Math 3 final

to convert from degrees to radians

Divide by 180, convert to fraction (MATH-->1) and add a pi sign to the numerator

(x-5) is a factor of x^3-7x^2+7x+15. Find the remaining factors.

Divide the equation (x^3-7x^2+7x+15) by the one factor (x-5) that you have and then factor the answer (x^2-2x-3) to get the other two (x-3)(x+1).

Complex Fractions (addition or subtraction)

Ex. ((x-2/x)-(2/2x))/((3/5)-(1/3x)) 1. Find the LCM of every denominator 2. Multiply every term by the LCM and simplify like before 3. Remaining terms on top are the numerator and on bottom are the denominator.

Simplifying logarithms

Ex. LOG2 8 - LOG2 4 (subtraction sign means you divide) 8/4= 2 so it's LOG2 2 Ex. LOG5 2 + LOG5 6 (addition sign means multiply) 2*6= 12 so it's LOG5 12 When simplifying move any number in front of the log to the back as an exponent, and when expanding a log you move the exponent to the front.

Rational equations

Ex. X+1/5=2x-3/6 Plug into Y1 and Y2 and use 2nd+trace (intercept) to get answer (x-value). Ex. Ana does volunteer work in 4 days. Fran does it in 6 days. How long would it take working together? 1/4+1/6=1/x Y1= 1/4+1/6 Y2= 1/x 2.4 hours

Determine the center and radius of a circle using completing the square

Ex. X^2 +Y^2 +8x -4y +30 =0 1. Move the number without a variable to the other side (30) 2. Separate the x and y variables and leave boxes at the end of each (X^2+8x+ Y^2-4y+ =-30) 3. Divide the middle term by 2 and square it to get the number the box (do this for the x's and y's) 4. Add the numbers in the boxes to the other side as well 5. Factor the X's and Y's The opposite of the factors is the center and the number on the other side is the radius squared.

Period, amplitude, midline

Ex. Y= 4cos3x+1 4 is the amplitude (distance up and down from the midline) 3 is for the period (360/3= 120) 120 is the period 1 is the midline distance above or below the x-axis that the graph is centered around

Write polynomials in factored form

Factor using the GCF and get the zeros Ex. x^3+7x^2+10x= x(x^2+7x+10x)= x(x+2)(x+5) zeros= 0,-2,-5

Literal Equations

First try to move the addition or subtraction to the other side and then divide any multiplication

A firework is shot from a height of 610ft, initial upward velocity of 88 ft/s. h= -16t^2 +88t+610

How long to max height? 2.75 seconds Max height= 731 ft How long until it hits the ground? 9.51 sec 1. Plug into Y= 2. Do Zoom 0 if you can't see it on the graph 3. Use 2nd+trace—> Maximum to find vertex. - (first number in vertex is the seconds to max height and second number is the max height) 4. Use 2nd+trace—> Zero to find the end seconds where it intersects with the x-axis.

Solving Natural Logs

Natural logs use Ln and e (2nd+Ln on calc) 1. Move the exponent to the front in front of the Ln after subtracting any constants from the left side or dividing by any number attached to the Ln/e through multiplication. 2. Divide both sides by the exponent 3. Move the Ln to the other side like the base of a log. Instead of writing Ln, write e on the other side and make the number the exponent. 4. Use calculator to find the value of the e and solve. If there was an exponent of 2 in the original equation then put a +/- in front of the value of e and solve for both. If you get e alone on one side and you are trying to solve for the exponent of the e, then you can multiply both sides by Ln because the Ln will cancel out the e. *Only use this as a last resort if you can't solve by putting the equation in Y1 and Y2 and getting an intercept.*

Writing expressions as polynomials

expression--> polynomial = box method Ex. (x+1)(x+2)(x+4)= do box for (x+1)(x+2) and get x^2+3x+2, and then do that times (x+4) in a box and get x^3+7x^2+14x+8

Cubic function

f(x)=x^3 3 lines on the graph= 3 zeros

95% confidence interval

mean +/-1.96 x standard deviation/√ sample 1. find 1.96*(s/√n) 2. add and subtract it from the mean

Arc length

(measure of the central angle/360) x (2 pi r)

Arc sectors (shaded area)

(measure of the central angle/360) x (r^2 pi) If there is a triangle inside of the circle then find the area of the triangle (bh/2) and subtract it from the area of the whole circle (use 360 as the central angle) to get the area of the excess space.

