Algebra

How to find the sum of the measures of the interior angles of an n-sided polygon

1) if a polygon has n sides, it can be divided into n-2 triangles 2) the sum of the measures of the interior angles of a triangle is 180° 3) so, the sum of the measures of the interior angles of an n-sided polygon is (n-2)(180°)

how to use the Pythagorean theorem to find the length of one side of a triangle (if the other two sides are known)

1) leg A has length 5 and the hypotenuse has length 8 2) 8²=5²+x² 64=25+x² 39=x² 3) take sq rt ...x = √39 ≈6.2

a quadratic function defined by g(x)=x² where the quadratic equation is y=x² a quadratic function defined by f(x)= -1/2 x+1 where the linear equation is y=-1/2 x+1 find the points on the graph algebraically

1) set g(x) = f(x) ... x² = -1/2 x+1 2) solve for x x² = -1/2 x+1 ... x² + 1/2 x-1 = 0 2x²+x-2=0 3) solution is (-1±√1+16)/4 which represents the x coordinates x= (-1+√17)/4 ≈.78 and x=(-1-√17)/4 ≈-1.28 4) use those input values to find the corresponding y coordinates (can use f or g) g(-1+√17 / 4)=(-1+√17 / 4)² ≈ .61 and g(-1-√17 / 4)=g(-1-√17 / 4)²≈1.64 5) thus, the two intersection points can be approximated by (.78, .61) and (-1.28, 1.64)

machine A takes 3 hr to produce a batch, it can produce 1/3 in 1 hour. machine B can produce 1/2 of the batch in 1 hr. let x = hours it takes both machines, working simultaneously, to produce the batch, the two machines will produce 1/x of the job in 1 hour.

1/3+1/2=1/x *find common denominator for first two fractions 2/6+3/6=1/x 5/6=1/x *flip both fractions 6/5=x working together, the machines will take 6/5 hours (1 hour 12 min) to produce a batch

steps to solve a linear equation in one variable

11x-4-8x = 2(x+4)-2x *combine like terms 3x-4 = 2x+8-2x *simplify 3x-4=8 *add 4 to both sides 3x=12 *divide both sides by 3 x=4

identify the terms: 2x²+7x-5

2 = the coefficient of the term 2x² 7 = the coefficient of the term 7x -5 = the constant term

a quadratic equation in one variable

20y²+6y-17=0

linear equation examples

2x+1=7x 10x-9y-z=3

solve quadratic equation by factoring 2x²-x-6=0

2x²-x-6=0 => (2x+3)(x-2)=0 *when a product is =0 at least one of the factors must be 0 ...so either 2x+3=0 pr x-2=0 2x+3 = 0 ... 2x=-3 ... x=-3/2 OR x-2=0 ... x=2 solutions are: -3/2 and 2

what is π

3.14 or 22/7

a linear equation in one variable

3x+5=-2

solve inequality 4x+9 / 11 < 5

4x+9 / 11 < 5 *multiply both sides by 11 4x+9<55 *subtract 9 from both sides 4x<46 *divide both sides by 4 x<11.5 solution set of original inequality consists of all real numbers less than 11.5

exponent rule 1 x^-a = 1/x^a

4⁻³ = 1/4³ = 1/64 x⁻⁹=1/x⁹ 1/2⁻⁴ = 2⁴

right triangles and the Pythagorean theorem

PythagThm states that in a right triangle the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs (EF)²=(DE)²+(DF)²

find area of sector of a circle central angle = 50° radius = 5

S/25π = 50/360 S = (50/360)(25π) = 125π/36 ≈ (125)(3.14)/36 ≈ 10.9

how can like terms be combined?

by adding their coefficients w³z + 5z² - z² +6 = w³z + 4z² + 6

congruent triangles

the vertices can be matched up so the corresponding angles and sides are congruent

two lines are perpindicular if...

