# Regents Review #5 What else do I need to know?.

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Regents Review #5 What else do I need to know?

Pythagorean Theorem a2 + b2 = c2 The two missing sides are 8 and 10.
For 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. 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 x: length of one leg x + 2: length of hypotenuse 8 8 + 2 = 10 62 + x2 = (x + 2)2 36 + x2 = x2 + 4x + 4 36 = 4x + 4 32 = 4x 8 = x The two missing sides are 8 and 10.

Shaded Area In 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 polynomials A = (9x2 + 18x – 7) – (2x2 + 5x) Distribute the - sign A = 9x2 + 18x – 7 – 2x2 – 5x Combine like terms A = 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 Represent the domain of f(x) = 2x + 3 graphed over the interval -4 < x < 6. [-4,6] (2) (-4,6) (3) [-4,6) (4) (-4,6] 2) Represent the domain and range of the function Domain: [-3, ∞) Range: [0, ∞) Remember: Infinity ∞ always uses )

Linear Functions Equation: y – 5 = 3(x – 2)
Linear 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… Calculate the slope Identify a point on the line 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 Functions Parallel 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 = 20 y = 4x – 10, m = 4 Slope of perpendicular line = m = point: (1, -6) y – y1 = m(x – x1) y + 6 = - ¼(x – 1)

Quadratic Functions y = – (x – 2) 2 + 4 y = a(x – h) 2 + k
Consider 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 Vertex: (2, 4) Parabola opens down, a = -1 y = – (x – 2) 2 + 4

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

Sequences Explicit 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) + 3n f(2) = f(2 – 1) + 3(2) f(2) = f(1) + 3(2) f(2) = f(2) = 18 f(n) = f(n – 1) + 3n f(3) = f(3 – 1) + 3(3) f(3) = f(2) + 3(3) f(3) = f(3) = 27 f(n) = f(n – 1) + 3n f(4) = f(4 – 1) + 3(4) f(4) = f(3) + 3(4) f(4) = f(4) = 39 The 4th term of the sequence is 39.

Rational Equations 8x(x + 12) = 2x(2x + 4) 8x2 + 96x = 4x2 + 8x
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. 8x(x + 12) = 2x(2x + 4) 8x2 + 96x = 4x2 + 8x 4x2 + 88x = 0 4x(x + 11) = 0 4x = x + 11 = 0 x = x = -11 FOO FOO Reject 0 because it makes the equation undefined. Solution: x = -11

Work Word Problems Suppose 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 Job Completed per Hour (rate) Combined Labor 1st painter: 12 hours 1st painter: per hour Equation: 2nd painter: 8 hours 2nd painter: per hour Together: x hours Together: per hour Together, the painters can complete the job in 4.8 hours (just under 5 hours).

Work Word Problems One 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 Job Completed per Hour (rate) Combined Labor Pipe A: x hours Pipe A: per hour Equation: Pipe B: 1.25x Pipe B: per hour Together: 5 hours Together: per hour It takes the fast pipe 9 hours. It takes the slow pipe hours (9 X 1.25). It would take 11 hours and 15 minutes to fill the pool if only the slow pipe is used.

Profit Word Problems Profit = 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. Write a function, C(b), to model the company’s weekly costs for producing b biscotti cookies. What is the total weekly cost in dollars if the company produces 5,000 biscotti cookies. 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) = b C(b) = (5000) = \$ 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 = 1000 The company will earn \$1000

Age Word Problems Sue 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 = (x + 6) – 11 x + 11 = 2(x + 6) – 11 x + 11 = 2x + 12 – 11 x + 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 Problems Joe has \$ 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) = or 5x + 10(7 + x) = 250 5x x = 250 15x + 70 = 250 15x = 180 x = 12 Joe has 12 nickels and 19 dimes. Check: 12 nickels = 60 cents 19 dimes = \$1.90 Total: \$ \$0.60 = \$2.50 Remember: (Value)(Quantity) = Total value of Coins \$ per coin x how many = total \$

Ratio Word Problems The 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) 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 1:2:3 Let x = the amount of almonds Let 2x = the amount of walnuts Let 3x = the amount of raisins x + 2x + 3x = 4 3x + 7x = 180 10x = 180 x = 18 The larger angle measures 126○

Consecutive Integer Word Problems
Find two consecutive integers whose sum is -35. x: 1st consecutive integer x + 1: 2nd consecutive integer -18 -17 ( ) Remember: Consecutive integers count by 1’s Ex: x, x+1, x+2, x+3…. Consecutive odd or even integers count by 2’s Ex: x, x+2, x+4, x+6… Negative integers doesn’t change anything

Statistics Quantitative 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

Statistics A 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. You ask every fifth person leaving a Taylor Swift concert about the type of music they listen to. You ask every fifth person leaving the local mall about the type of music they listen to. Biased Unbiased