Regents Review #5

What else do I need to know?

Pythagorean TheoremFor any right triangle with legs (shorter sides) a and b and hypotenuse (the longest side) c, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

a2 + b2 = c2

The length of the shortest side of a right triangle is 6 inches. The lengths of the other two sides are consecutive even integers. a)Write an equation that can be used to find the missing sides.

b) Solve the equation for the two missing sides.

a2 + b2 = c2

62 + x2 = (x + 2)2

62 + x2 = (x + 2)2

36 + x2 = x2 + 4x + 436 = 4x + 432 = 4x 8 = x

The two missing sides are 8 and 10.

x: length of one legx + 2: length of hypotenuse

88 + 2 = 10

Shaded AreaIn the diagram, the dimensions of the large rectangle are 3x – 1 by 3x + 7 units. The dimensions of the cut-out rectangle are x by 2x + 5 units. Represent the area of the shaded region as a simplified polynomial expression.Area of Shaded Region = Area of Big Rectangle – Area of Small Rectangle “Whole Shape” – “Inside Shape”

Area = (3x – 1)(3x + 7) – (2x + 5)(x)Multiply each pair of polynomialsA = (9x2 + 18x – 7) – (2x2 + 5x)Distribute the - signA = 9x2 + 18x – 7 – 2x2 – 5x Combine like termsA = 7x2 + 13x – 7

The area of the shaded region is 7x2 + 13x – 7 square units.

Interval Notation[ means to include < or >( means do not include < or >For Example: [12,16) means…. All Real Numbers 12 through 16, including 12 but not 16 1) Represent the domain of f(x) = 2x + 3 graphed over the

interval -4 < x < 6.

(1) [-4,6] (2) (-4,6) (3) [-4,6) (4) (-4,6]

2) Represent the domain and range of the function .

Domain: [-3, ∞)

Range: [0, ∞)

3xf(x)

Remember:Infinity ∞ always uses )

Linear FunctionsLinear Functions written in slope-intercept form identify the slope (rate of change) and y-intercept of the function.

Linear Functions written in point-slope form identify the slope (rate of change) and one point that lies on the function.

In order to write a linear function in point-slope form, follow these steps…

1)Calculate the slope2)Identify a point on the line3)Replace m with the slope and x1 and y1 with the coordinates of a point on the function.

Equation: y – 5 = 3(x – 2)

Linear FunctionsParallel lines have the same slope and different y-intercepts.

Perpendicular lines intersect at a 90⁰ angle. These lines have opposite reciprocal slopes.

Write the equation of a line in point slope form that is perpendicular to 8x – 2y = 20 and passes through the point (1, -6).

8x – 2y = 20y = 4x – 10, m = 4

Slope of perpendicular line =

m = point: (1, -6)

y – y1 = m(x – x1)

y + 6 = - ¼(x – 1)1

4

1

4

Quadratic FunctionsConsider the parent function y = x2 with the vertex (0,0).

The function y = (x – 4)2 shifts the parent function to the right 4 units. The vertex of the new function is (4, 0).

The function y = (x – 4)2 – 5 shifts the parent function to the right 4 units and down 5 units. The vertex of the new function is (4, -5).

The graph pictured to the left represents a transformation of y = x2. Write an equation to represent this graph.

y = a(x – h) 2 + k

y = – (x – 2) 2 + 4Vertex: (2, 4) Parabola opens down, a = -1

SequencesExplicit vs. Recursive

An explicit formula that defines a sequence can tell you any term of the sequence.

A recursive formula that defines a sequence can tell you the next term provided that you know the previous term.

Consider the sequence 8, 24, 72, 216, …Explicit Formula: s(n) = 8 3n-1

Find the 5th terms(5) = 8 35-1

s(5) = 648The 5th term is 648

Recursive Formula: s(1) = 8 s(n) = s(n – 1) 3Find the 5th terms(5) = s(5 – 1) 3s(5) = s(4) 3s(5) = 216 3S(5) = 648The 5th term is 648

SequencesExplicit vs. Recursive

Explicit Formulas:

Arithmetic: an = a1 + d(n – 1)

Geometric: an = a1 rn – 1

Recursive Formulas:

Arithmetic: an = an-1 + d

Geometric: an = an-1 r

Sometimes Recursive Formulas are more complicated.Find the 4th term given the recursive formula: f(1) = 12 and f(n) = f(n – 1) + 3n

f(n) = f(n – 1) + 3nf(2) = f(2 – 1) + 3(2)f(2) = f(1) + 3(2)f(2) = 12 + 6f(2) = 18

f(n) = f(n – 1) + 3nf(3) = f(3 – 1) + 3(3)f(3) = f(2) + 3(3)f(3) = 18 + 9f(3) = 27

f(n) = f(n – 1) + 3nf(4) = f(4 – 1) + 3(4)f(4) = f(3) + 3(4)f(4) = 27 + 12f(4) = 39

The 4th term of the sequence is 39.