Equation of a circle

(x-h)²+(y-k)²=r², (h, k) is the center, r is the radius. Can be used to write an equation of a circle given center and radius.

sampling error (margin of error)

+/- 1/√n N is the sample Find the decimal and convert to a percent Add and subtract the percent from the percentage in the problem to get the interval

Inscribed angles

- *Vertex is on the edge of the circle* - inscribed angle is 1/2 of the corresponding arc - Opposite interior angles are equal

Parallelograms

- 2 sets of parallel sides - opposite sides are congruent (equal) - opposite angles are congruent - adjacent angles are supplementary - diagonals bisect each other

Graph behaviors

- Each "hump" in the parabola has 2 zeros (one on either side), either real or imaginary. *If the parabola intersects the x-axis then that is a real zero*, if it does not then the zeros are imaginary. *Imaginary zeros travel in pairs*. - If the arrow farthest to the right is pointing down then it is negative and if it is pointing up then it is positive. - If both arrows are pointing in the same direction then it is even, if not, odd. - know how to do end behavior

If it asks you to find the SA of a half sphere on top of a cylinder

- Use 2pi(r^2) for the surface area of the sphere (it would be 3pi(r^2) if the half sphere had the bottom circle showing) - use the regular 2r(r+h)pi to find to find the SA of the cylinder but subtract the area of one base If solving for volume- - find the volume of the cylinder (Bh) - find the volume of a sphere and divide it by 2 for the half sphere (4/3pi(r^3))

isosceles trapezoid

- angles on the top are equal and angles on the bottom are equal - angles on the sides are supplementary - legs are congruent (equal)

Long division of polynomials

- divide the first number (largest term) in the dividend by the first number (largest term) in the divisor and put on top of the quotient "house." Multiply that number by the whole divisor and then put under the dividend. Change the signs and find the difference. Repeat. *** Remember to put in zeros for any powers that are not in the dividend*. Ex. X^3-1, write as x^3+0x^2+0x-1

To graph radians/degrees

- if it's a radian then convert it to a degree - if it's a degree that is negative then add 360 until you get a positive co-terminal angle. If it's over 360 then subtract 360 to get a co-terminal (line on graph will look like a swirl- one rotation for each time 360 was subtracted) - if the degree was originally positive then the arrow moves counter-clockwise - if the degree was originally negative then the arrow moves clockwise

Tangent triangles

- line through the center from the middle of the circle would be the hypotenuse - radius and tangent lines are sides

Rhombus

- parallelogram with four congruent sides - diagonals bisect corners - diagonals intersect to form four 90 degree angles - opposite angles are equal - angles on the same side are supplementary

Rectangle

- parallelogram with four right angles (not angle bisectors) - diagonals are congruent - made of four isosceles triangles (2 matching angles)

Chords of circles

-A chord connects two points inside a circle -The radius or diameter acts as the hypotenuse -Use Pythagorean's theorem

Trapezoid

-A quadrilateral with exactly one pair of -parallel sides (bases) the other two are legs. -A median runs through the middle and connects the midpoints of the legs. -The length of the median is 1/2 the sum of the bases.

Horizontal and vertical asymptotes

-An asymptote is a line that will never touch zero -Vertical asymptote can be found in the denominator by setting it equal to zero (may have to do this after factoring, so there can be more than one) -Horizontal asymptote is either located to the right of the fraction (added or subtracted) or found by dividing the first term of the numerator and denominator *(a higher degree on the bottom means 0 and a higher degree in the top means none)*. -A hole is created when a term on the top and bottom matches and cancels out (Ex. (X+2) would be a hole of -2)

Arcs and central angles

-an arc is the portion of the outside circle that lies within an angle -a circle totals 360 degrees all the way around - a straight line running through the center of a circle is 180 degrees

Inscribed quadrilateral

-opposite angles are supplementary -outside circle adds up to 360 -inscribed angles (arc is 2X the corresponding angle)

Factoring with GCF

1. Bring the GCF outside of the parenthesis for the whole equation 2. Factor the parenthesis To get the factors of a number enter the number/x into Y= and go to the table. Whole numbers are factors.

Radian rotated by a degree

1. Convert the radian to a degree 2. If counterclockwise then add the degree that it's rotated by and if its clockwise then subtract 3. Convert back to radian

Expanding logarithms

1. Determine how many logs you need. Each variable or number is a separate log. If it is in parenthesis then it is one part. 2. Write "log" for each part of the expression (Ex. X^3 Y^4 would be 2 parts/logs) 3. Move any numbers/variables to the back of their logs and move any exponents to the front of their logs. 4. Put a plus or minus sign in front of each one depending on what they were doing in the original expression.

Simplifying expressions- Multiplication

1. Factor the top and bottom. 2. State restrictions (All factors in the bottom/denominators are restrictions Ex. (x+2) has a restriction of -2. If there is an x outside a parenthesis then the restriction for it is 0). 3. Simplify (Cancel out the factors that are the same on the top and bottom and carry over everything that was not cancelled out by restrictions).

