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3.3 SLOPE AND RATE OF CHANGE Geometry R/H What does each of the following look like? Positive Slope Negative Slope Zero Slope Undefined Slope When given 2 points (x1, y1) and (x2, y2) plug them into our slope formula: y2 y1 m x2 x1 Ex1: (4,3)and (2,5) From a graph! Find the slope of the line. Blue Line: Red Line: Rate of Change COLLEGE ADMISSIONS In 2004, 56,878 students applied to UCLA. In 2006, 60,291 students applied. Find the rate of change in the number of students applying for admission from 2004 to 2006. X – independent variable Y – Dependent variable Let's try One More • Find the rate of change for the data in the table. Pairs of Lines • Given two lines in the coordinate plane, they can intersect, coincide, or be parallel. y = 5x + 8 y = 2x – 5 y = 5x – 4 y = 4x + 3 Parallel lines Intersecting Same slope Different y-intercept Different Slopes y = 2x – 4 y = 2x – 4 Coinciding Same slope Same y-intercept We can use this information to compare lines and then classify them. Perpendicular Lines • With intersecting lines, we can also tell if they are perpendicular or not. Example: y = 2x + 4 and y = -½x - 3 Perpendicular lines The product of the slopes is -1 The y-intercepts can be any number. More Parallel and Perpendicular lines Determine whether AB and CD are parallel, perpendicular, or neither for the given set of points. Ex 1: A(1, -3) B(-2, -1) C(5, 0) and D(6, 3) Ex 2: A(3, 6) B(-9, 2) C(5, 4) and D(2, 3) Using Slope to Solve Problems Determine whether triangle ABC is a right triangle. y Slope of AB is: 2 3 Slope of AC is: 3 2 B A 0 Since the slopes are C the opposite reciprocals of each other, the line segments are perpendicular and the triangle is a right triangle. x Classifying Pairs of Lines Determine whether the lines are parallel, intersect, coincide, or are perpendicular. 2y – 4x = 16, y – 10 = 2(x - 1) GRAPHING Graphing Equations Determine which two lines are parallel and then graph the lines. y 1 y x 2 3 y 3x 1 1 y x 1 3 0 x Graphing Equations of Lines Graph this equation: y 2 y x2 3 Step 1: Graph the yintercept Step 2: Use the slope to find the next point. Step 3: Draw a line through the two points. 0 x Graphing Equations Graph this line: 2 y 1 x 2 3 The equation is given in the point-slope form, with 2 3 a slope of through the point (–2, 1). Plot the point (–2, 1) and then rise –2 and run 3 to find y another point. Draw the line containing the points. 0 x WRITING EQUATIONS OF LINES The Forms of Equations of Lines • The equation of a line can be written in several different forms. • The standard form is Ax + By = C • The slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. • The point-slope form is y – y1 = m(x – x1), where m is the slope and ( x1, y1) is a point on the line. The Forms of Equations of Lines • Some special cases are the following: • The equation of a vertical line is x = a, where a is the x- intercept. The slope is undefined. • The equation of a horizontal line is the y-intercept. The slope is zero. y = b, where b is Writing Equations in the SlopeIntercept Form: y = mx +b 1. Given a slope of 0 and a y-intercept at (0, 5). 2. Given an undefined slope and passing through (3, 0). 3. Given a slope of -4 and through the point (5, 26). Given a Slope and a Point Write the equation of the line with slope 6 through (3, –4) in slope-intercept form y y1 m x x1 Start with Point-slope form y 4 6 x 3 Substitute 6 for m, 3 for x 1, and -4 for y 1. y 4 6 x 18 y 6 x 22 Simplify Convert to slope-intercept form. Write equation in point-slope form. • (4, 7) and (6, 13) • (2, -1) and (5, -13) • (-8, 5) and (-5, 5) Writing equations of parallel/Perpendicular Lines • Original Line (-6, 12) and (3, 6) • Parallel through (12, -12) • Perpendicular through (6, 14) Parallel/Perpendicular Ex 2 • Original line: (5, 27) and (9, 47) • Parallel through (8, 36) • Perpendicular through (-20, 11) MIDPOINT AND EQUATION OF A PERPENDICULAR BISECTOR Review of Midpoint Formula • You can find the midpoint of a segment by using the coordinates of its endpoints. • The midpoint of the segment joining the points A(x1, y1) and B(x2, y2) has these coordinates: x1 x2 y1 y2 , 2 2 Example: Find the midpoint of A (-1, 4) & B (3, 5). 1 2 9 1 3 4 5 , 2 , 2 1, 4 2 2 2 Example 2 • S is the midpoint of RT . R has coordinates (-6, -1), and S has coordinates (-1, 1). Find the coordinates of T. Step 1: Let the coordinates of T equal (x, y). Step 2: Use the Midpoint Formula: 1, 1 6 x 1 y , 2 2 T 4, 3 Step 3: Find the x- and y-coordinate: 6 x 1 x4 2 1 y 1 y 3 2 Find equation of the perpendicular bisector • Write the equation for the perpendicular bisector of segment AB . • Step 1: Find the midpoint of AB. 04 53 4 8 2 , 2 2,2 2,4 6 A 4 B 2 5 Step 2: Find the slope of AB . y 2 y1 3 5 2 1 x 2 x1 4 0 4 2 Find equation of the perpendicular bisector • Step 3: Find the slope of perpendicular bisector. • The slope of AB is -½, so the slope of ⊥ bisector is 2. (opposite reciprocal) • Step 4: Use (2, 4) and slope of 2 to find equation of line. 6 A 4 B 2 Use the point slope form: y 4 2(x 2) y y1 m(x x1 ) y 2x 5 Now You Try! Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints A(7, 9) and B(–3, 5) . Step 1: Find the midpoint of AB Step 2: Find the slope of AB Step 3: Find the slope of the perpendicular line. Step 4: Find the equation of the perpendicular bisector. Median of a Triangle • A median of a triangle is a segment whose end-points are a vertex of the triangle and the mid-point of the opposite side. C • What would you need to find an equation of a median? A D • Every triangle has three medians – one from each vertex. B Altitude of a Triangle • An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. • Every triangle has three altitudes – one from each vertex • An altitude can be inside, outside, or on the triangle. A B D C Examples of Altitudes • These examples of altitudes show how the altitude can be inside the triangles, outside the triangle or on (one of the sides) of a triangle. • What would you need to find an equation of an altitude? a a a