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Understanding Point-Slope Equation in Mathematics

Learn how to derive and use the point-slope equation for lines, solve examples, graph lines, determine parallel and perpendicular lines, and solve problems using point-slope form. Understand the concept with clear examples and solutions.

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Understanding Point-Slope Equation in Mathematics

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  1. Chabot Mathematics §2.5 Line EqnPoint-Slope Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  2. MTH 55 2.4 Review § • Any QUESTIONS About • §’s2.4 → Slope-Intercept Eqn, Modeling • Any QUESTIONS About HomeWork • §’s2.4 → HW-06

  3. The Point-Slope Equation • The equation y−y1 = m(x−x1) is called the point-slope equation for the line with slope m that contains the point (x1,y1). • Note that (x1,y1) is a KNOWN point • e.g.; (x1,y1) = (−7,11) • Sometimes (x1,y1) is called the ANCHOR Point

  4. Point-Slope Derivation • Suppose that a line through (x1, y1) has slope m. Every other point (x, y) on the line must satisfy the equation • Because any two points can be used to find the slope. Multiply both sides by (x −x1) yielding: • which is the point-slope form of the equation of the line.

  5. Find m & b for the line through (−1, −1) and (3, 4). Find m by y-chg divided by x-chg Use Pt-Slope Eqn and Solve for y to reveal b Example  Point-Slope Eqn • Last Line to shows both m & b m b

  6. Example  Point-Slope • Write a point-slope equation for the line with slope 2/3 that contains the point (4, 9) • SOLUTION: Substitute 2/3 for m, and 4 for x1, and 9 for y1 in the Pt-Slope Eqn:

  7. Example  Pt-Slope → Slp-Inter • Write the slope-intercept equation for the line with slope 3 and point (4, 3) • SOLUTION: There are two parts to this solution. First, write an equation in point-slope form:

  8. Example  Pt-Slope → Pt-Inter • slope 3 & point (4, 3) → y = mx + b • SOLUTION: Next, we find an equivalent equation of the form y = mx + b: By Distributive Law Add 3 to Both Sides to yield Slope-Intercept Line:y = mx + b = 3x + (−9)

  9. Graphing and Point-Slope Form When we know a line’s slope and a point that is on the line (i.e., an ANCHOR Point), we can draw the graph.

  10. SOLUTION: Since y− 3 = 2(x − 1) is in point-slope form, we know that the line has slope 2 and passes through the point (1, 3). We plot (1, 3) and then find a second point by moving up 2 units and to the right 1 unit. Example  Graph y− 3 = 2(x− 1) right 1 up 3 (1, 3)AnchorPoint • Connect the Dots to Draw the Line

  11. Example  Graph y+3 = (−4/3)(x+2)

  12. SOLUTION: Find an equivalent equation: Example  Graph y+3 = (−4/3)(x+2) (2, 3) down 4 • The line passes through Anchor-Pt (−2, −3) and has slope of −4/3 right 3

  13. Parallel and Perpendicular Lines • Two lines are parallel (||) if they lie in the same plane and do not intersect no matter how far they are extended. • Two lines are perpendicular (┴) if they intersect at a right angle (i.e., 90°). E.g., if one line is vertical and another is horizontal, then they are perpendicular.

  14. Para & Perp Lines Described • Let L1 and L2 be two distinct lines with slopes m1 and m2, respectively. Then • L1 is parallel to L2 if and only if m1 = m2and b1≠b2 • If m1 = m2. and b1 = b2 then the Lines are CoIncident • L1 is perpendicular L2 to if and only if m1•m2 = −1. • Any two Vertical or Horizontal lines are parallel • ANY horizontal line is perpendicular to ANY vertical line

  15. Parallel Lines by Slope-Intercept • Slope-intercept form allows us to quickly determine the slope of a line by simply inspecting, or looking at, its equation. • This can be especially helpful when attempting to decide whether two lines are parallel • These Lines All Have the SAME Slope

