SlopeIntercept Equation. Example


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1 1.4 Equations of Lines and Modeling Find the slope and the y intercept of a line given the equation y = mx + b, or f(x) = mx + b. Graph a linear equation using the slope and the yintercept. Determine equations of lines. Given the equations of two lines, determine whether their graphs are parallel or whether they are perpendicular. Model a set of data with a linear function. Fit a regression line to a set of data; then use the linear model to make predictions. SlopeIntercept Equation The linear function f given by f(x) = mx + b has a graph that is a straight line parallel to y = mx. The constant m is called the slope, and the yintercept is (0, b). f(x) = mx + b (0, b) y = mx (0, 0) Slide 111 Find the slope and yintercept of the line with equation y = 0.6x Solution: y = 0.6x Slope = 0.6; yintercept = (0, 4.2) Slide 112
2 Find the slope and yintercept of the line with equation 4x + y 12 = 0. 4x + y 12= 0 Solution: We solve for y: y= 4x+ 12 ( y) = ( 4x+ 12) y= x+ 4 Thus, the slope is 4 and the yintercept is (0, 4). Slide 11 A line has slope and contains the point ( 2, 5). Find an equation of the line. Solution: We use the slopeintercept equation, y = mx + b, and substitute for m. y = x + b. Using the point ( 2, 5), we substitute for x and y and solve for b. 5 = ( 2) + b 5 = 6 + b 11 = b The equation of the line is y = x Slide Graph y = x 2 2 Solution: The equation is in slopeintercept form, y = mx + b. The yintercept is (0, 2). rise change in y 1 move 1 unit up m = = = run change in x 2 move 2 units right Slide 115
3 PointSlope Equation The pointslope equation of the line with slope m passing through (x 1, y 1 ) is y y 1 = m(x x 1 ). : Find the equation of the line containing the points (2, 7) and ( 1, 8). 8 7 m = 1 2 Solution: First determine the slope: 15 = Using the pointslope equation, = 5 substitute 5 for m and either of the points for (x 1, y 1 ): y y1 = m( x x1) y 7 = 5( x 2) y 7= 5x 10 y= 5x Slide 116 Parallel Lines Vertical lines are parallel. Nonvertical lines are parallel if and only if they have the same slope and different yintercepts. y = x + 2 y = x 4 Slide 117 Perpendicular Lines Two lines with slopes m 1 and m 2 are perpendicular if and only if the product of their slopes is 1: m 1 m 2 = 1. 1 y = x+ 2 y = x 4 Lines are also perpendicular if one is vertical (x = a) and the other is horizontal (y = b). Slide 118
4 s Determine whether each of the following pairs of lines is parallel, perpendicular, or neither. a) y 4x =, 4y 8 = x (perpendicular) b) 2x + y = 4, x + 2y = 8 (neither) c) 2y = 4x + 12, y 8 = 2x (parallel) Slide 119 Write equations of the lines (a) parallel and (b) perpendicular to the graph of the line y + 4 = 18x and containing the point (1, 2). Solve the equation for y: y+ 4= 18x y = 18x 4 y = 6x 4 (a) The line parallel to the given line will have the same slope. We use either the slopeintercept or pointslope equation for the line. Slide 120 continued Substitute and solve the equation. y y = m( x x ) 1 1 y+ 2= 6( x 1) y+ 2= 6x 6 y = 6x 8 Slide 121
5 continued (b) For a line perpendicular: m = 1 6 y y1 = m( x x1) 1 y+ 2 = ( x 1) y+ 2 = x y= x 6 6 Slide 122 Curve Fitting In general, we try to find a function that fits, as well as possible, observations (data), theoretical reasoning, and common sense. : Model the data given in the table on foreign travel on the next slide with two different linear functions. Then with each function, predict the number of U.S. travelers to foreign countries in Of the two models, which appears to be the better fit? Slide 12 Curve Fitting continued Model I: Choose any two points to determine the equation m = = y 5.08= ( x 1) y = x Predict the number of travelers: y = x y = ( 12) y = 6.92 Slide 124
6 Curve Fitting continued Model II: Choose any two points to determine the equation m = y = 0.28x Predict the number of travelers: y = 0.28x y = 0.28( 12) y = Slide 125 Curve Fitting continued Using model I, we predict that there will be about 6.92 million U.S. foreign travelers in 2006, and using model II, we predict about 7.51 million. Since it appears from the graphs that model II fits the data more closely, we would choose model II over model I. Slide 126 Textbook Problems Section 1.4: #11, 1, 19, 2, 7, 41, 5, 55, 61, 67 Slide 127
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