review homework page 225 # 3 page 226 # 4 page 234 page 235 #9-12

Review Homework

•Page 225 # 3•Page 226 # 4•Page 234 •Page 235 #9-12

Algebra Quiz Thursday November 19th pages 208-235Extra credit –mixture problem13 questions

Page 225 #3

Equation(s): y=0.5x+2x y-4 0-2.0 10 22.0 3

226#4

Equation(s):y=4x+2x y-3.0 -10.0-2.0 -6.0-1.0 -2.00 2.01.0 6.02.0 10.0

Page 2341. not a function D={0,1,2,3,4}

R={0,2,3,4,5,6}2. Function D={0,1,2,3,4,22} R= {3,22}3. f(1)=5, f(-3)=-34. f(0)=5, f(3)=20 f(-3)=265. y=4x 6. y=2x+17. y=2x-2 8. y=2x+4

Page 235•  t

There are less than 100 apples in a basket. If they are counted by 3’s, there are two left over.

If they are counted by 4’s, there are 3 left over. If they are counted by 5’s, there are 4 left over. How many apples are in the basket?

59

Direct Variation정비례　관계

What is it and how do I know when I see it?

Definition:Y varies directly as x means that y = kx where k is the constant of variation.

Y varies directly as x means that y = ax where a is the constant of variation. ( a or k ≠ 0)

(see any similarities to y = ax + b?)

Another way of writing this is a = In other words:

The ratio between the two variables is constant.

yx

Direct Variation The total price you pay for video games

you buy is directly related to how many you buy. If video games cost 50,000 won and you buy 5, the cost is 250,000 W. If you buy 10, the cost is 500,000W. This is an example of direct variation y = ax where a is a constant. In this example, a is the price of the video games, 50,000 W each.

We say y varies directly with x. y=ax a is called the constant of variation.

Write the equation of the direct variation that includes the point (2,6)….

Start with the equation y=ax and substitute the values of y and x so you can determine the value a.

Rewrite the equation y=ax and only substitute the value of a to write the equation of variation.

y= ax 6 = a(2) a=3 y=3x is the equation of variation.

Complex problems

Examples of Direct Variation:

X Y 6 12 7 14 8 16

Note: X increases,

6 , 7 , 8

And Y increases.

12, 14, 16

What is the constant of variation of the table above?

2 is the constant of variation

y = 2x is the equation!

X Y -4 -1 -16 -4 -40 -10

Note: X decreases,

-4, -16, -40

And Y decreases.

-1,-4,-10

What is the constant of variation of the table above?

Since y = ax we can say Therefore:

-1/-4=a or a = ¼ -4/-16=a or a = ¼

-10/-40=a or a = ¼ Note a stays constant.

¼ is the constant of variation

y = ¼ x is the equation!

Examples of Direct Variation:

What is the constant of variation for the following direct variation?

X Y 4 -8 8 -16 -6 12 3 -6

1. 22. -23. -½4. ½

Is this a direct variation? If yes, give the constant of variation (a) and the equation.

X Y 4 6 8 12

12 18 18 27

Yes!

a = 6/4 or 3/2

Equation?

y = 3/2 x

X Y 10 25 6 15 4 10 2 5

Yes!

a = 25/10 or 5/2

a = 10/4 or 5/2

Equation?

y = 5/2 x

Is this a direct variation? If yes, give the constant of variation (a) and the equation.

X Y 15 5 3 26 1 75 2 150

No!

The values of a are different!

Is this a direct variation? If yes, give the constant of variation (a) and the equation.

Which of the following is a direct variation?

Using Direct Variation to find unknowns (y = ax)

Given that y varies directly with x, and y = 28 when x=7, Find x when y = 52. HOW???

2 step process X Y

7 28 ? 52

1. Find the constant variation

a = y/x or a = 28/7 = 4

a=4

2. Use y = ax. Find the unknown (x).

52= 4x or 52/4 = x

x= 13

Therefore:

X =13 when Y=52

Given that y varies directly with x, and y = 3 when x=9,

Find y when x = 40.5. HOW???

2 step process X Y 9 3

40.5 ?

1. Find the constant variation.

a = y/x or a = 3/9 = 1/3

a = 1/3

2. Use y = ax. Find the unknown (x).

y= (1/3)40.5

y= 13.5

Therefore:

X =40.5 when Y=13.5

Using Direct Variation to find unknowns (y = ax)

Given that y varies directly with x, and y = 6 when x=-5,

Find y when x = -8. HOW???

2 step process X Y

-5 6 -8 ?

1. Find the constant variation.

a = y/x or a = 6/-5 = -1.2

a = -1.2

2. Use y =ax. Find the unknown (x).

y= -1.2(-8)

x= 9.6

Therefore:

X =-8 when Y=9.6

Using Direct Variation to find unknowns (y = ax)

Using Direct Variation to solve word problems

Problem:

A car uses 10 gallons of gasoline to travel 250 miles. How much gasoline will the car use to travel 400 miles?

Step One: Find points in table

X (gas) Y (miles) 10 250 ? 400

Step Two: Find the constant variation and equation:

a = y/x or a = 250/10 or 25

y = 25 x

Step Three: Use the equation to find the unknown.400 =25x 400 =25x25 25or x = 16

Direct Variation and its graphDirect Variation and its graph

y = ax +b, a = slope and b = y-intercept

With direct variation the equation is y = ax

Note: m = a or the constant and b = 0 therefore the graph will always go through…

the ORIGIN!!!!!

No Yes

No No

Tell if the following graph is a Direct Variation or not.

Practice Page 2401. Yes2. No3. Yes4. No5. Not a function D={1,2,3,4} R={3,5,6}6. Function D={0,1,2,3}, R={0,1}7. Function D={0,1,2,3,4,5} R={0,1,2,3,4,5}8. No, it fails the vertical line test

Homework Page 241