their slopes are negative reciprocals of each other ex: y=2x+5 is perpendicular to y= -1/2x+9

diameter

twice the radius any chord that passes through the center of the circle

determining if triangles are congruent 3

two angles and the included side of one triangle are congruent to two angles and the included side of another

a system of equation

two equations with the same variables

graphing functions explained

use the x-axis for the input and the y-axis for the output y = f(x)

use substitution to solve 4x + 3y = 13 x + 2y = 2

x + 2y = 2 *express x in terms of y, as x= 2 - 2y; then substitute 2 - 2y for x in the first equation to fine y's value 4(2 - 2y) + 3y = 13 8 - 8y + 3y = 13 *subtract 8 from both sides -8y + 3y = 5 *combine like terms -5y = 5 *divide both sides by -5 y = -1 *use -1 as y in either equation to find x x + 2(-1) = 2 x - 2 = 2 x = 4 ANSWER: (4,-1)

solve quadratic equation 2x² - x - 6 = 0 a=2, b=-1, c=-6

x = -(-1)±√(-1)²-4(2)(-6) / 2(2) x = 1±√49 / 4 x = 1±7 / 4 *solve this using + and - x = 2 or -3/2

quadratic formula

x = -b±√b²-4ac / 2a * can use either + or -

in a driving competition, Jeff and Dennis drove the same course at avg 51 mph and 54 mph, respectively. if it took Jeff 40 min to drie the course, how long did it take Dennis?

x = min for Dennis to drive the course d = distance, r = product of the rate in mph t = hours (d=rt) 40 min = 40/60 of 1 hour distance traveled by Jeff = (51)(40/60) distance traveled by Dennis = 54(x/60) (51)(40/60)=(54)(x/60) (51)(40)=54x x=(51)(40)/54 ≈37.8 min

a mixture of 12oz of vinegar and oil is 40% vinegar, where all of the measurements are by weight. how many oz of oil must be added to the mixture to produce a new mixture that is only 25% vinegar?

x = oz of oil to be added (.40)(12)/ 12+x = .25 (.40)(12)=(12+x)(.25) 4.8=3+.25x 1.8=.25x 7.2=x

Ellen has received the following scores on 3 exams: 82, 74, 90. what score will ellen need to recieve on the next exam so the average score of the 4 exams will be 85?

x = score of 4th exam 82+74+90+x / 4 => 246+x / 4 = 85 multiply both sides by 4 => 246+x=340 x=94

a linear equation in two variables

x-3y=10

simplify x²-9 / 4x-12 with an identity

x²-9 / 4x-12 (x+3)(x-3) / 4(x-3) x+3/4 (where x≠3)

Pythagorean theorem to find the ratios of the lengths of the sides of special right triangles - isosceles right triangle two congruent angles = 45°

y²=x²+x² y²=2x² y=√2 ratio is 1 to 1 to √2

define slope of a line passing through Q(x₁,y₁) and R(x₂,y₂) where x₁≠x₂

y₂-y₁ / x₂-x₁

if $10,000 is invested at a simple annual interest rate of 6% what is the value of the investment after half a year?

$10,000(1+.06(1/2)) $10,000(1.03) = $10,3000

a college student expects to ear at least $1,000 in interest on an initial investmetn of $20,000. if the money is invested for one year at interest compounded quarterly, what is the least annual interest rate that would achieve the goal?

$20,000(1+r/400)⁴ ≥ $21,000 (1+r/400)⁴≥1.05 *taking the positive 4th root of each side here preseves the direction of the inequality 1+r/400≥√1.05 - 1) *to compute the 4th root, take the sqrt twice r≥400(⁴√1.05 - 1) = 400(√√1.05 - 1) ≈ 4.91

exponent rule 5 (x^a)(y^a)=(xy)a

(2³)(3³)=6³=216 (10z)³=10³z³=1000z³

exponent rule 7 (x^a)^b = x^ab

(2⁵)² = 2¹⁰ = 1024 (3y⁶)² = (3²)(y⁶)² = 9y¹²

exponent rule 6 (x/y)^a = x^a/y^a

(3/4)² = 3²/4² = 9/16 (r/4t)³ = r³/64t³

exponent rule 2 (x^a)(x^b) = x^a+b

(3²)(3⁴) = 3²⁺⁴ = 3⁶ = 729 (y³)(y⁻¹) = y²

identity examples

(a+b)² = a+2ab+b² (a-b)³ = a³-3a²b+3ab²-b³ a²-b²= (a+b)(a-b)

find the length of an arc

*remember the ratio of the length of an arc to the circumference = the ratio of the degree measure of the arc to 360° length of arc ABC / 10π = 50/360 length of arc ABC = (50/360)(10π) = 25π/18 ≈ (25)(3.14) / 18 ≈ 4.4

describe reflections/symmetries about axis on a graph

- P' is a reflection of P about the x-axis, or P' and P are symmetric about the x-axis - P'' is a reflection of P about the y axis, or P'' and P are symmetric about the y-axis - P''' is the reflection of P about the origin, or P''' and P are symmetric about the origin

