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Unit 4 Seminar Agenda 4.1 Variation 4.2 Linear Inequalities 4.3 Graphing Linear Equations

3 Direct Variation If a variable y varies directly with a variable x, then y = kx, where k is the constant of proportionality. 3

4 The resistance, R, of a wire varies directly as its length, L. If the resistance of a 30 ft length of wire is 0.24 ohm, determine the resistance of a 40 ft length of wire? R = kL 0.24 = k(30) 0.24/30 = k = k R = 0.008L R = 0.008(40) R = 0.32 ohm 4

5 Inverse Variation If a variable y varies inversely with a variable x, then y = k/x, where k is the constant of proportionality. 5

6 The time, t, for an ice cube to melt is inversely proportional to the temperature, T, of the water in which the ice cube is placed. If it takes an ice cube 2 minutes to melt in 75 degree F water, how long will it take an ice cube of the same size to melt in 80 degree F water? t = k/T 2 = k/ = k t = 150/T t = 150/80 t = minutes 6

7 Joint Variation The general form of a joint variation, where y varies directly as x and z, is y = kxz, where k is the constant of proportionality. 7

8 The volume, V, of a pyramid varies jointly as the volume of its base, B, and height, h. If the volume of a pyramid is 12 cubic feet when the volume of the base is 4 square feet and the height is 9 feet, find the volume of a pyramid when the area of the base is 16 square feet and the height is 9 feet. V = kBh 12 = k(4)(9) 12 = 36k 12/36 = k 1/3 = k V = (1/3)Bh V = (1/3)(16)(9) V = 48 cubic feet 8

9 F varies jointly as M1 and M2 and inversely as the square of d. If F = 20 when M1 = 5, M2 = 10 and d = 0.2, determine F when M1 = 10, M2 = 20 and d = 0.4 F = kM1 M2/d 2 20 = k(5)(10)/ = 50k/ = 1250k = k F = M1 M2/d 2 F = 0.016(10)(20)/0.16 F = 20 9

10 EVERYONE: C varies inversely as J. If C = 7 when J = 0.7, determine C when J = 12.

11 C = k/J 7 = k/ = k C = k/J C = 4.9/12 C = EVERYONE: C varies inversely as J. If C = 7 when J = 0.7, determine C when J = 12.

12 Inequality signs < is less than ( parenthesis) ≤ is less than or equal to [ brackets] > is greater than ( ) ≥ is greater than or equal to [ ] 12

13 Graphing Inequalities x < 0 x ≤ 0 13

14 Graphing Inequalities x > 0 x ≥ 0 14

15 4x + 9 > 25 4x > x > 16 x > 4 15

x ≤ x - 10 ≤ x ≤ 11 -3x/-3 ≥ 11/-3 x ≥ -11/3 16

17 -(1/4)x - 5 > 9 -(1/4)x ≥ (1/4)x ≥ 14 (-4)(-1/4)x ≤ (-4)(14) x ≤

18 56 > -9x + 2 > > -9x > > -9x > 20 54/-9 < -9x/-9 < 20/-9 -6 < x < -2 2/9 18

19 3 < (4x - 1)/5 ≤ 21 (3)(5) < [(4x - 1)/5](5) ≤ (21)(5) 15 < 4x - 1 ≤ < 4x ≤ < x ≤ 26 1/2 19

20 Coordinate Grid x axis y axis origin Quadrant IQuadrant II Quadrant III Quadrant IV A B C D

21 List the Ordered Pairs (x,y) x axis y axis A B C D

22 Plotting Points Plot the following points: (0,7), (-2, -4), (5, -3), (2,2), (-5, 4)

Graphing a Line EX: Graph x + y = 5 Pick 3 points for x and solve for y NOTE: MML will only allow you to plot 2 points. However, it is good to choose 3 points to double check yourself. 23

Graphing a Line xx + y = 5y(x,y) y = 5 y = 7 7(-2, 7) 00 + y = 5 y = 5 5(0, 5) 22 + y = 5 y = 3 3(2, 3) 24

25 Example: x + y = 5. x = 0 and y = 5 gives ordered pair (0,5) x = 1 and y = 4 gives ordered pair (1, 4) x = 2 and y = 3 gives ordered pair (2, 3) x = 3 and y = 2 gives ordered pair (3,2) x = 4 and y = 1 gives ordered pair (4, 1) x = 5 and y = 1 gives ordered pair (5, 1)

26 Graphing Linear Equations x + y = 5 (-2,7) (0,5) (2,3)

27 Example: Graph 2x – 3y = 12 First, solve equation for y. (It will make it easier to graph) Y = 2/3x – 4 When graphing an equation with a fraction, choose points that will eliminate the fraction. Choose 0, and multiples of 3. 27

28 xy = 2/3x – 4Y(x,y) -3y = 2/3(-3) – 4 Y = -2-4 Y = -6 -2(-3, -6) 0Y = 2/3(0) – 4 Y = 0 – 4 Y = -4 -4(0, -4) 3Y = 2/3(3) - 4 Y = (3, -2)

29 Graphing Linear Equations Y = 2/3x - 4 (-3,-6) (0,-4) (3,-2)

30 Example: What are the x- and y- intercepts of 3x – 7y = 21? To find the x- intercept, let y = 0. 3x – 7(0) = 21 3x – 0 = 21 3x = 21 x = 7 (7,0) To find the y-intercept, let x = 0. 3(0) – 7y = 21 0 – 7y = 21 -7y = 21 y = -3 (0, -3)

Positive Slope As x values increase, y values also increase

Negative Slope As x values increase, y values decrease

Horizontal Line Form: y = constant number

Vertical Line Form: x = constant number

Finding Slope Graphically (-6, -1) (2, 4) Up 5 Right 8 Slope is 5/8

Finding Slope Graphically (-4, 1) (3, -3) Right 7 Down 4 Slope is -4/7

Finding slope given 2 points Find the slope between (1,3) and (4,5) m = 2 3

Example: Find the slope of the line that passes through the points (-2, 6) and (-1, -2).

EVERYONE: Find the slope of the line that passes through the points (0, 3) and (7, -2) m = -2 – 3 7 – 0 m = -5 7

Example Find the slope of the line that passes through the points (1, 7) and (-3, 7). m = 0

Example find the slope of the line that passes through the points (6, 4) and (6, 2). m = UNDEFINED (Division by zero is undefined)

Finding slope and y-intercept given an equation First solve for y When you have it in the form y = mx + b then m is your slope and b is your y- intercept 3x + y = 7 y = -3x + 7 m = -3 is your slope b = (0, 7) is your y-intercept