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Lesson 1.1: Lines

Lesson 1.1: Lines. AP Calculus Mrs. Mongold. Definition of Increment. If coordinates change from (x 1 , y 1 ) to (x 2 , y 2 ) the increments in the coordinates are Δ x = x 2 – x 1 and Δ y = y 2 - y 1 where Δ x and Δ y are increments of x and y respectively.

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Lesson 1.1: Lines

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  1. Lesson 1.1: Lines AP Calculus Mrs. Mongold

  2. Definition of Increment • If coordinates change from (x1, y1) to (x2, y2) the increments in the coordinates are Δx = x2 – x1 and Δy = y2- y1 where Δx and Δy are increments of x and y respectively. • Increments can be positive, negative, or zero

  3. Finding Increments • 1. ( 4, -3) to (2, 5) • 2. (5, 6) to (5, 1)

  4. Slope • Let P1 (x1, y1) and P2 (x2, y2) be points on a non-vertical line, L. The slope of L is • Line going up hill from L to R has + slope • Line going down hill from L to R has – slope • Horizontal lines have zero slope • Vertical lines have no slope (undefined)

  5. Parallel and Perpendicular Lines • Parallel lines for equal angles with the x-axis…. i.e. the lines have the same slope • Perpendicular lines form 900 angles with the each other…. i.e. the lines have opposite reciprocal slopes

  6. Parallel Lines m1 m2 ɵ1 ɵ2

  7. Perpendicular Lines C m2 m1 h ɵ1 ɵ2 a D B A ADC is similar to CDB so ɵ1 is also the upper angle in CDB, where tan ɵ1 = a/h And m2 = tan ɵ2 = -h/a. Hence m1m2 = (a/h)(-h/a) = -1.

  8. Equations of Lines • Vertical Lines, x = a • Horizontal Lines, y = b • Point-slope Form y – y1 = m(x – x1) • Slope-Intercept Form, y = mx + b • General Linear Equation (Standard Form), Ax + By = C

  9. Write the equation of the line with the given information in point-slope form • 1. (2, 3) and slope -3/2 • 2. (-2, -1) and (3, 4)

  10. Write an equation for the line with the given information in slope-intercept form • 1.Through (-1, 2) so its parallel to y=3x – 4 • 2. Repeat for perpendicular line

  11. Use Standard Form to answer the following questions • If 8x + 5y = 20 find the slope and y-intercept

  12. Application Example #1 • Find relationship between Fahrenheit and Celsius temperature. Then find the Celsius equivalent of 900F and the Fahrenheit equivalent of -50 C. • F = mC + b • Freezing point for water is 320 F and 00 C, while boiling point of water is 2120 F and 1000C.

  13. Application Example #2 • Regression Analysis: • Build a linear model for the growth of the world population and then use the model to predict the world population in 2015. • Keep all decimals, rounding can seriously alter your data!!!

  14. Steps for Regression Analysis • 1. Plot Data (scatter plot) • 2. Find regression equation in y = mx+b • 3. Superimpose the graph of equation on scatter plot • 4. Use equation to predict y-values for particular x-values

  15. Homework • Pages 7-9/ 3-45 multiples of 3, 49, & 51

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