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Graphing and Writing Equations of Lines (2-4) Objective: Graph linear equations and write the equations of lines in slopeintercept and point-slope form. Forms of Linear Equations O There are three main forms for the equation of a line O Slope-Intercept Form y = mx + b O m = slope O b = y-intercept O Standard Form Ax + By = C O A, B and C integers O A > 0, A and B not both 0 O Point-Slope Form y - y1 = m(x – x1) O m = slope O (x1, y1) is any point on the line Graphing Lines O From slope intercept form O Determine the y-intercept and plot it on the graph FIRST O Determine the slope and use it to find one or more additional points O Connect the points. Don’t forget to put arrows on the ends. Graphing Lines O Example: 2 Graph y x 5 3 O Graph the y-intercept of -5 O From the y-intercept, use the slope to rise 2 and run 3. This will give you a second point. Repeat. O Use a ruler to connect the points and create a line. y x . . . Graphing Lines O From Standard Form O Method 1: convert to slope-intercept form O Method 2: find x- and y-intercepts Graphing Lines O Graph 3x + 4y = 24 O x – intercept O 3x + 4(0) = 24 O y-intercept O 3(0) + 4y = 24 O 0 + 4y = 24 O y=6 . O 3x = 24 O x=8 y . 0 . x Graphing Lines O From point-slope form O Determine the point (x1, y1) O Graph the point O Use the slope to find additional points O Draw the line Graphing Lines O Graph y – 5 = -2(x + 1) O Rewrite as y – 5 = -2(x - -1) O The point is (-1, 5) O Graph the point . O The slope is -2/1, so rise -2 and run 1 y x Writing Equations of Lines O You must be given two pieces of information to write the equation of a line. This information can be O Slope and y-intercept O Slope and one point O Two points O If you are given the graph of a line, you can get that information from the graph. Writing Equations of Lines 2 y x 1 3 m rise 2 run 3 b = -1 b = -1 y = mx + b 2 -1 y = ___x + ___ 3 rise = -2 run = 3 Writing Equations of Lines O Write the equation of a line with slope -3 and one point at (-5, 2) O When given the slope and one point, it is O O O O O easiest to use the point-slope form y – y1 = m(x – x1) y – 2 = -3(x - -5) y – 2 = -3(x + 5) covert to slope-intercept y – 2 = -3x – 15 y = -3x - 13 Writing Equations of Lines O When given two points you need to: O Use the slope formula to determine the slope O Use the calculated slope and one of the two points to determine the equation of the line Writing Equations of Lines O Write the equation of the line passing through (-3, 5) and (6, -2). O Find the slope m y2 y1 x2 x1 2 5 6 3 7 9 O Use the slope and one of the two points to write the equation O y – y1 = m(x – x1) 7 (x 3) 9 O y 5 7 (x 3) 9 O y 5 Parallel and Perpendicular Lines O Parallel lines have the SAME slope O Perpendicular lines have slopes that are OPPOSITE RECIPROCALS Parallel and Perpendicular Lines O Find the equation of the line parallel to 3x + 5y = 7 and passing through (10, -2) O Find the slope of the original line: O From standard form, slope is –A/B O The slope of the line above is -3/5, so the slope of the parallel line is also -3/5 O Use the slope and the point to write the equation 3 O y 2 (x 10) 5 3 O y 2 (x 10) 5 Parallel and Perpendicular Lines O Find the equation of the line perpendicular to 8x – 2y = 9 and with y-intercept -4 O Find the slope of the line: -A/B = -8/-2 = 4 O The slope of the line perpendicular to the given line would be 1 4 O Use the slope and the y-intercept to write the equation of the line O y 1 x4 4