I



I. Lines:

a) Slopes

[pic]

[pic] (same slope)

[pic] (opposite reciprocals)

Horizontal m = 0

Vertical m = undefined

On a graph: Read off 2 points and use formula or find rise/run. Watch sign value!

b) Equations of lines

Point slope form: [pic]

Slope intercept form: [pic]

Standard form: [pic]

Horizontal: y = k

Vertical: x = k

To find the equation of any line, we always need a point and a slope!

c) Graphs of lines:

x-int: point where graph crosses x-axis. Plug in y=0 and solve for x. Always give answer as a point (___, 0)

y-int: point where graph crosses y-axis. Plug in x=0 and solve for y. Always give answer as a point (0, ___). Note the y-coordinate is the value of b in slope intercept form!

Slope: [pic]

d) Applications:

Given information (either a rate and point) or (given two points) find the equation

Use the equation to answer or interpret.

Example: Suppose you mastered 99 topics in ALEKS after 3 hours of practice. You completed 131 topics after 7 hours of practice. A) Find a formula that relates the number of topics mastered as a linear function of the number of hours of practice.

B) If you continue at this rate, how many hours will it take to master all 240 topics?

Note (3, 99) and (7, 131) are on the graph.

To find the equation, we need to first find the slope:

[pic]

Equation: y = m x + b

y = 8x + b

Plug in point and solve for b

99 = 8(3) + b

b = 75

Equation y= 8x + 75

B) y = 240, solve for x

240 = 8x + 75

x = 165/8 = 20.625 hours.

II. Functions:

Domain: set of all inputs

Range: set of all outputs

To be a function, every value in the domain must have exactly one value in the range.

VTL: If a vertical lines hits more than once, the relation is NOT a function.

To find DOMAIN of equation: exclude all points where denominators are zero and/or where stuff under the square root is negative. (set radicand >= 0)

Remember, if your answer in an interval, it must¸be put in interval notation!

Evaluation of functions [pic]

III: Systems

a) Types of solutions

i) Unique: Solution is a unique point

(must give final answer as a point)

System is consistent (has solution)

System in independent (different lines)

ii) No solution

system in inconsistent

system is independent (different lines)

lines are parallel.

When solving with substitution

or Addition, you get a contradiction.

iii) Infinite solution

solution set is the set of all points

on the line.

System is consistent (has solution)

System in dependent (same lines).

When solving with substitution

or Addition, you get a # = itself.

b) Methods of solving

i) Graphing

Graph both lines and find intersection

ii) Substitution

Solve one equation for one variable and plug into other. Solve for remaining variable. Plug back into first equation to find second variable.

Method is often preferred if one equation is already solved for a variable or one equation is easy to solve for one variable without fractions.

iii) Addition: Put both equations in standard form. Multiple one or both equation by constants to get one of the variables to cancel out. Solve for remaining variable. Plug back into first equation to find second variable.

Method is often preferred if substitution will result in equations with messy fractions!

c) Applications:

i) interest

ii) value problems

iii) Mixture

iv) D = rt

v) Geometry

IV: polynomials:

a) terminology:

standard form

degree

leading coefficient

b) Addition and subtraction

c) Multiplication:

ALWAYS remember to re-write squares as a produce and F.O.I.L

i) multiplication by monomial (distribute)

ii) multiplication when both expressions have more than one term.

c) Division:

i) division by monomial

ii) long division

Examples From Quiz 2

#2 Given 2x + 3y = 7. Find

a) the slope

First solve for y. The slope is the coefficient of x

3y = -2x + 7

y = -2/3 x + 7/3

slope = ___-2/3

b) x-intercept ( give answer as a point!)

first plug in y = 0

2x + 3(0) = 7

x = 7/2

Must write as point x-int __(7/2, 0)

c) y-intercept (give answer as a point!)

plug in x = 0

2(0) + 3y = 7

y = 7/3

Must write as point y-int ____(0, 7/3)

d) the slope of the line parallel to this line

Same as slope of given line

slope of parallel line = __-2/3

e) the equation of the line perpendicular to this line going through (4,- 5)

[pic]

Point slope form: y – (-5) = 3/2 ( x – 4) or :

y +5 = 3/2 ( x – 4)

Or slope intercept form y = 3/2 x + b

-5 = 3/2 (4) + b

b = -5-6 = -11

y = 3/2 x -11

Or solve y +5 = 3/2 ( x – 4)for y

y + 5 = 3/2 x – 3/2 * 4

y = 3/2 x - 11

Equation: ___ y = 3/2 x - 11

#3 Given the graph of y = f(x) below

[pic]

a) is this a function? Why or why not?

No, a vertical line hits the graph more than once, so it does not pass the VLT.

b) What is the domain?

Domain: ___[-2, 5]

c) What is the range?

Range:______(-4, 10]

Note: You must use interval notation because the answer is an interval.

Writing the answer in set notation would mean that there were no values between the integers. This is not true!

Examples From Quiz 3

Quiz 3 #1. (Form A) Set up a system of equations and then solve the system algebraically.

A jar of face cream contains 18% moisturizer, and another contains 45% moisturizer. How many ounces of each should be combined to get 16 oz of cream that is 36% moisturizer?

| |18% |45% |36% mix |

|Amt cream | x |y |16 |

|Amt moisturizer |0.18x |0.45y |0.36 ( 16) |

So the two equations are

x + y = 16

0.18x + 0.45 y = 0.36 ( 16)

Because the first equation is easy to solve, substitution is easier.

Solve for y: y = 16 – x

Plug into second equation:

0.18x + 0.45 (16 – x) = 0.36 ( 16)

Then solve for x. . .

Quiz 3 #1 (Form B) Set up a system of equations and then solve the system algebraically.