how to find the height of a triangle

- any side of triangle can be the base - the height that corresponds to the base is the perpendicular line segment from the opposite vertex to the base

isosceles triangles

- at least two congruent sides - the angles opposite the two congruent sides are also congruent - the converse is also true ∠A and ∠C measure 50°, then AB = BC 50 + 50 + x = 180, the measure of ∠B is 80°

to multiply two algebraic expressions

- each term of the first expression is multiplied by each term of the second expression - results are added (x+2)(3x-7) = x(3x)+x(-7)+2(3x)+2(-7) = 3x²-7x+6x-14 = 3x²-x-14

Pythagorean theorem to find the ratios of the lengths of the sides of special right triangles - the 30°-60°-90° right triangle

- the length of the shortest side x is 1/2 the length of the longest side 2x; thus this is half an equilateral triangle x²+y²=(2x)² x²+y²=4x² y²=4x² - x² y²=3x² y=√3x ratio of x to y is 1 to √3, so the ratio of the 3 sides is 1 to √3 to 2

equilateral triangles

- three congruent sides - measures of the interior angles are all equal, measuring 60° each

solve inequality -3x+5≤17

-3x+5≤17 *subtract 5 from both sides -3x≤12 *reverse direction of inequality when dividing by -3 -3x/3≥12/3 x≥-4 the solution set of the original inequality consists of all real numbers greater than or equal to -4

find the slope, x and y intercepts of the line passing through Q(-2,-3) and R(4,1.5)

1) 1.5 - (-3) / 4 - (-2) = 4.5/6 = .75 2) y=.75x+b -3=(.75)(-2)+b b= -3+(.75)(2) b= -1.5 equation of line QR is y=.75x-1.5 3) set y to 0 0 = .75x-1.5 1.5=.75x x= 1.5/.75 x=2

exponent rule 3 x^a / x^b = x^a-b = 1/x^b-a

5⁷/5⁴ = 5⁷⁻⁴ = 5³ = 125 t³/t⁸ = t⁻⁵ = 1/t⁵

exponent rule 4 x⁰=1

7⁰=1 (-3)⁰=1

area of trapezoid

A = 1/2(b₁+b₂)(h) ex: bases of length 10 & 18, height of 7.5 1/2(10+18)(7.5) = 105

parallelogram area

A = bh -any side can be base, the height corresponding to the base is the perpendicular line segment from any point of a base to the opposite side

area of triangle formula

A = bh/2 b = base h= height

are of circle

πr² if r = 5 ... π(5)² = 25π

circumference formula

C/d=π or C = 2πr

if an amount P is to be invested at an annual interest rate of 3.5% compounded annually, what should be the value of P so that the value of the investment is $1,000 at the end of 3 years?

P(1+0.035)³ = $1,000 *divide both sides by (1+0.035)³ P= $1,000 / (1+0.035)³ ≈ $901.94

similar triangles

if their vertices can be matched up so that the corresponding angles are congruent, or the lengths of corresponding sides have the same ratio (scale factor of similarity) AB/DE = BC/EF = AC/DF ... by cross multiplication we obtain the other proportion such as AB/BC = DE/EF

Simple Interest formula

V = P(1+ rt/100) - if the amount P is invested at an annual interest rate of r percent, then the value of V of the investment are the end of t years is given

Compound interest formula (annually)

V = P(1+r/100)^t -if the amount P is invested at an annual interest rate of r percent, compounded annually, then the value of V of the investment at the end of t years is given

Compound interest formula (x times per year)

V = P(1+r/100n)^t -if the amount P is invested at an annualy interest rate of r percent, compounded n times per year, then the value of the investment at the end of t years is given

determining if triangles are congruent 2

if two sides and the included angle of one triangle are congruent to two sides and the included angle of another

tangent to a circle

a line that intersects the circle at exactly one point (point of tangency) - if a line is tangent to a circle then a radius drawn to the point of tangency is perpendicular to the tangent line; also if a line is perpendicular to a radius at its endpoint on a circle then the line is a tangent to the circle at that endpoint

the coefficient of a term is

a number that is multiplied by variables

sector of a circle

a region bounded by an arc and two radii - has a central angle, found from where the two radii extend from the circles center point - the ratio of the area of a sector of a circle to the area of the entire circle is equal to the ratio of the degree measure of its arc to 360°

what is an identity?