Standard deviation

1. Find the mean 2. For each number- subtract the mean and square the result 3. Find the mean of the squared numbers 4. Find the square root of that In calc- 1. STAT-->edit 2. STAT-->calc 1 First number in the list is the mean and the last one (with the o-ponytail symbol) is the standard deviation.

Write the equation of a circle using coordinates of the circle (diameter)

1. Find the midpoint if not provided - add the x-values together and divide by two to get the x-value of the midpoint (do the same for y) 2. Use midpoint to plug into circle equation 3. Plug one of the other coordinates back in to the equation to find the radius squared.

Solving exponential equations with logarithms

1. Insert your logs in front of each side of the equation. 2. Move the exponent to the front. 3. In order to isolate the numbers/variables in the front (from the exponent) you divide both sides by the log that it is in front of. 4. The log on one side should cancel out and you use the log button on the calculator to divide the other side. 5. Finish with basic solving.

Solving logarithmic equations

1. Move the base of the log to the other side and make the number on the other side the exponent. (If the log has no base then use 10). 2. Solve If there is a number in front of the logarithm then divide both sides by that number before moving over the log base. When the log base is X- 1. Move X to the other side and make the number on that side the exponent. 2. Multiply the exponent by the reciprocal (both sides).

Simplifying expressions- addition

1. Multiply each side by the LCM of the denominators - cancel out whatever matches the denominator from the LCM on each side. 2. Multiply whatever is left in the LCM on each side by the numerator 3. This is the numerator of your new fraction and the denominator is the LCM 4. Simplify (must be able to divide every number by the same thing) For subtraction just change the sign to addition and put a negative sign in front of the numerator on the right term.

Heron's Formula (area of triangle w/ no 90 degree angle)

1. S= (a+b+c)/2 2. A= √(s(s-a)(s-b)(s-c))

If given points, how do you use your calculator to make a quadratic?

1. STAT 2. Edit 3. Stat —> Calc 5

Synthetic division of polynomials

1. Set the divisor equal to zero and solve. Put that number in a box. 2. Take each number and line them up (keep any signs but don't include Xs or powers). Draw a line underneath 3. Carry the first number down below the line. 4. Multiply it by the number in the box and put it under the next number. Find the difference with the and put it below the line. 5. Repeat until you get the remainder. If the number in the box is a fraction then divide the answer by the denominator (don't divide the remainder). Ex. 3x^3 + 8x^2 - 9x + 2 divided by x-1, the number in the box is what x in the divisor (x-1) equals, (in this case 1).

Bell curve with standard deviation

1. Stat—>enter 2. Stat—>calc—>1 First value in the list is the mean and the last value (o) is the standard deviation. Use them to build the bell curve

Inverse functions

1. Switch x and y 2. Solve for y 3. Put +/- in front of square roots 4. Replace y with (f)^-1

Surface area of a prism

2(area of base) + (perimeter of base x height) 2B + ph

Pi radians

2pi radians= 360 Pi radians= 180

Surface area of a cylinder

2r(r+h)pi

Surface area of a sphere

4(pi)r^2

Volume of a sphere

4/3 pi r^3

Surface area of a cube

6s^2

Square

A quadrilateral with 4 sides that are equal and have all 90 degree angles Each angle is bisected into 45 degree angles Diagonals are equal and form 90 degree angles in the middle

Area of a regular polygon

A= (apothem x perimeter)/2

Area of a kite

A= (d1 x d2)/2

Area of a triangle (w/ 90 degree angle)

A= bxh/2

Area of a trapezoid

A=(b1+b2)h/2

Area of an equilateral triangle

A=(√3/4)s^2

Continuously compounded interest

A=Pe^rt P= money in account e= e (press 2nd+LN) r= annual interest rate (decimal form) t= time in years Ex. $1,300. Rate of 4.3% After 3 years? 1,300e^.043*3 = $1,478.99

Area of a parallelogram

A=bh

Area of a rectangle

A=lw perimeter= 2(l+w)

Area of a square

A=s² perimeter= 4s

Area and circumference of a circle

Area: pi x r^2 Circumference: 2 x pi x r

Surface area of a pyramid

B + Pl/2 *l is the slant height

Volume of a prism/cylinder

Bh (B is the area of the base)

Volume of a pyramid/cone

Bh/3

Growth and decay equations

For growth or decay A= P(1 +/- r/n)^nt P is the initial amount 1- if it is decay and 1+ is it is growth r is the percentage of growth/decay (must be a decimal) n is the times per year t is the years Ex. Car is $25,000. Depreciates by 2.5% After 7 years? 25,000(1-.025/1)^7(1) Ex. Wants to save $5,000 to buy a car in 6 years. 4% interest quarterly. Initial investment? 5000/(1+.04/4)^6*4 = $3,937.83 (solving for P) If you have a list and are told to find the rate of growth/decay r= final value-initial value/initial value

Chords with distance lines

If a circle has chords (possibly shaped like a triangle) with distance lines to the center- - equal distance lines mean the same distance from the center and equal length chords - longer the distance line the shorter the chord bc it is farther from the center

Interior chords circle

If two chords of a circle intersect, then the product of the parts of one chord equals the product of the parts of the other chord.