Alina invested $ 2400 in 2 accounts one that pays 2% simple interest and one that pays 5% simple interest. At the end of 4 years, her total return was $305.70. How much was invested in each account.? Round answer to the nearest cent.

| |2% |5% |total |

|Amt invested |x |y |2400 |

|Interest |x*(0.02)*4 |y(0.05)4 |305.70 |

We know t = 4 years NOTE NOT 1 year!

I = p* r* t

System:

x + y = 2400

x(0.02)(4) + y (0.05)(4)= 305.70

Note again, this is easy to solve with substitution.

Solve 1st equation for y. y = 2400 - x

Plug into second equation:

0.08x + 0.2 (2400 – x) = 305.70

. . .

Quiz 3 #1 (Form C) Mr Cote invested 4 times as much money in a stock fund that returned 8% interest after one year than he did in a bond find that earned 3% interest. If his total earnings came to 858.27 after 1 year, how much did he invest in each fund? Round answer to the nearest cent.

| |8% |3% |total |

|Amt invested |x |y | |

|Interest |x(0.08)(1) |y(0.03) |858.27 |

We know t = 1!

I = p* r* t

Amt in 8% act is 4 times amt in 3% act.

x = 4 * y

System:

x = 4 y

0.08x + 0.03 y = 858.27

Using substitution

0.08(4y) + 0.03 y = 858.27

…

Quiz 3 # 1 (Form D) Set up a system of equations and then solve the system algebraically.

It takes a boat 2 hrs to travel 16 mi downstream with the current and 3.2 hrs to return against the current. Find the speed of the boat in still water and the speed of the current.

| |D |r |t |

|Upstream |16 |x – c |3.2 |

|downstream |16 |x+ c |2 |

Let x be the speed of the boat in still water and c be the speed of the current.

D = r*t

System:

16 = 3.2(x-c)

16 = (x+c)*2

Or simplifying

5 = x – c

8 = x + c

This system is easier to solve with ADDITION.

Quiz 3 (Form E) Set up a system of equations and then solve the system algebraically.

A credit union offers 5.5% simple interest on a certificate of deposit (CD) and 1.5% simple interest on a savings account. If My Levy invested $200 more in the CD than in the savings account and the total interest was $ 97.45, how much was invested in each account. Round answer to the nearest cent.

| |CD 5.5% |Savings 1.5% |Total |

|Amt invested |x |y | |

|Interest |0.055x(1) |0.015y(1) |97.45 |

We know t = 1

I = Prt

200 more in CD than in savings

x = y + 200

System:

x = y + 200

0.055x + 0.015y = 97.45

Quiz 3 (Form F) Set up a system of equations and then solve the system algebraically.

Mickey brought lunch for his fellow office workers on Monday. He spend $20.70 on 4 hamburgers and 3 chicken sandwiches. Chole brought lunch on Tuesday and spent $21.20 on 2 hamburgers and 5 chicken sandwiches. What is the cost of 1 hamburger and one sandwich? Round answer to the nearest cent.

Let x = cost hamburger

y = cost of chicken sandwich

System:

4 x + 3 y = 20.70

2 x + 5 y = 21.20

Quiz 3 #1 (Form G) A moving sidewalk in the airport moves people between gates. It takes Mike 20 seconds to travel 96 ft walking with the sidewalk. It takes him 30 sec to travel 72 ft walking against the sidewalk (in the opposite direction). Find the speed of the moving sidewalk and Mike’s speed on nonmoving ground.

| |D |r |t |

|With sidewalk |96 |x+s |20 |

|Against sidewalk |72 |x-s |30 |

Let x be the Mike’s speed on non-moving ground and s be the speed of the sidewalk.

D=rt

System:

96 = (x+s)20

72 = (x-s)30

Simplify first

4.8 = x + s

2.4 = x-s

. . . x = 3.6 ft/sec., y = 1.2 ft/sec.

Note: Every one of the applications from the different versions of the quiz was just a homework problem from the textbook with minor number changes! Remember, it is EXTREMELY IMPORTANT that you practice ALL textbook homework problems until you can master them! Remember, don’t forget units!

Quiz 3 #2 (form A) Solve the system algebraically

[pic]

[pic]

Easiest to use substitution!

[pic]

[pic]

[pic]

Always true

Answer: infinite solution

[pic]

Note: You can also clear fractions from equation #1 first!

Quiz 3 #2 (form B) Solve the system algebraically

Solve the system algebraically

[pic]

[pic]

Easiest to use substitution!

[pic]

[pic]

[pic]

Never true [pic]

Answer: No solution!

Note: You can also clear fractions from equation #1 first!

Multiplying Eqn 1 by LCD = 6 results in 4x-y=3

Plugging in second equation would give

4x-(4x+7)=3

-7 = 3 (again a contradiction so no solution)

Quiz 3 # 4 (Form F)

Simplify [pic] [pic] completely.

First ALWAYS re-write squares as products.

[pic]

F.O.I.L

[pic]

[pic]

Distribute! [pic]

[pic]

Quiz 3 # 4. Solve algebraically. [pic]

Easiest to first clear fractions

Multiple eqn #1 by lcd=15

6x -5y = -4

21y=-28x-6

Because solving either equation or any variable will result in messy fractions, it is easier to use ADDITION Method

6x – 5y = -4

28 x + 21 y = -6

Eqn 1 * 21: 126x – 105y = -84

Eqn 2 * 5 140 x +105 y = -30

Add equations: 266 x = -114

x = -114/266 = -3/7

6(-3/7) – 5y = -4

-5y = -4 + 18/7

-5y = -10/7

y = 2/7

Answer: ____(-3/7, 2/7)

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