U2 – Linear Graphs and Applications Notes 1

Introduction We recommend that you actually play this presentation to get the full value You can also print these slides if you prefer to work on paper 2

Graphs Review – Part 1 Notes with Answers 3

Explanation A French mathematician, Rene Descartes, developed a system for graphing ordered pairs on a grid. This system is called the Cartesian Co-ordinate System. In this system, ordered pairs are graphed on a grid made up of two perpendicular number lines. The lines meet at a point called the origin. The horizontal number line is called the x-axis. The vertical number line is called the y-axis. The x- and y-axes divide the plane into 4 quadrants. 4

Explanation (con’t) Read along the x-axis to identify the x-value. This is called the x-coordinate. Read along the y-axis to identify the y-value. This is called the y-coordinate. For point A, the x-coordinate is –4 and the y- coordinate is 3. The ordered pair is (-4, 3) (See next page for graph). 5

Example 1 6

Now you try! State the coordinates of the following points. PointCo-ordinates B C D E F G H 7

Answers State the coordinates of the following points. PointCo-ordinates B(0,3) C(2,1) D(1,0) E(3,-2) F(0,-4) G(-3,-2) H(0,0) 8

Example 2 y = x + 7 xy d = -3t - 5 td A) B) Complete these tables of values. 9

Example 2 - Answers y = x + 7 xy d = -3t - 5 td -5 -3(-5)-5= (-3)-5=4 2 -3(2)-5= (5)-5= (11)-5=-38 A) B) Complete these tables of values. 10

Example 3 xy Solution: Choose any values of x and work out the value of y for each that you chose. Sometimes, it helps to isolate y first. i.e. y=6 – 2x Complete the table of values if 2x +y =6 and graph 11

Example 3 – Graphing Complete the table of values if 2x +y =6 and the graph is shown below. 12

Example 3 - Answers xy

Example 4-Application Jackie has decided to sell stained glass Canadian flags for Canada Day. It costs $300 to start her glass business (glass cutters, grinder…) and she determines that each flag will cost $10 to make. She plans to sell the flag for $15 each. 14

Example 4 (con’t) Represent this information on the grid after filling in the table of values provided on the next slide. Find the slope of each line and explain what the slope means in words. 15

Example 4 (con’t) Costs Flags Cost Flags Revenue Fill in the blanks 16

Example 4 (con’t) - Solution Costs Flags Cost Flags Revenue Fill in the blanks 17

Example 4 - Graph 18

Example 4 (con’t) Find the equation of the line for: i)the Cost line; and ii)ii) the revenue line Graph of R: starts at (0,0). Graph of C: starts at (0,300) Break-even at 60 flags, $900 19

Example 4 Solution (con’t) The point of intersection is the break-even point, when cost of flags=revenue from flags. To find the equation (formula) for a line, this is found by using y=mx+b, where m is the slope, and b is the y- intercept (where the graph crosses the y-axis). 20

Fill in the blanks Summary for finding the equation of a line: Formula for slope: m= To find the y-intercept (b), let ____________ To find the x-intercept, let ______________ Equation of a line: ______________ 21

Fill in the blanks Summary for finding the equation of a line: Formula for slope: m= To find the y-intercept (b), let x=0 and find y value To find the x-intercept, let y=0 and find x value Equation of a line: y=mx +b 22

Example 5 Find the equation of a line which contains the two points A(1,5) and B(-2,3). Find also the x and y intercept of the line (ie where does the line intersect the coordinate axes?) 23

Example 5 (con’t) 24

Example 5 (con’t) 25

Example 6 It is known that the profit for my company follows a linear model. My company sells an electric car, called the Gasnomore. When we sell 100 cars in a month, we lose $200,000. When we sell 2,000 cars, I make $180,000. Find: a)How much profit will we make on each car that we sell? b)What are my fixed monthly costs? c)How many cars do we need to sell to break even? 26

Example 6 (con’t) Solution: Since this is a linear model, the slope is constant. Also, the slope= profit/car (look at units) x number of cars y profit 100-$200,000 2,000$180,000 27

Example 6 (con’t) So, again we are given two points. We follow the same steps. (You can write the slope as a decimal or a fraction) 28

Example 6 (con’t) 29

Example 6 (cont.) a)How much profit will we make on each car that we sell? b)What are my fixed monthly costs? c)How many cars do we need to sell to break even? 30

Answers a)As we know the slope is $200/car so that is the answer to how much we make per car. b) My fixed monthly costs are $200,000 so to make a profit we need to recoup this amount. c) This is finding the x intercept as we are interested in breaking even which on a profit graph is the x intercept. The algebra is on the next slide. 31

Example 6 (cont) To find where we break even: 32

Example 7 State the equations of the three lines (A,B,C) shown below 33

Example 7 (con’t) A: Answer: y=2 (note that a horizontal line has slope 0, so it is actually y=0x+2) B: Solution: y=mx+ 4 (y intercept=4) C: Solution: 34

Example 8 a)Find the slope of the line passing through A(1, -3) and B (-1, 5) Answer: b) Find the equation of the line passing through A (1, -3) and B( -1, 5) 35

Example 8 (con’t) b) Solution: 36

Example 8 (con’t) c) Find the y-intercept and x-intercept of this line. 37

Example 9 A soccer team to raise money for an upcoming trip. The cost (expenses) and the revenue (total sales, or income) of selling the candy bars are represented on the graph below. (note: on the cost line, assume that (0,25) is a point, as is (300,125)) 38

Example 9 (con’t) - Graph 39

Example 9 (con’t) a) How many candy bars must the team sell to break even? b) At what price is the team selling each chocolate bar? c)Determine an equation to represent the cost to the team of this fund raising project. d)How many candy bars must the band sell for the revenue to be $200. How much of this revenue would be profit? 40

Example 9 (con’t) a)How many candy bars must the team sell to break even? Answer: 60 b)At what price is the team selling each chocolate bar? Answer: about 67 cents/bar or perhaps 3 bars for $2 is better as it is exact. c)Determine an equation to represent the cost to the team of this fund raising project. Answer: d)How many candy bars must the band sell for the revenue to be $200. How much of this revenue would be profit? Answer: Looking at the graph, 300 bars. 41

Summary Problem #10 X minutes usedY – Total Monthly Cost 400$50 475$ $ A cell phone plan bills by the minute and also adds a fixed cost per month. The table represents some examples. Assume a linear model.

Summary Problem #10 cont. a)Find a formula for the relationship in the form y=mx+b b) Find the y-intercept and explain what it represents in words c) State the slope and explain what it represents d) Find the missing values in the chart provided 43

Summary Problem #10 cont. e) A rival company charges $0.05 per minute and a fixed cost of $51 per month. For what number of minutes do the two plans have the same total monthly charge? 44

Summary Problem #10 Solution a) Find a formula for the relationship in the form y=mx+b 45

Summary Problem #10 Solution b) Find the y-intercept and explain what it represents in words The y-intercept is 18 and it represents the monthly fixed cost of the plan ($18) c) State the slope and explain what it represents the slope is 0.08 and represents the cost per minute ( 8 cents per minute) 46

Summary Problem #10 Solution d) Find the missing values in the chart provided Let x=1000, y=0.08(1000) +18=$98 Let y= =0.08x =0.08x x=104/0.08=

Summary Problem #10 Solution e) A rival company charges $0.05 per minute and a fixed cost of $51 per month. For what number of minutes do the two plans have the same total monthly charge? Let 0.05x+51=0.08x x=33 x=33/0.03=1100 The plans charge the same amount for 1100 minutes 48