2 Graphs and Functions Copyright 2013 2009 2005

2 Graphs and Functions Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1

2. 5 Equations of Lines; Curve Fitting • Point-Slope Form • Slope-Intercept Form • Vertical and Horizontal Lines • Parallel and Perpendicular Lines • Modeling Data • Solving Linear Equations in One Variable by Graphing Copyright © 2013, 2009, 2005 Pearson Education, Inc. 2

Point-Slope Form The point–slope form of the equation of the line with slope m passing through the point (x 1, y 1) is Copyright © 2013, 2009, 2005 Pearson Education, Inc. 3

Example 1 USING THE POINT-SLOPE FORM (GIVEN A POINT AND THE SLOPE) Write an equation of the line through (– 4, 1) having slope – 3. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 4

Example 2 USING THE POINT-SLOPE FORM (GIVEN TWO POINTS) Write an equation of the line through (– 3, 2) and (2, – 4). Write the result in standard form Ax + By = C. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 5

Slope-Intercept Form As a special case, suppose that a line passes through the point (0, b), so the line has y-intercept b. If the line has slope m, then using the pointslope form with x 1 = 0 and y 1 = b gives the following. Slope Copyright © 2013, 2009, 2005 Pearson Education, Inc. y-intercept 6

Slope-Intercept Form The slope-intercept form of the equation of the line with slope m and y-intercept b is Copyright © 2013, 2009, 2005 Pearson Education, Inc. 7

Example 3 FINDING THE SLOPE AND y-INTERCEPT FROM AN EQUATION OF A LINE Find the slope and y-intercept of the line with equation 4 x + 5 y = – 10. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 8

Example 4 USING THE SLOPE-INTERCEPT FORM (GIVEN TWO POINTS) Write an equation of a line through (1, 1) and (2, 4). Then graph the line using the slopeintercept form. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 9

Example 5 FINDING AN EQUATION FROM A GRAPH Use the graph of the linear function shown to complete the following. (a) Find the slope, y-intercept, and x-intercept. – 3 – 1 y = (x) Copyright © 2013, 2009, 2005 Pearson Education, Inc. 10

Example 5 FINDING AN EQUATION FROM A GRAPH Use the graph of the linear function shown to complete the following. (b) Write the equation that defines . – 3 – 1 y = (x) Copyright © 2013, 2009, 2005 Pearson Education, Inc. 11

Equations of Vertical and Horizontal lines An equation of the vertical line through the point (a, b) is x = a. An equation of the horizontal line through the point (a, b) is y = b. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 12

Parallel Lines Two distinct nonvertical lines are parallel if and only if they have the same slope. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 13

Perpendicular Lines Two lines, neither of which is vertical, are perpendicular if and only if their slopes have a product of – 1. Thus, the slopes of perpendicular lines, neither of which are vertical, are negative reciprocals. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 14

Example 6 FINDING EQUATIONS OF PARALLEL AND PERPENDICULAR LINES Write the equation in both slope-intercept and standard form of the line that passes through the point (3, 5) and satisfies the given condition. (a) parallel to the line 2 x + 5 y = 4 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 15

Example 6 FINDING EQUATIONS OF PARALLEL AND PERPENDICULAR LINES Write the equation in both slope-intercept and standard form of the line that passes through the point (3, 5) and satisfies the given condition. (b) perpendicular to the line 2 x + 5 y = 4 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 16

Equation y = mx +b Description Slope-Intercept Form Slope is m. y-intercept is b. y – y 1 = m(x – x 1) Point-Slope Form Slope is m. Line passes through (x 1, y 1) When to Use Slope and y-intercept easily identified and used to quickly graph the equation. Also used to find the equation of a line given a point and the slope. Ideal for finding the equation of a line if the slope and a point on the line or two points on the line are known. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 17

Equation Ax + By = C Description Standard Form (If the coefficients and constant are rational, then A, B, and C are expressed as relatively prime integers, with A ≥ 0). When to Use The x- and yintercepts can be found quickly and used to graph the equation. The slope must be calculated. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 18

Equation Description When to Use y=b Horizontal Line Slope is 0. y-intercept is b. If the graph intersects only the y-axis, then y is the only variable in the equation. x=a Vertical Line Slope is undefined. x-intercept is a. If the graph intersects only the x-axis, then x is the only variable in the equation. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 19

Example 7 FINDING AN EQUATION OF A LINE THAT MODELS DATA Average annual tuition and fees for in-state students at public four-year colleges are shown in the table for selected years and graphed as ordered pairs of points where x = 0 represents 2005, x = 1 represents 2006, and so on, and y represents the cost in dollars. This graph of ordered pairs of data is called a scatter diagram. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 20

Example 7 Year 2005 2006 2007 2008 2009 2010 FINDING AN EQUATION OF A LINE THAT MODELS DATA Cost (in dollars) 5492 5804 6191 6591 7050 7605 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 21

FINDING AN EQUATION OF A LINE THAT MODELS DATA (a) Find an equation that models the data. Example 7 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 22

FINDING AN EQUATION OF A LINE THAT MODELS DATA (b) Use the equation from part (a) to predict the cost of tuition and fees at public 4 -year colleges in 2012. Example 7 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 23

Guidelines for Modeling Step 1 Make a scatter diagram of the data. Step 2 Find an equation that models the data. For a line, this involves selecting two data points and finding the equation of the line through them. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 24