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Do Now: • What do you know about the following: – Slope – x-intercept, y-intercept – Linear equations – Calculus Lines Increments • Calculus has proven to be useful for relating the rate of change of a quantity to the graph of the quantity. • In order to begin explaining this relationship we must begin with the slopes of lines. Increments • When a particle in the plane moves from one point to another we must use the starting point and stopping point to discuss change. Increments • If a particle moves from point (x1,y1) to the point (x2,y2), the increments in its coordinates are: x = x2 – x 1 and y = y2 – y1 Increments Let a particle move from P1(x1, y1) to P2(x2, y2). The increments in its coordinates are x = (x2 – x1) and y= (y2 – y1). Ex. The coordinate increments from (4, -3) to (2, 5) are: Solution x = (x2 – x1) = 2 – 4 = -2 and y = (y2 – y1) = 5- -3 = 8. Ex. The coordinate increments from (5, 6) to (5, 1) are: Solution x = (x2 – x1) = 5 – 5 = 0 and y = (y2 – y1) = 1- 6 = -5. Slope • Each nonvertical line has a slope, and we can use increments to calculate our slope. Slope • Let P1(x1, y1) and P2(x2, y2) be points on a nonvertical line, L. The slope of L is rise y y2 y1 m run x x2 x1 Parallel and Perpendicular lines m 1 = m2 PARALLEL PERPENDICULAR Equations of vertical & horizontal lines Ex. Find the equations of the vertical and horizontal lines that pass through the point (2, 3) Solution (2, 3) Y=3 x=2 Point-slope Equation The equation y - y1 = m(x –x1) is the point slope formula with m = slope and (x1, y1) is a pt on line Ex. Find the equation of the line that passes through (-1, 2) and is a)Parallel to y = 3x – 4 b)Perpendicular to y = 3x - 4 Solution a)y – 2 = 3(x - -1) => y = 3x + 5 b) Y – 2 = (-1/3)*(x - -1) => y = -x/3 + 5/3 Slope-intercept Equation The equation y = mx + b is the slope-intercept formula with m = slope and b = y-intercept Ex. Find the equation of the line through (-1, 2) that passes through (0, 5). Solution m = (5 – 2)/(0 - –1) => m = 3. Since (0, 5) is the yintercept, we get y = 3x + 5. General Linear Equation The equation Ax + By = C (A and B not both 0) is the general linear equation. Sketch the line 8x + 5y = 40 Solution Substitute x = 0 => y = 8; substitute y = 0 => x = 5. Y 8 X X 5 X Linear Regression Using the best fitting line to predict future trends Ex Use a linear model of the data in the table to predict the population in the year 2010. Year Population (mil) 1986 4936 1987 5023 1988 5111 1989 5201 1990 5329 1991 5422 Method 1 Draw a scatter plot by hand and overlap the best fitting straight line. 5500 5400 5300 5200 5100 5000 Best fit 4900 1984 1986 1988 1990 1992 Method 1, continued Find the equation of the line and use its equation to predict population in 2010. Eqn: Use point slope formula. Use 2 pts on best line. (1986, 4936mil), (1990, 5300mil) m = (5300-4936)/(1990-1986) = 364/4 = 91 (mil/yr) y – 4936mil = 91mil(x – 1986) y = 91mil*x - 175790mil. In 2010, the population will be 91(2010) – 175790 Pop = 7120 mil <= INACCURATE Method 2 Use the capabilities of the TI-89 calculator. To simplify, let x = 0 represent 1986, x = 1 represent 1987 etc. {0, 1, 2, 3, 4, 5} -> L1 ENTER 2nd { 0, 1, 2, 3, 4, 5 STO (Upper case L used for clarity.) 2nd alpha } L 1 ENTER Method 2, continued {4936, 5023, 5111, 5201, 5329, 5422} -> L2 ENTER 2nd { 4936, 5023, 5111, 5201, 5329, 5422 STO alpha (Upper case L used for clarity.) 2nd } L 2 ENTER LinReg L1, L2 ENTER 2nd MATH 6 3 2 alpha LinReg Statistics Regressions L 1 , alpha L 2 The calculator should return: ENTER Done Method 2, continued ShowStat ENTER 2nd MATH 6 Statistics 8 ENTER ShowStat The calculator gives you an equation and constants: y = ax + b a = 98.228571, b = 4924.761905, corr = 0.997826 R2 = 0.995658 Method 2, continued Use calculator - plot new curve & the original points: Y= 2nd Plot 1 y1=regeq(x) VAR-LINK x ) regeq ENTER Use alpha etc. Type in L1 and L2 ENTER WINDOW Method 2, continued WINDOW Xmin = 0 Xmax = 5 Xsc = 1 Ymin = 4936 Ymax = 5422 Ysc = 1 Xres = 1 GRAPH produces the graph Method 2, continued Go to the homescreen 2nd QUIT To get the population in 2010, note that 2010 is 24 years after 1986. So we enter y1(24) at homescreen and obtain 7282.25 (mil). Compare this value with the value obtained using the “by hand” method p