common core math 3 - unboundthe mathematician's patterns, like the painter's or the poet's must be...

!

The mathematician's patterns,

like the painter's or the poet's must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics. !

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Common Core Math 3 Unit 2A – Modeling with Linear Functions

A P E X H I G H S C H O O L

1501 L A U R A D U N C A N R O A D

A P E X , N C 27502

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1

Linear function: a function whose graph is a line

Linear functions do NOT contain:

! variables with exponents

! variables in the denominator of a fraction

! the product of variables

Forms of linear equations:

1) Slope intercept form: y = mx+b

! “m” is the slope (the rate of change)

! “b” is the y-intercept (the y value when x=0, the “start”,)

2) Standard form: Ax + By = C where A, B and C are integers and A is

positive.

Slope:!!!!!

m =rise

run=

Dy

Dx=y2

- y1

x2

- x1

!!!!!!!"#$%!&'()*!+,()-'.

A horizontal line has a “0” slope.

! horizontal lines cross the y-axis

! their equations look like y = #

A vertical line does not have slope, slope is “undefined” or (null)

! vertical lines cross the x-axis

! their equations look like x = #

! a vertical line is not a function

Parallel lines have the same slope.

Perpendicular lines have slopes that are opposite reciprocals.

o “opposite” means the opposite sign

o “reciprocals” means a pair of numbers that, when multiplied

together, equal 1. Ex. 2/3 and 3/2 are reciprocals.

!

2

To graph a linear equation:

1) use one point and the slope

! the point can be any point

o the y-intercept is a point with x=0

o the x-intercept is a point with y=0

2) use two points

! any 2 points,

o given an equation make a table (pick an x find the y)

To write a linear equation:

1) given the slope and y-intercept

1) use y = mx + b (slope-intercept form)

! Plug slope in for m

! Plug y-intercept in for b

2) given the slope and a point

Use slope intercept form and Solve for b

Step 1 Start with y = mx + b

Step 2 Plug in slope for m

Step 3 Plug in numbers for x and y

(x value from the point, y value from the point)

Step 4 Solve equation for b

Step 5 Use y = mx + b and plug in m and b

3) given two points

1) Find the slope

2) Choose one of the points

3) Proceed same as for “B. Given the Slope and a Point.”

4) from a graph

1) Find the slope m from the graph

2) Find the y-intercept b from the graph

3) Use slope intercept form y = mx + b

3

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More Practice with L ines F ill in the missing information:

1. Equation: 5x ! 8y = 40 G raph: Slope: _____________

y-intercept: ______________

Table:

2. Equation: _______________ G raph: Slope: 4/3

y-intercept: _____________

Table:

3. Equation: _______________ G raph: Slope: __________

y-intercept: _____________

Table:

x y

2

-2

x y

-4 3

2

x y

Writing Linear Equations Practice

State the slope and y-intercept of the graph of each equation.

1. 5x – 4y = 8 2. 3x – y = -11

3. 4. 3y = 7

Find the slope-intercept form of each equation. (y = mx + b)

5. 3x – 4y = 15 6. 4x + 7y = 12

7. 9y = -15 8. 2x = -8

Write an equation for the line that satisfies each of the given conditions in slope-intercept

form.

9. slope = -5, passes through (-3,-8) 10. slope = , passes through (10, -3)

11. passes through(4,3) and (7,-2) 12. passes through(-6,-3) and (-8,4)

Write an equation for the line satisfying the following conditions in slope-intercept form.

13. passes through (3,11) and (-6,5) 14. passes through (7,2) and (3,-5)

15. x-intercept = -5, y – intercept = -5 16. x-intercept = -5, y – intercept = 7

Write an equation in Standard form for the following lines:

17. parallel to y = -2x + 6, through (9,-3) 18. perpendicular to y = 2/3x – 5,

through (-4, 7)

19. x intercept = 2 and parallel to y = 7 20. Passes through (3, -2) and ( 4,0)

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< If Y is

Dashed Line

If Y is >, shade ABOVE boundary line

! If Y is !, shade BELOW boundary line

" Solid Line

If Y is ", shade ABOVE boundary line

If you are unsure, use a test point.

If the point makes the inequality TRUE,

shade where the point is located.

If the point makes the inequality FALSE,

shade the other side of the line.

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Linear Programming Graphing Practice #1

Solve using the graphs from #1 & 2

Linear Programming Graphing Practice #2

Do all work on a separate sheet of graph paper.

1. p = x + 5y, find the maximum profit under these constraints:

x + y £ 5

x + 2y £ 8

x ! 0

y ! 0

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2. p = 4x – y, find the maximum profit under these constraints.

x + y £ 6

2x + y £ 10

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y ! 0

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3. c = 2x + 2y, find the minimum costs under these constraints:

2x + y £ 6

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4. If cost is represented by c = x + 3y, find the minimum costs under these constraints:

x + 2y £ 8

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5. If profit is represented by p = 3x + 4y, find the maximum profit under these constraints:

x + y £ 3

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6. If cost is represented by c = 2x + 3y, find the minimum costs under these constraints:

