 3 + x – 7

§4.1 – Functions Using function notation Ex: Given f(x) = x 2 – 3, find: f(7) f(–2) f(z) f(a + b)

§4.1 – Functions Ex: Given f(t) = 5 – 2t + t 2 and g(t) = t 2 – 4t + 4 find: f(4) g(0) f(t) + g(t)

TMAT 103 §4.2 Graphing Equations

§4.2 – Graphing Equations Cartesian Coordinate System –Descartes –Rectangular coordinate system x – axis y – axis origin quadrants

§4.2 – Graphing Equations Cartesian Coordinate System

§4.2 Graphing Equations Plot each of the following points on the Cartesian coordinate system: A(3,1) B(2, –3) C(–4,–2) D(–3, 0) E(–6, 2) F(0, 2)

§4.2 Graphing Equations Examples: –Graph y = –3x – 2 –Graph y = x –Graph y = –3x 2 – x + 2

§4.2 Graphing Equations Solving equations by graphing –Ex: Given the graph of y = x 3 + 4x 2 – x – 4 below, solve the equation y = x 3 + 4x 2 – x – 4 when: y = 0 y = 3 y = 6

§4.2 Graphing Equations Ex: Solve the equation y = 2x 2 – 5x – 3 graphically for y = 1, –2, and –10

TMAT 103 §4.3 The Straight Line

§4.3 – The Straight Line Slope of a line –If P 1 (x 1, y 1 ) and P 2 (x 2, y 2 ) represent any two points on a straight line, then the slope m of the line is:

§4.3 The Straight Line Examples: –Find the slope of the line passing through (1, 7) and (4, –3) –Find the slope of the line passing through (1, 5), and (3, 2)

§4.3 The Straight Line Properties of the slope of a line 1.If a line has positive slope, then the line slopes upward from left to right (rises) 2.If the line has negative slope, then the line slopes downward from left to right (falls) 3.If the line has zero slope, then the line is horizontal (flat) 4.If the line is vertical, then the line has no slope since x 1 = x 2 in all cases

§4.3 The Straight Line Examples –Graph the line with slope 3 that passes through (1, 4) –Graph the line with slope –2 that passes through (0, 7) –Graph the line with slope 0 that passes through (–1, 2) –Graph the line with no slope that passes through (3, 5)

§4.3 The Straight Line Point slope form of a straight line –If m is the slope and (x 1, y 1 ) is any point on a non-vertical line, its equation is: y – y 1 = m(x – x 1 )

§4.3 The Straight Line Examples: –Find the equation of the line with slope –2 and which passes through (4, –1) –Find the equation of the line passing through (10, 3), and (3, 0)

§4.3 The Straight Line Slope-intercept form of a straight line –If m is the slope and (0, b) is the y-intercept of a non-vertical line, its equation is: y = mx + b

§4.3 The Straight Line Examples: –Find the equation of the line with slope –2 and which passes through (0, –1) –Find the equation of the line with slope 5 and y-intercept 16

§4.3 The Straight Line Equation of a horizontal line –If a horizontal line passes through the point (a, b), its equation is: y = b

§4.3 The Straight Line Equation of a vertical line –If a vertical line passes through the point (a, b), its equation is: x = a

§4.3 The Straight Line Examples: –Find the equation of the line parallel to and 7 units below the x-axis –Graph the line x = 4

TMAT 103 §4.4 Parallel and Perpendicular Lines

§4.4 – Parallel and Perpendicular Lines Parallel Lines –Two lines are parallel if either of the following conditions holds: 1.They are both parallel to the x-axis 2.They both have the same slope

§4.4 – Parallel and Perpendicular Lines Parallel Lines

§4.4 – Parallel and Perpendicular Lines Examples: –Determine if l 1 and l 2 are parallel: l 1 : y = 3x – 15 l 2 : y = 3x + 7 –Determine if l 3 and l 4 are parallel: l 3 : y = –2x – 15 l 4 : 2y – 4x = 7

§4.4 – Parallel and Perpendicular Lines Perpendicular Lines –Two lines are perpendicular if either of the following conditions holds: 1.One line is vertical with equation x = a, and the other line is horizontal with equation y = b 2.Neither is vertical and the slope of one line is the negative reciprocal of the other.

§4.4 – Parallel and Perpendicular Lines Perpendicular Lines

§4.4 – Parallel and Perpendicular Lines Examples: –Determine if l 1 and l 2 are perpendicular: l 1 : y = 2x – 15 l 2 : y = –½x + 7 –Determine if l 3 and l 4 are perpendicular : l 3 : y = –3x – 15 l 4 : 9y – 3x = 7

TMAT 103 §4.5 The Distance and Midpoint Formulas

§4.5 The Distance and Midpoint Formulas The Distance Formula

§4.5 The Distance and Midpoint Formulas Distance Formula –The distance between two points P(x 1, y 1 ) and Q(x 2, y 2 ) is given by the formula

§4.5 The Distance and Midpoint Formulas Examples: –Find the distance between the points (1, 2) and (7, 14) –Find the distance between the points (–3, 2) and (4, –7)

§4.5 The Distance and Midpoint Formulas The Midpoint Formula

§4.5 The Distance and Midpoint Formulas Midpoint Formula –The coordinates of the point Q(x m, y m ) which is midway between the two points P(x 1, y 1 ) and R(x 2, y 2 ) are given by:

§4.5 The Distance and Midpoint Formulas Examples: –Find the midpoint of the points (1, 2) and (7, 14) –Find the midpoint of the points (–3, 2) and (4, –7)