Unit 3: Visualizing the Change Section 5: Starting from Over There Algebra 1Predicting Patterns & Examining Experiments This lesson we will be examining the point-slope form. Unit 3: Visualizing the Change Section 5: Starting from Over There

Previously: Two trips Solution Plot this information on the same set of axes and draw two lines connecting the points in each set of data. What is the slope of each line? What is the meaning of these slopes in the context of this problem? What is the equation of each line? At what times will each have driven 600 miles? (Teacher Lecture): First, we revisit the Two Trips problem from lesson 3-2, where (without much mention) we added the column “hours after 8:00”. Here we are going to explain why and what other options we have. time hours after 8:00 car distance 8:00 9:00 1 52 10:00 2 104 11:00 3 156 12:00 4 208 time hours after 8:00 truck distance 8:00 9:00 1 46 10:00 2 92 11:00 3 138 12:00 4 184 0 1 2 3 4 1 hr 52 miles 1 hr 46 miles slope = 52/1 mph = 52 mph slope = 46/1 mph = 46 mph 600 miles 600 = 52x 52 52 11.5 ≈ x 600 miles 600 = 46x 46 46 13 ≈ x m = 52 b = 0 y = 52x m = 46 b = 0 y = 46x The car will have driven 600 miles at 7:30 pm. The truck will have driven 600 miles at 9:00 pm.

Previously: Two trips Solution Plot this information on the same set of axes and draw two lines connecting the points in each set of data. What is the slope of each line? What is the meaning of these slopes in the context of this problem? What is the equation of each line? At what times will each have driven 600 miles? transition slide time hours after 8:00 car distance 8:00 9:00 1 52 10:00 2 104 11:00 3 156 12:00 4 208 time hours after 8:00 truck distance 8:00 9:00 1 46 10:00 2 92 11:00 3 138 12:00 4 184 0 1 2 3 4 1 hr 52 miles 1 hr 46 miles slope = 52/1 mph = 52 mph slope = 46/1 mph = 46 mph 600 miles 600 = 52x 52 52 11.5 ≈ x 600 miles 600 = 46x 46 46 13 ≈ x m = 52 b = 0 y = 52x m = 46 b = 0 y = 46x The car will have driven 600 miles at 7:30 pm. The truck will have driven 600 miles at 9:00 pm.

What are the advantages of each method of describing the answer graphically? vs. transition slide 0 1 2 3 4 time hours after 8:00 car distance truck distance 8:00 9:00 1 52 46 10:00 2 104 92 11:00 3 156 138 12:00 4 208 184 hours after departure time of day

What are the advantages of each method of describing the answer graphically? vs. (Large Class Discussion) We have here two different ways of representing the data from the problem. Be sure to point out that each situation will give different equations of the lines, but either can be used to find our solutions. 0 1 2 3 4 time hours after 8:00 car distance truck distance 8:00 9:00 1 52 46 10:00 2 104 92 11:00 3 156 138 12:00 4 208 184 hours after departure time of day

Name Another There is a line that has a slope of 2 and goes through the point ( -1 , 3 ). Name another point on that line. m=2 ( -1 , 3 ) (Large class discussion) We are going to develop the point-slope form of a line.

Name Another There is a line that has a slope of 2 and goes through the point ( -1 , 3 ). Name another point on that line. Now, name another point. ( 0, 5 ) m=2 +2 ( -1 , 3 ) +1 (Large class discussion) To find one point, you can just use the slope to go right one and up two.

Name Another There is a line that has a slope of 2 and goes through the point ( -1 , 3 ). Name another point on that line. Now, name another point. ( 0, 5 ) m=2 +2 ( -1 , 3 ) +1 -2 ( -2, 1 ) (Large class discussion) To find another point, one could continue to the right (right one, up two) or go to the left one and down two. -1

Name Another There is a line that has a slope of 2 and goes through the point ( -1 , 3 ). Name another point on that line. Now, name another point. Now, give me an equation that represents all points. ( 0, 5 ) m=2 ( -1 , 3 ) ( -2, 1 ) (Large class discussion) transition slide

Name Another There is a line that has a slope of 2 and goes through the point ( -1 , 3 ). Name another point on that line. Now, name another point. Now, give me an equation that represents all points. y = 2x + 5 ( 0, 5 ) m=2 ( -1 , 3 ) (Large class discussion) Since we have a slope and the y-intercept, we can find the equation, but.... (next slide)

Name Another There is a line that has a slope of 2 and goes through the point ( -1 , 3 ). Name another point on that line. Now, name another point. NOW, give me an equation that represents all points (without the y-intercept being known). y = 2x + 5 m=2 ( x, y ) ( -1 , 3 ) ( x, y ) ( x, y ) (Teacher Lecture) ... we should be able to find an equation using any two points (another problem) or ANY slope and a point (the point doesn’t have to be the y-intercept). (next slide for the method) ( x, y )

Name Another There is a line that has a slope of 2 and goes through the point ( -1 , 3 ). Name another point on that line. Now, name another point. NOW, give me an equation that represents all points (without the y-intercept being known). y = 2x + 5 m=2 ( -1 , 3 ) ( x, y ) (Teacher Lecture) What we want is an equation that represents all points (x,y) that are on the line. So, we’ll use the slope formula...