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Linear Programming Practice #1 1. Suppose you want to buy a number of CDs & tapes. You can afford up to ten tapes and seven CDs. You want at least 4 CDs and 10 hours of recorded music. Each tape holds about 45 minutes of music and each CD holds about an hour of music. If each tape costs $8 and each CD costs $12, how many of each can you buy with minimal cost? 2. A concession stand makes $0.85 on each regular hot dog and $1.33 on each foot-long hot dog. On a typical Saturday, it sells between 25 and 40 regular hot dogs and between 30 and 50 foot-longs. The total sales have never exceeded 80 hot dogs. How many of each type should be prepared to maximize profit? 3. You are in charge of decorating the school gym for graduation. You need to buy gold and green rolls of crepe paper. Gold crepe paper costs $5 per roll and green costs $3 per roll. You will need at least 10 rolls of crepe paper. You want no more than 7 rolls of green and 6 rolls of gold. How many rolls of each color crepe paper should you buy to minimize your cost? What is the minimum cost? 4. A small computer company manufactures two models of computers. One model is for business use and the other model is for personal use. The company can make no more than 8 computers per day. They want to build no more than five business computers and no more than 6 personal computers. The company makes a profit of $75 on each personal computer and $100 on each business computer. How many of each type of computer should they make to maximize their profit? What is the maximum profit? 5. You own a company that makes furniture. Your company makes end tables and coffee tables. Each week you must make at least 6 end tables and at least 4 coffee tables. Your company can make at most 16 tables per week. The profit on a coffee table is $40 and an end table is $30. How many tables of each should you make per week to maximize your profit? What is your maximum profit? 6.A tray of corn muffins requires 4 c milk and 3 c wheat flour. A tray of bran muffins takes 2 c milk and 3 c wheat flour. There are 16 c milk and 15 c wheat flour available. And the baker makes $1 profit per tray of corn muffins and $2 per tray of bran muffins. The baker must make both kinds of muffins. How many trays of each should he make in order to maximize profits? 7. Kay grows and sells tomatoes and green beans. It costs $1 to grow a bushel of tomatoes and it takes 1 yd squared of land. It costs $3 to grow a bushel of beans and 6 yd squared of land. Kay’s budget is $15 and she has 24 yd squared of land available. If she makes $1 profit on each bushel of tomatoes and $4 profit on each bushel of beans, how many bushels of each should she grow in order to maximize profits? 8. You need to buy some filing cabinets. You know that Cabinet X costs $10 per unit, requires six square feet of floor space, and holds eight cubic feet of files. Cabinet Y costs $20 per unit, requires eight square feet of floor space, and holds twelve cubic feet of files. You have been given $140 for this purchase, though you don't have to spend that much. The office has room for no more than 72 square feet of cabinets. How many of which model should you buy, in order to maximize storage volume?

Linear Programming – Practice #2

1) You are about to take a test that contains questions of type A worth 4 points and of type B worth 7 points. You must answer at least 5 of type A and 3 of type B, but time restricts answering more than 10 of either type. In total, you can answer no more than 18. How many of each type of question must you answer, assuming all of your answers are correct, to maximize your score?

2) Mrs. Wood’s Biscuit Factory makes two types of biscuits, Biscuit Jumbos and Mini Mint

Biscuits. The oven can cook at most 200 biscuits per day. Jumbos each require 2 ounces of flour. Minis each require 1 ounce of flour. There are 300 ounces of flour available. The income from Jumbos is 10 cents each. The income from Minis is 8 cents each. How many of each type should be baked to earn the greatest income?

3) Wheels Inc. makes mopeds and bicycles. Experience shows they must produce at least

10 mopeds. The factory can produce at most 60 mopeds and 120 bicycles per month. The profit on a moped is $134 and on a bicycle, $20. They can make at most 160 units combined. How many of each should they make per month to maximize profit?

4) Lois makes banana bread and nut bread to sell at a bazaar. A loaf of banana bread

requires 2 c flour and 2 eggs. A loaf of nut bread takes 3 c flour and 1 egg. Lois has 12 c flour and 8 eggs on hand. She makes $2 profit per loaf of banana bread and $2 per loaf of nut bread. To maximize profit, how many loaves of each type should she bake?

5) Juan makes two types of wood clocks to sell at local stores. It takes him 2 h to assemble

a pine clock, which requires 1 oz of varnish. It takes 2 h to assemble an oak clock, which takes 4 oz pf varnish. Juan has 16 oz of varnish in stock, and he can work 20 hours. If he makes $3 profit on each pine clock and $4 profit on each oak clock, how many of each type should he make to maximize profits?

6) A biologist needs at least 40 fish for her experiment. She cannot use more than 25 perch

or more than 30 bass. Each perch costs $5, and each bass costs $3. How many of each fish should she use in order to minimize the cost?

7) A biologist is developing two new strains of bacteria. Each sample of Type I bacteria

produces 4 new viable bacteria, and each sample of Type II produces 3 new viable bacteria. Altogether, at least 240 new viable bacteria must be produced. At least 30, but not more than 60, of the original samples must be Type I. Not more than 70 of the samples are to be Type II. A sample of Type I costs $5 and a sample of Type II costs $7. If both types are to be used, how many samples of each should be used to minimize the cost?

8) In order to ensure optimal health (and thus accurate test results), a lab technician needs

to feed the rabbits a daily diet containing a minimum of 24 grams (g) of fat, 36 g of carbohydrates, and 4 g of protein. But the rabbits should be fed no more than five ounces of food a day. Rather than order rabbit food that is custom-blended, it is cheaper to order Food X and Food Y, and blend them for an optimal mix. Food X contains 8 g of fat, 12 g of carbohydrates, and 2 g of protein per ounce, and costs $0.20 per ounce. Food Y contains 12 g of fat, 12 g of carbohydrates, and 1 g of protein per ounce, at a cost of $0.30 per ounce. What is the optimal blend to minimize cost?

common core math 3 - unboundthe mathematician's patterns, like the painter's or the poet's must be...

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