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Understanding Equations and Graphs in Mathematics

Learn about functions, relations, graphing equations, and properties of lines in this comprehensive guide. Understand functions, slopes, and graphing methods with practical examples.

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Understanding Equations and Graphs in Mathematics

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  1. TMAT 103 Chapter 4 Equations and Their Graphs

  2. TMAT 103 §4.1 Functions

  3. §4.1 – Functions • Relations • Set of ordered pairs (x, y) • Independent variable • x • Domain • Dependent variable • y • Range

  4. §4.1 – Functions • Ex: Find the domain and range of the relation y + x = 2 • Ex: Find the domain and range of the relation

  5. §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?

  6. §4.1 – Functions • Ex: Is y + x = 2 a function? • Ex: Is x = y2 a function?

  7. §4.1 – Functions • Functional notation • Isolate y and replace it with f(x)

  8. §4.1 – Functions • Using function notation • Ex: Given f(x) = x2 – 3, find:f(7)f(–2)f(z)f(a + b)

  9. §4.1 – Functions • Ex: Given f(t) = 5 – 2t + t2 and g(t) = t2 – 4t + 4 find:f(4)g(0)f(t) + g(t)

  10. TMAT 103 §4.2 Graphing Equations

  11. §4.2 – Graphing Equations • Cartesian Coordinate System • Descartes • Rectangular coordinate system • x – axis • y – axis • origin • quadrants

  12. §4.2 – Graphing Equations Cartesian Coordinate System

  13. §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)

  14. §4.2 Graphing Equations • Examples: • Graph y = –3x – 2 • Graph y = x2 + 3 • Graph y = –3x2 – x + 2

  15. §4.2 Graphing Equations • Solving equations by graphing • Ex: Given the graph of y = x3 + 4x2 – x – 4 below, solve the equation y = x3 + 4x2 – x – 4 when:y = 0y = 3y = 6

  16. §4.2 Graphing Equations • Ex: Solve the equation y = 2x2 – 5x – 3 graphically for y = 1, –2, and –10

  17. TMAT 103 §4.3 The Straight Line

  18. §4.3 – The Straight Line • Slope of a line • If P1(x1, y1) and P2(x2, y2) represent any two points on a straight line, then the slope m of the line is:

  19. §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)

  20. §4.3 The Straight Line • Properties of the slope of a line • If a line has positive slope, then the line slopes upward from left to right (rises) • If the line has negative slope, then the line slopes downward from left to right (falls) • If the line has zero slope, then the line is horizontal (flat) • If the line is vertical, then the line has no slope since x1 = x2 in all cases

  21. §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)

  22. §4.3 The Straight Line • Point slope form of a straight line • If m is the slope and (x1, y1) is any point on a non-vertical line, its equation is:y – y1 = m(x – x1)

  23. §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)

  24. §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

  25. §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

  26. §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

  27. §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

  28. §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

  29. TMAT 103 §4.4 Parallel and Perpendicular Lines

  30. §4.4 – Parallel and Perpendicular Lines • Parallel Lines • Two lines are parallel if either of the following conditions holds: • They are both parallel to the x-axis • They both have the same slope

  31. §4.4 – Parallel and Perpendicular Lines Parallel Lines

  32. §4.4 – Parallel and Perpendicular Lines • Examples: • Determine if l1 and l2 are parallel: • l1: y = 3x – 15 • l2: y = 3x + 7 • Determine if l3 and l4 are parallel: • l3: y = –2x – 15 • l4: 2y – 4x = 7

  33. §4.4 – Parallel and Perpendicular Lines • Perpendicular Lines • Two lines are perpendicular if either of the following conditions holds: • One line is vertical with equation x = a, and the other line is horizontal with equation y = b • Neither is vertical and the slope of one line is the negative reciprocal of the other.

  34. §4.4 – Parallel and Perpendicular Lines Perpendicular Lines

  35. §4.4 – Parallel and Perpendicular Lines • Examples: • Determine if l1 and l2 are perpendicular: • l1: y = 2x – 15 • l2: y = –½x + 7 • Determine if l3 and l4 are perpendicular : • l3: y = –3x – 15 • l4: 9y – 3x = 7

  36. TMAT 103 §4.5 The Distance and Midpoint Formulas

  37. §4.5 The Distance and Midpoint Formulas The Distance Formula

  38. §4.5 The Distance and Midpoint Formulas • Distance Formula • The distance between two points P(x1, y1) and Q(x2, y2) is given by the formula

  39. §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)

  40. §4.5 The Distance and Midpoint Formulas The Midpoint Formula

  41. §4.5 The Distance and Midpoint Formulas • Midpoint Formula • The coordinates of the point Q(xm, ym) which is midway between the two points P(x1, y1) and R(x2, y2) are given by:

  42. §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)

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