Rational Equations

21

x6

8x4

41

When solving rational equations (equations with algebraic fractions), combine fractions and set up a proportion. Remember: A common denominator is needed to add or subtract fractions.

8x4

8x2x

11

8x4

41

2x2x

8x42x

2xx

2x12

xx

21

x6

22

2x12x

2x12x

8x42x

8x(x + 12) = 2x(2x + 4) 8x2 + 96x = 4x2 + 8x 4x2 + 88x = 04x(x + 11) = 0 4x = 0 x + 11 = 0 x = 0 x = -11 Reject 0 because it makes the equation undefined.

Solution: x = -11

FOO FOO

Work Word ProblemsSuppose one painter can paint the entire house in twelve hours, and the second painter takes eight hours. How long would it take the two painters together to paint the house?

Hours to Complete the Job

1st painter: 12 hours12

1

2nd painter: 8 hours8

1

1st painter: per hour

Together: x hours

2nd painter: per hour

Together: per hourx

1

Equation:

x

1

8

1

12

1

x

1

24

5

8.4

245

x

xTogether, the painters can complete the job in 4.8 hours (just under 5 hours).

Job Completed per Hour (rate) Combined Labor

x

1

24

3

24

2

Work Word ProblemsOne pipe can fill a pool 1.25 times faster than a second pipe. When both pipes are opened, they fill the pool in five hours. How long would it take to fill the pool if only the slower pipe is used?

Hours to Complete the Job

Pipe A: x hoursx

1

Pipe B: 1.25xx25.1

1

Pipe A: per hour

Together: 5 hours

Pipe B: per hour

Together: per hour5

1

Equation:

5

1

25.1

1

25.1

25.15

1

25.1

11

xx

xx

5

1

25.1

25.2x

9

25.1125.1

x

x

It takes the fast pipe 9 hours. It takes the slow pipe 11.25 hours (9 X 1.25). It would take 11 hours and 15 minutes to fill the pool if only the slow pipe is used.

Job Completed per Hour (rate) Combined Labor

Profit Word ProblemsProfit = Income – Expenses (P = I – E)Net Profit Revenue Cost Gross Profit

The cost of operating Hannah’s Biscotti Company is $750 per week plus $0.05 to make each biscotti cookie.a)Write a function, C(b), to model the company’s weekly costs for producing b biscotti cookies.

b)What is the total weekly cost in dollars if the company produces 5,000 biscotti cookies.

c)Hannah’s company makes a gross profit of $0.40 for each biscotti cookie they sell. If they sold all 5000 biscotti cookies, would they make money or lose money?

C(b) = 750 + .05b

C(b) = 750 + .05(5000) = $1000 It costs the company $1000 to make 5000 cookies

Gross Profit = 0.40(5000) = $2000$ earned before expenses are subtracted

Net Profit = Income – Expenses = 2000 – 1000 = 1000The company will earn $1000

Age Word ProblemsSue is 5 years older than Ann. In 6 years, Sue’s age will be 11 years less than twice Ann’s age then. How old is each person now?

Person Age Now Age In 6 Years

Ann x x + 6

Sue x + 5 (x + 5) + 6 = x + 11

Future Sue will be 11 years less than twice Future Ann x + 11 = 2(x + 6) – 11

x + 11 = 2(x + 6) – 11x + 11 = 2x + 12 – 11x + 11 = 2x + 1 11 = x + 1 10 = x

Right now, Ann is 10 years old and Sue is 15 years old.

Remember:It is helpful to organize information in a table prior to creating an equation.

Coin Word ProblemsJoe has $2.50. He has 7 more dimes than nickels. How many of each does he have?

