SLOPE of a LINE Objective To determine the

SLOPE of a LINE Objective: To determine the slope of a line given a table, graph or two points on the line

What is different about these three roofs?

Rate of Change The rate of change is a ratio that compares, on average, how much one quantity changes in relation to another. If x is the independent variable, and y is the dependent variable, then the rate of change is or

Rate of Change The rate of change is often not constant for an entire set of data. In this case, we often look at the average rate of change over an interval.

Rate of Change The graph below shows data on the fastest growing restaurant chain the U. S. from 2001 -2006. Find the rate of change in the number of stores during this time. Δy = change in stores Δx = change in time The chain grew at a rate of 5. 4 stores per year

Slope of a Line � The slope of a line is a measure of the steepness of the line. � The letter “m” is often used to represent slope � Slope is also called: ◦ rate of change or ◦ ◦ m=

Finding Slope from a Table Find the slope of each line given a table of points on the line Find the change in x change = 2 Slope (m) = x y 2 4 4 5 6 6 change = 1 8 7 10 8 change = 1 = Find the change in y change = 1

Finding Slope from a Table Find the slope of each line given a table of points on the line Find the change in x change = 3 change = 6 change = 9 Slope (m) = x y -5 5 -2 7 1 9 change = 2 7 13 16 19 change = 4 Find the change in y change = 2 change = 6 = In each case, the fraction will reduce to the same number.

Finding Slope from a Table Find the slope of each line given a table of points on the line Find the change in x change = 2 change = 4 change = 2 change = 6 Slope (m) = x y 1 10 3 4 7 -8 9 -14 15 -32 = Find the change in y change = -6 change = -12 change = -6 change = -18 = -3

Finding Slope from a Graph To find the slope from a graph, you will need to: Pick two points on the line Count the rise (the vertical distance) Count the run (the horizontal distance) Write the slope in the form m =

Finding Slope from a Graph Find the slope of the line below: run = 5 rise = 3

Finding Slope from a Graph rise = -9 run = 6

Finding Slope Given Two Poins To find the slope given two points Label the points as (x 1, y 1) and (x 2, y 2) Use the formula m =

Finding Slope Given Two Points Find the slope of the line containing the points (5, 9) and (-2, 3) x 1 y 1 x 2 y 2

Finding Slope Given Two Points If the slope of a line is and the line contains the points (a, -2) and (7, 3), what is the value of x 1 y 1 a? m= x 2 y 2 -5(7 – a) = 4(5) -35 + 5 a = 20 +35 5 a = 55 a = 11

Special Slopes Horizontal lines ◦ A horizontal line is FLAT (like the horizon) ◦ The slope of any horizontal line is 0 (because it has no steepness) ◦ If you do for a horizontal line, the rise will be zero and anything divided by zero is zero Vertical lines ◦ A vertical line is straight up-and-down ◦ The slope of any vertical line is UNDEFINED (it is too steep to give it a number) ◦ If you do for a vertical line, the run is zero, and dividing by zero is not allowed

Special Slopes Find the slope of the line containing the points (4, 7) and (-2, 7) x 1 y 1 x 2 y 2 The slope of this line is 0, so it is a HORIZONTAL line

Special Slopes Find the slope of the line containing the points (8, 2) and (8, -6) x 1 y 1 x 2 y 2 The slope of this line is undefined, so it is a VERTICAL line

Positive and Negative Slopes A line with a POSITIVE slope goes UP from left to right A line with a NEGATIVE slope goes DOWN from left to right

Slopes Determine if the slope of each line is positive, negative, zero or undefined negative undefined zero positive negative

Applications of Slope The Americans with Disabilities Act (ADA) states that wheelchair ramps can have a rise of no more than 1” for every 12” of run. If you need for the ramp to be 15” high, how long should the ramp be? 1 x = 180 Set up a proportion comparing length to height. The ramp should be 180” long.

Applications of Slope If you only have 6 feet available for your wheelchair ramp, what is the maximum height you can make the ramp? The code states the rise and run in INCHES so 6 feet is 6 x 12 or 72” Solve by cross multiplying 12 x = 1(72) 12 x = 72 x=6 The ramp can be at most 6” or ½ a foot high.

Applications of Slope In order to get proper drainage from the water pipes in your home, plumbing code tells the plumber what the minimum slope has to be for different size drainage pipes. The table below shows the minimum slope for different sizes of pipes. Table 704. 1 Slope of Horizontal Drainage Pipe Size (inches) Minimum Slope (inch per foot) 2 ½ or less ¼ 3 to 6 1/8 8 or larger 1/16

Applications of Slope A plumber using a 2” drainage pipe needs for the pipe to fall 4”. How many feet of pipe does the plumber need to use? The code table tells us that for a 2” pipe, the minimum slope is ¼. 1 x = 4(4) x = 16 Because the code table tells us that the slope is in inches per FOOT, the plumber will need to use 16 feet of pipe.

Applications of Slope The plumber from the previous problem decides to use 4” pipe instead of 2” pipe. How long does the 4” pipe need to be? The code table tells us that for a 4” pipe, the minimum slope is 1/8. The plumber will need to use 32 feet of pipe. 1 x = 4(8) x = 32