a statement of equality between two algebraic expressions that is true for all possible values of the variables involved

a constant term is

a term that has no variable

function

an algebraic expression in one variable can be used to define a function of that variable f(x) is the value of f at x (x can be substituted by a number)

central angle of a circle

an angle with its vertex at the center of the circle

the solution to a linear equation in two variables is...

an ordered pair that makes the equation true when the values of x and y are substituted into the equation

chord

any line segment joining two points on the circle

linear equation in two variables example

ax + by = c a, b, c are real numbers a & b are not both zero

quadratic equation example

ax² + bx + c = 0 a, b, c are real numbers a≠0

a linear equation

involves one or more variables in which each term in the equation a constant term or variable multiplied by a coefficient *no variables are multiplied together or raised to a power greater than 1

the domain of a function

is the set of all permissible inputs -all permissible values of the x variable

how to use the Pythagorean theorum to find distance between two points on a graph Q(-2,-3) and R(4, 1.5)

construct a right triangle, note the two shorter sides have lengths QS = 4 - (-2) = 6 and RS = 1.5 - (-3) = 4.5 line segment QR is the hypotenuse AR = √6²+4.5² = ²56.25 = 7.5

define y=mx+b

it is a straight line in the xy-plane, m = slope of the line and b = y-intercept

radius

distance from center point of circle to outer point any line segment joining a point on the circle to the center

let f be the function defined by f(x) = 2x / x-6

f is not defiend as x = 6 because 12/0 is undefined the domain of f consists of all real numbers EXCEPT 6

solve quadratic equation by factoring 5x²+3x-2=0

factors into (5x-2)(x+1)=0 therefore... 5x-2=0 ... x=2/5 OR x+1=0 ... x=-1 solutions are: 2/5 and -1

what is needed to solve a system of two equations

finding an ordered pair that satisfies both equations in the system

let g be the function defined by f(x)=x³ + √x+2 - 10

g(x) is not a real number if x < -2 the domain of g consists of all real numbers x such that x≥-2

congruent circles

have equal radii

a polygon inscribed in a circle

if all its vertices lie on the circle - the circle will appear to be around the polygon

the length of each side of a triangle must be

less than the sum of the lengths of the other two sides - a triangle cannot have lengths 4, 7, 12 because 12 is greater than 4+7

solving systems of linear equations using elimination

make the coefficients of one variable the same in both equations so that one variable can be eliminated by adding the equations or by subtracting one from the other

measure of an arc

measure of its central angle, which is the angle formed by two radii that connect the center of the circle to the two endpoints of the arc

use elimination to solve 4x + 3y = 13 x + 2y = 2

multiply both sides of x + 2y = 2 by 4 4(x+2y)=4(2) => 4x+8y=8 (now both equations have same coefficient of x) 4x+3y=13 SUBTRACT 4x+8y=8 -5y=5 y=-1 ANSWER: (4,-1)

Find the sum of the interior angles of a 4 sided polygon

n = 4 (4-2)(180°) = 360°

Find the sum of the interior angles of a hexagon

n = 6 (6-2)(180°) = 720°

Find the measures of each angle of a regular octagon

n = 8 (8-2)(180°) = 1080° sum of all interior angles 1080°/8 = 135° each angle

solving systems of linear equations using substitution

one equation is manipulated to express one variable in terms of the other, then the expression is substituted in the other equation

arc

part of the circle containing the two points on a circle and all the points between them

define rise over run

rise is the change in y when moving from Q to R run is the change in x when moving from Q to R

when both sides of the inequality are multiplied or divided by the same nonzero constant...

the direction of the inequality is preserved if the constant is positive but the direction is reversed if the constant is negative regardless, it is equivalent to the original

let h be the function defined by h(x)= |x|

the domain of h is the set of all real numbers also h(x) = h(-x) for all real numbers x, which reflects the property that on the number line the distance between x and 0 is the same as the distance between -x and 0

to produce an equivalent equation, 1)same constant is added or subtracted form both sides and 2)both sides are multiplied or divided by the same nonzero constant so...

the equality is preserved and the new equation is equivalent to the original

simultaneous equations

the equations in the system of equation

why is the slope of a vertical line not defined?

the run is 0 the equation of every vertical line has x = a, where a is the x-intercept

determining if triangles are congruent 1

the three sides of one triangle are congruent to the three sides of another

why are 5z² and z² like terms?

the variables and the variables' exponents are the same