Coin Value Quantity Total Value

Nickels .05 x .05x

Dimes .10 7 + x .10(7 + x)

.05x + .10(7 + x) = 2.50 or 5x + 10(7 + x) = 250 5x + 70 + 10x = 250 15x + 70 = 250 15x = 180 x = 12

Joe has 12 nickels and 19 dimes.Check: 12 nickels = 60 cents 19 dimes = $1.90Total: $1. 90 + $0.60 = $2.50 Remember: (Value)(Quantity) = Total value of Coins

$ per coin x how many = total $

Ratio Word ProblemsThe measures of two supplementary angles are in the ratio of 3:7. What is the measure of the larger angle?

Let 3x = the measure of the smaller angle

Let 7x = the measure of the larger angle

smaller angle = 54 degrees (3)(18)

larger angle = 126 degrees (7)(18)

The larger angle measures 126○

3x + 7x = 180

10x = 180

x = 18

Remember:Include an x in each part of the ratio.

Ex: Donna wants to make 4lbs of trail mix made up of almonds, walnuts and raisins. She wants to mix one part almonds, two parts walnuts, and three parts raisins. Ratio 1:2:3Let x = the amount of almondsLet 2x = the amount of walnutsLet 3x = the amount of raisinsx + 2x + 3x = 4

Consecutive Integer Word ProblemsFind two consecutive integers whose sum is -35.

x: 1st consecutive integer

x + 1: 2nd consecutive integer

18x

362x

3512x351)(xx

-18

-17 (-18 + 1)

Remember:

Consecutive integers count by 1’sEx: x, x+1, x+2, x+3….

Consecutive odd or even integers count by 2’sEx: x, x+2, x+4, x+6…

Negative integers doesn’t change anything

StatisticsQuantitative Data is numerical, meaning it can be counted or measured.Ex: height of a flagpole, weight of a backpack

Categorical Data (Qualitative) is not numerical, meaning it can be observed.Ex: type of toppings on a pizza, favorite ice cream flavor

Univariate refers to single variable data.Ex: the number of pets each student owns

Bivariate refers to two variable data.Ex: a person’s shoe size compared to their height

StatisticsA population is a group that you want information about.A sample is part of a population that is used to make estimates about the population.In a random sample, each member has an equal chance of being selected, and the sample is representative of the entire population.A biased sample favors one or more parts of the population over others.Ex: You want to conduct a survey to find out what type of music people listen to.Determine which of these scenarios is biased.A.You ask every fifth person leaving a Taylor Swift concert about the type of music they listen to.B.You ask every fifth person leaving the local mall about the type of music they listen to.

Biased

Unbiased

Test Taking Tips1) Don't rush through the exam. You have 3 hours…use them!2) If you get stuck on a problem, move on and come back to it later.3) After completing the exam once, take a little mental break, then re-do ALL the problems in a different order.

Part I (Multiple Choice):oEliminate choices that don't make senseoIf you're not sure how to solve a problem, try testing each choice by substitutingoWhen checking correct choices…make sure you understand why the other choices are incorrect

Part II-IV (Open-Ended Response)oShow all work! (a correct answer with no work only receives 1 point)oRead each problem carefully. Read it a couple of times and underline key words and phrases.oAlways draw a picture or diagram if one is not provided.oWhen asked to sketch a graph, always set up a table of values , plot points, and label.oRounding should never be done until the end of the problem.oMake sure your answer makes sense and ask yourself, "Did I answer the question completely?"oWith all written explanations, make sure you are specific and use appropriate mathematical reasoning. Provide mathematical evidence for everything!

How do you know when you are completely done with the exam?1) You have answered every problem at least twice.2) You have checked that all of your work for Parts II-IV is in ink (except for any graphs).3) You have made sure that every answer you wrote or bubbled (in ink) is correct.

Reminders

• Eat BREAKFAST! Testing is periods 2 – 5.

• Report to the Auditorium at 7:35 am

• No pencil cases, only clear plastic zip-lock bags with graphing calculator,

extra batteries, pens, pencils, eraser and a ruler

• All cell phones, electronic devices and book bags in lockers (Absolutely no

reading material allowed)

• Water bottles must be clear plastic (no label) and can only sit under the

desk

• Upon the start of the exam, take time to write down any formulas or main

ideas you may forget on the front or back of the reference sheet

THE REGENTS IS TUESDAY, JUNE 3rd

STUDY! STUDY! STUDY!

END OF REGENTS REVIEWSNow it’s your turn to review on your own!

Review the study guides, practice sets, power points, mini-quizzes and green book.Everything is on the REGENTS REVIEW page on halgebra.org.