Name Another There is a line that has a slope of 2 and goes through the point ( -1 , 3 ). Name another point on that line. Now, name another point. NOW, give me an equation that represents all points (without the y-intercept being known). y = 2x + 5 m=2 ( -1 , 3 ) ( x, y ) (Teacher Lecture) We know our slope and the point (-1,3) and the other point is the unknown (x,y). [next slide]

Name Another There is a line that has a slope of 2 and goes through the point ( -1 , 3 ). Name another point on that line. Now, name another point. NOW, give me an equation that represents all points (without the y-intercept being known). y = 2x + 5 m=2 ( -1 , 3 ) ( x, y ) (Teacher Lecture) We know our slope and the point (-1,3) and the other point is the unknown (x,y). [next slide]

Name Another There is a line that has a slope of 2 and goes through the point ( -1 , 3 ). Name another point on that line. Now, name another point. NOW, give me an equation that represents all points (without the y-intercept being known). y = 2x + 5 m=2 ( -1 , 3 ) ( x, y ) (Teacher Lecture) We know our slope and the point (-1,3) and the other point is the unknown (x,y). [next slide]

Name Another There is a line that has a slope of 2 and goes through the point ( -1 , 3 ). Name another point on that line. Now, name another point. NOW, give me an equation that represents all points (without the y-intercept being known). y = 2x + 5 m=2 ( -1 , 3 ) ( x, y ) (Teacher Lecture) By multiplying each side by the LCD, we get a simpler equation (in point-slope form), now we just need to flip it around... [next slide]

Name Another There is a line that has a slope of 2 and goes through the point ( -1 , 3 ). Name another point on that line. Now, name another point. NOW, give me an equation that represents all points (without the y-intercept being known). y = 2x + 5 m=2 ( -1 , 3 ) ( x, y ) (Teacher Lecture) We have our new form of a line. [next slide]

Name Another There is a line that has a slope of 2 and goes through the point ( -1 , 3 ). Name another point on that line. Now, name another point. NOW, give me an equation that represents all points (without the y-intercept being known). y = 2x + 5 m=2 ( -1 , 3 ) ( x, y ) (Teacher Lecture) Notice that with a distribution and solving for y, we have the same exact equation as before (in slope-intercept form). So... [next slide] Notice:

Name Another y = 2x + 5 There is a line that has a slope of 2 and goes through the point ( -1 , 3 ). Name another point on that line. Now, name another point. NOW, give me an equation that represents all points (without the y-intercept being known). y = 2x + 5 m=2 ( -1 , 3 ) ( x, y ) (Teacher Lecture) Equivalence. same line (different form)

Name Another y = 2x + 5 There is a line that has a slope of 2 and goes through the point ( -1 , 3 ). Name another point on that line. Now, name another point. NOW, give me an equation that represents all points (without the y-intercept being known). y = 2x + 5 ( 0, 5 ) m=2 ( -1 , 3 ) ( x, y ) (Teacher Lecture) Equivalence, with values clarified. same line (different form) slope slope y-intercept point on the line

Name Another y = 2x + 5 There is a line that has a slope of 2 and goes through the point ( -1 , 3 ). Name another point on that line. Now, name another point. NOW, give me an equation that represents all points (without the y-intercept being known). y = 2x + 5 ( 0, 5 ) m=2 ( -1 , 3 ) slope y-intercept ( x, y ) (Teacher Lecture) Two forms of lines explicitly defined.... same line (different form) slope slope y-intercept point on the line

Name Another y = 2x + 5 There is a line that has a slope of 2 and goes through the point ( -1 , 3 ). Name another point on that line. Now, name another point. NOW, give me an equation that represents all points (without the y-intercept being known). y = 2x + 5 ( 0, 5 ) m=2 ( -1 , 3 ) ( x, y ) point on the line (Teacher Lecture) Two forms of lines explicitly defined.... slope same line (different form) slope slope y-intercept point on the line

Name Another y = 2x + 5 There is a line that has a slope of 2 and goes through the point ( -1 , 3 ). Name another point on that line. Now, name another point. NOW, give me an equation that represents all points (without the y-intercept being known). y = 2x + 5 ( 0, 5 ) m=2 ( -1 , 3 ) slope y-intercept ( x, y ) point on the line (Teacher Lecture) Two forms of lines explicitly defined.... slope same line (different form) slope slope y-intercept point on the line

Parallel Problem What is the equation of the line that has the same slope as the line below, but goes through the point (2 , 4) ? (2 , 4)

Parallel Problem What is the equation of the line that has the same slope as the line below, but goes through the point (2 , 4) ? Our line is graphed to the left. (2 , 4)

Parallel Problem What is the equation of the line that has the same slope as the line below, but goes through the point (2 , 4) ? Our line is graphed to the left and has the slope of 1/2. So, using the point-slope form*, we see that our equation is: (2 , 4) *

Parallel Problem What is the equation of the line that has the same slope as the line below, but goes through the point (2 , 4) ? Our line is graphed to the left and has the slope of 1/2. So, using the point-slope form*, we see that our equation is: and if you change it to slope-intercept form, you find: (2 , 4) *

Which of the graphs below could fit the equation: y-3 = -2(x-1) (Think, pair, share) The answer is ‘d’. Notice that the slope is -2 and the point is (1,3)

Depreciated A 4-year old car is currently worth $10,800. The car is known to depreciate by $1200 per year. a) What was the original price for the car when it was new? b) How much will the car be worth in two years? c) When will the car be worth nothing? (Homework) FYI: The Citroen is worth more than stated.

Disclaimer All photos contained are used under creative commons rights. Citroen C2 VTR 002 by someonessomewhereslife http://www.flickr.com/photos/someonesomewhereslife/2350463819/ the fine print