3.6 and 3.7 slopes of ll and Lines

Standard/Objectives: Standard 3: Students will learn and apply geometric concepts. Objectives: Find slopes of lines and use slope to identify parallel lines in a coordinate plane.  Write equations of parallel lines in a coordinate plane.

The slope of a nonvertical line is the ratio of vertical change (the rise) to the horizontal change (the run). The slope of a line is represented by the letter m. Just like any 2 points determine a line, any 2 points are all that are needed to determine the slope. The slope of the line is the same regardless of which 2 points are used

Slope of a line  In algebra, you learned that the slope of a nonvertical line is the ratio of the vertical change (rise) to the horizontal change (run).  If the line passes through the points (x 1, y 1 ) and (x 2, y 2 ), then the slope is given by slope = rise run m = y 2 – y 1 x 2 – x 1

Ex. 2: Finding Slope of a line  Find the slope of the line that passes throug the points (0,6) and (5, 2).  m = y 2 – y 1 x 2 – x 1 = 2 – 6 5 – 0 = - 4 5

Find the slope of a line passing thru (-3,5) & (2,1)

Classification of lines by slope  A line with positive slope: rises from left to right (m>0)  A line with negative slope: falls from left to right (m<0)  A line with slope of zero is horizontal (m=0)  A line with undefined slope is vertical (m is undefined)

Show slopes on board

Ex. 1: Finding the slope of train tracks  COG RAILWAY. A cog railway goes up the side of Mount Washington, the tallest mountain in New England. At the steepest section, the train goes up about 4 feet for each 10 feet it goes forward. What is the slope of this section? slope = rise = 4 feet =.4 run 10 feet

W/out graphing, tell whether the line thru the points rises, falls, horizontal, or vertical.  1) (3,-4), (1,-6)  2) (2,-1), (2,5) Undefined: The line is vertical m>0, the line rises

 Not only does slope tell you whether the line rises, falls, is horizontal, or is vertical; it also tells you the steepness of the line

Tell which line is steeper  Line 1 thru (2,3) & (4,7)  or Line 2 thru (-1,2) & (4,5) Since the lines have positive slope & m 1 >m 2 line 1 is steeper

 Two lines are parallel if they don’t intersect.  Two lines are perpendicular if they intersect to form a right angle.

Postulate 17 Slopes of Parallel Lines  In a coordinate plane, two non-vertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel. k1k1 k2k2 Lines k 1 and k 2 have the same slope.

Ex. 3 Deciding whether lines are parallel  Find the slope of each line. Is j 1 ║j 2 ? M 1 = 4 = 2 2 M 2 = 2 = 2 1 Because the lines have the same slope, j 1 ║j 2.

Ex. 1: Deciding whether lines are perpendicular  Find each slope.  Slope of j =  Slope of j 2 3-(-3) = 6 = 3 0-(-4) 4 2 Multiply the two slopes. The product of -2 ∙ 3 = -1, so j 1  j 2 3 2

Ex.2 Deciding whether lines are perpendicular  Decide whether AC and DB are perpendicular.  Solution:  Slope of AC= 2-(-4) = 6 = 2 4 – 1 3  Slope of DB= 2-(-1) = 3 = 1 -1 – The product of 2(-1/2) = -1; so AC  DB

Ex.4: Deciding whether lines are perpendicular  Line r: 4x+5y=2 4x + 5y = 2 5y = -4x + 2 y = -4/5 x + 2/5 Slope of line r is -4/5  Line s: 5x + 4y = 3 5x + 4y = 3 4y = -5x + 3 y = -5/4 x + 3/5 Slope of line s is -5/4 -4 ∙ -5 = The product of the slopes is NOT -1; so r and s are NOT perpendicular. Or remember not (flipped and opposite) If Negative reciprocals of each other, then they are ┴.

Tell whether the lines are װ, ┴, or neither  L 1 : thru (-3,3) & (3,-1)  L 2 : thru (-2,-3) & (2,3) Negative reciprocals Of each other : They are ┴.

Ex. 4 Identifying Parallel Lines M 1 = 0-6 = -6 = M 2 = 1-6 = -5 = -5 0-(-2) M 3 = 0-5 = -5 = (-6) k3k3 k2k2 k1k1

Solution: Compare the slopes. Because k 2 and k 3 have the same slope, they are parallel. Line k 1 has a different slope, so it is not parallel to either of the other lines.

Writing Equations of parallel lines  In algebra, you learned that you can use the slope m of a non-vertical line to write an equation of the line in slope-intercept form. slopey-intercept y = mx + b The y-intercept is the y-coordinate of the point where the line crosses the y-axis.

Determine rather parallel or perpendicular or neither  Y = 5X-3 and Y= 1/5X -3  Y = 2X -4 and Y = 2X +4  Y = 2/3X – 5 and Y= - 3/2X +6  Y = 3 and X = 5  Y = 5 and Y = - 6  Y = 3X and Y = - 1/3X - 5

Write the equation  If m = - 1/3 and b = 3, an equation for this line is y = -1/3x + 3  Using Y=mX + b the slope intercept form

Ex. 5: Writing an Equation of a Line  Write an equation of the line through the point (2, 3) that has a slope of 5. y = mx + b 3 = 5(2) + b 3 = 10 + b -7= b  Steps/Reasons why Slope-Intercept form Substitute 2 for x, 3 for y and 5 for m Simplify Subtract.

Given the slope, m, and the y-intercept, b, use the equation y=mx+b The y-intercept is -3 b=-3 The slope is 4/3 The equation is: y = 4/3x – 3 4 3

Use Point Slope Form: If you are given slope, m, and a point (x 1,y 1 ) on the line  y – y 1 = m ( x – x 1 )

Write an equation of a line containing the point (1,2) with slope of -1/2.  Use (x 1,y 1 ) = (1,2) & m = -1/2  y – 2 = -1/2 ( x – 1)  Now you can simplify to the slope intercept form  y – 2 = -1/2 x + ½  y = -1/2 x + 5/2

If you are given two points on the line  Find the slope using the two points  Then plug this slope and either one of the points into the point slope formula.

Given two points (-2,2) & (3,7)  Find the slope:  m=1  Plug this slope and one of the points into the point slope formula.  y – 2 = 1 ( x – (-2))  y – 2 = x + 2  y = x + 4 (put the equation into slope intercept form) (-2,2) (3,7)

Graph y = 3x

You Try Use the board and marker  Graph Y = -2/3 X + 1  Graph Y = 2  Graph X = -3  Graph Y = ½ X + 1 and Y = -2 X – 3 Determine rather they are ll or perpendicular or neither.