Any questions on the Section 3 3 homework Now please CLOSE YOUR LAPTOPS and turn off and put away your cell phones Sample Problems Page Link Dr Bruce Johnston Section 3 4 The Slope of a Line Slope of a line Informally slope is the tilt of a line It is the ratio of vertical change to horizontal change or rise y change y2 y1 m run x change x2 x1 x2 x1 Example Find the slope of the line through 4 3 and 2 2 If we let x1 y1 be 4 3 and x2 y2 be 2 2 then 2 3 5 m 2 4 2 Note If we let x1 y1 be 2 2 and x2 y2 be 4 3 then we get the same result 3 2 5 m 4 2 2 Given the graph of a line how do you find the slope Find 2 points on the graph then use those points in the slope formula Which points do you use 8 It s your choice but it s much easier if you pick points whose x and ycoordinates are both integers 6 4 2 2 2 10 5 5 10 2 4 0 4 6 8 Slope 4 2 6 3 3 0 2 2 1 Slope intercept form of a line y mx b has a slope of m and has a y intercept of 0 b This form is useful for graphing since you have a point and the slope readily visible Example Find the slope and y intercept of the line 3x y 5 First we need to solve the linear equation for y By adding 3x to both sides y 3x 5 Once we have the equation in the form of y mx b we can read the slope and y intercept slope is 3 y intercept is 0 5 Example Find the slope and y intercept of the line 2x 6y 12 First we need to solve the linear equation for y 6y 2x 12 subtract 2x from both sides 1 3 y x 2 divide both sides by 6 Since the equation is now in the form of y mx b slope is 1 3 y intercept is 0 2 Example from today s homework For any 2 points the y values will be equal to the same real number The numerator in the slope formula 0 the difference of the y coordinates but the denominator 0 two different points would have two different x coordinates So the slope 0 For any 2 points the x values will be equal to the same real number The denominator x2 x1 in the slope formula 0 so the slope is undefined since you can t divide by 0 Example from today s homework Summary of relationship between graphs of lines and slope If a line moves up as it moves from left to right the slope is positive If a line moves down as it moves from left to right the slope is negative Horizontal lines have a slope of 0 Vertical lines have undefined slope or no slope Two lines that never intersect are called parallel lines Parallel lines have the same slope unless they are vertical lines which have no defined slope Vertical lines are also parallel to each other even though their slope is undefined Two lines that intersect at right angles are called perpendicular lines Two nonvertical perpendicular lines have slopes that are negative reciprocals of each other The product of their slopes will be 1 Horizontal and vertical lines are perpendicular to each other Example Determine whether the following lines are parallel perpendicular or neither 5x y 6 and x 5y 5 First we need to solve both equations for y In the first equation y 5x 6 add 5x to both sides In the second equation 5y x 5 subtract x from both sides 1 y 5x 1 divide both sides by 5 The first equation has a slope of 5 and the second equation 1 has a slope of 5 so the lines are perpendicular Example Determine whether the following lines are parallel perpendicular or neither x 2y 2 and 2x 4y 3 In the first equation 2y x 2 add x to both sides 1 x 2 y 1 divide both sides by 2 In the second equation 4y 2x 3 subtract 3 from both sides 1 3 y 2x 4 divide both sides by 4 1 2 Both lines have a slope of so the lines are parallel Example from today s homework Web site that shows the effect of changing slope and y intercept http www mathsisfun com graph straight line graph html A quick way to check if you goofed on a negative sign Graph the two points and see if the slope should be positive or negative Watch out for this problem Hint To start this problem convert both distances to the same units Also make sure you type a 0 before decimals 1 e g type 0 08 not just 08 Practical application of slope to construction Web site that shows how to calculate the pitch of a roof http roofgenius com Roof Pitch xamples asp Pitch of a roof rise run Just make sure you measure both rise and run in the same units e g feet meters inches Problem Is it safe to walk on this roof Note You will be given sheet containing formulas to use on quizzes tests You can print off a copy of the formula sheet to use while you do your homework and practice quizzes tests by clicking on the Formula sheet menu button It is a good idea to print a copy of this sheet or keep a yellow copy that we ve handed out in class to use while you do your homework assignments and practice quizzes tests so you can get used to the location of the formulas and know which ones will be available to you on the sheet You will be given a clean copy of the sheet to use at each quiz and test so don t plan to rely on any notes you might write on your copy of the formula sheet Note The homework for this section shouldn t take too long so you might want to look ahead at section 3 5 on slope intercept equations of lines The homework for that section is a longer assignment so you might want to get a head start Reminder This homework assignment on Section 3 4 is due at the start of next class period You may now OPEN your LAPTOPS and begin working on the homework assignment