  16. Example  Parallel Lines • Determine whether the graphs of the lines y = −2x− 3 and 8x + 4y = −6 are parallel. • SOLUTION • Solve General Equation for y • Thus the Eqns are • y = −2x− 3 • y = −2x− 3/2

  17. Example  Parallel Lines • The Eqns y = −2x− 3 & y = −2x− 3/2 show that • m1=m2 = −2 • −3 = b1≠b2 = −3/2 • Thus the LinesARE Parallel • The Graph confirmsthe Parallelism

  18. Example  ║& ┴ Lines • Find equations in general form for the lines that pass through the point (4, 5) and are (a) parallel to & (b) perpendicular to the line 2x − 3y + 4 = 0 • SOLUTION • Find the Slope by ReStating the Line Eqn in Slope-Intercept Form

  19. Example  ║& ┴ Lines • SOLUTION cont. • Thus Any line parallelto the given line must have a slope of 2/3 • Now use the GivenPoint, (4,5) in thePt-Slope Line Eqn • Thus ║- Line Eqn

  20. Example  ║& ┴ Lines • SOLUTION cont. • Any line perpendicularto the given line must have a slope of −3/2 • Now use the GivenPoint, (4,5) in thePt-Slope Line Eqn • Thus ┴ Line Eqn

  21. Example  ║& ┴ Lines • SOLUTION Graphically

  22. Estimates & Predictions • It is possible to use line graphs to estimate real-life quantities that are not already known. To do so, we calculate the coordinates of an unknown point by using two points with known coordinates. • When the unknown point is located BETWEEN the two points, this process is called interpolation. • Sometimes a graph passing through the known points is EXTENDED to predict future values. Making predictions in this manner is called extrapolation

  23. Example  Aerobic Exercise • A person’s target heart rate is the number of beats per minute that bring the most aerobic benefit to his or her heart. The target heart rate for a 20-year-old is 150 beats per minute (bpm) and for a 60-year-old, 120 bpm • Graph the given data and calculate the target heart rate for a 46-year-old • Calculate the target heart rate for a 70-year-old

  24. Solution a) We draw the axes and label, using a scale that will permit us to view both the given and the desired data. The given information allows us to then plot (20, 150) and (60, 120) Example  Aerobic Exercise

  25. Example  Aerobic Exercise • Find the Slope of the Line • Use One Point to Write Line Equation

  26. Solution a) To calculate the target heart rate for a 46-year-old, we sub 46 for x in the slope-intercept equation: Example  Aerobic Exercise 130 46 • The graph confirms the target heart rate INterpolation

  27. Solution b) To calculate the target heart rate for a 70-year-old, we substitute 70 for x in the slope-intercept equation: Example  Aerobic Exercise 112 70 • The graph confirms the target heart rate EXtrapolation

  28. WhiteBoard Work • Problems From §2.4 Exercise Set • PPT-example 14, 22, 26, 40, 52, 62 • The LineEquations

  29. Given Column Chart DropOut Rates  Scatter Plot • Read Chart to Construct T-table • Use T-table to Make Scatter Plot on the next Slide

  30. Zoom-in to more accurately calc the Slope

  31. Intercept  15.2% (x1,y1) = (8yr, 14%) “Best” Line(EyeBalled)

  32. Calc Slope from Scatter Plot Measurements DropOut Rates  Scatter Plot • Thus the Linear Model for the Data in SLOPE-INTER Form • To Find Pt-Slp Form use Known-Pt from Scatter Plot • (x1,y1) = (8yr, 14%) • Read Intercept from Measurement

  33. Thus the Linear Model for the Data in PT-SLOPE Form DropOut Rates  Scatter Plot • X for 2010 → x = 2010 − 1970 = 40 • In Equation • Now use Slp-Inter Eqn to Extrapolate to DropOut-% in 2010 • The model Predicts a DropOut Rate of 9.2% in 2010

  34. 9.2%

  35. All Done for Today CommunityCollegeEnrollmentRates

  36. Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –

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