Interior and exterior secants circle

If two secants intersect on the exterior of a circle, the product of the length of the entire secant times its exterior part is equal to the other. a(a+b)=c(c+d)

Simplifying expressions- Division

Keep the first fraction the same, change the sign from division to multiplication, and flip the second fraction before factoring. Check the bottom and the top right side for restrictions.

find inside angle with chord arcs

Larger arc + smaller arc /2= inner angle a+b/2

Logarithmic functions

Logarithmic functions are the inverse of an exponential function. y= b^x then LOGb y= x 25=5^2 then LOG5 25= 2 Adding/Subtracting/Exponent properties LOGb M(N) —> LOGb M + LOGb N LOGb M/N --> LOGb M - LOGb N LOGb M^x --> xLOGb M *** They must have the same bases*

sine, cosine, tangent

SOH CAH TOA, use box, circle, box/box to set up problems (use inverse key for finding angles) *calculator must be in degree mode*

Cosine graphs

Shaped like U, start at the top of the amplitude

Sine graphs

Shaped like sideways S, start on the midline

Tangents of circles

The hypotenuse is made of the radius and another number since it leaves the circle. The radius is one side of the triangle and makes a 90 degree angle with the tangent.

Parent function

The simplest, most general function in a family of functions. Keep any sign/number on the outside and exponents (Example -(x-3)^2, parent is -x^2)

Exterior tangent circle

The square of the tangent is equal to the product of the exterior part of the secant times the entire chord length. a^2=b(b+c)

piecewise functions

Use Y= to find table of each function and graph according to the parameters of each line. Less than/greater than- open circle Less than or equal to/greater than or equal to= closed circle

Factoring without GCF

Use box method 1. Draw a box and put the first term in the upper left-hand side and the last term in the lower right-hand side. 2. Multiply the first and the last term together 3. Find factors of that number that can add or subtract to get the middle term. (You know what signs they will have to be depending on the signs in the original problem. -Negative before the last term= +,- in factors -Positive before the last term= +,+ if both terms are positive and -,- if the middle term is negative). 4. Put the factors into the two spots in the box (be sure to add an x to each) 5. Pull out the GCF of each row and column (4 total). The numbers/variables on the same side are in the same parenthesis.

Dividing polynomials

Use synthetic or long division - Synthetic is fast - long division works for any polynomials

Unit circle

Use to find the value of a degree or radian (the first value in the parenthesis is the cosine and the second is the sine)

Horizontal/Vertical translations

Y= +/- A(X +/- H) +/- K Positive A= graph opens up Negative A= graph opens down Positive H= graph shifts left Negative H= graph shifts right Positive K= shifts up Negative K= shifts down A over 1 = compression A under 1 (fraction) = stretch (rise/run)

median of a triangle

a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side

Laws of Sines

a/SinA=b/SinB=c/SinC

Lines of Cosines

a^2=b^2+c^2-2bc(cosA) b^2=a^2+c^2-2ac(cosB) c^2=a^2+b^2-2ab(cosC)

to convert from radians to degrees

multiply fraction by 180 (excluding the pi sign)

Surface area of a cone

r(r+l)pi

Surface area of a regular tetrahedron (pyramid w/ all equilateral sides)

s^2 √3

Volume of a cube

s^3

angles formed by tangents and secants

the larger arc minus the smaller arc divided by 2 equals the inner angle that is outside of the circle

incenter of a triangle

the point of concurrency of the angle bisectors of a triangle The distance from the incenter to each side is the same *(PY=PX=PZ)*

circumcenter of a triangle

the point of concurrency of the perpendicular bisectors of a triangle -a perpendicular bisector is a midpoint at a 90 degree angle *(don't exit at vertexes)* - the distance from the center to the vertexes is equal

orthocenter of a triangle

the point of concurrency of the three altitudes of a triangle -An altitude must *come from the vertex* of a triangle and make a 90 degree perpendicular angle

centroid of a triangle

the point of concurrency of the three medians of a triangle Cuts each median into two pieces, *the shorter piece is half the length of it's longer counterpart.*

coterminal angles

two angles in standard position that have the same terminal side Add 360 to a degree to get the positive co-terminal and subtract 360 to get the negative

Quadratic formula

x = -b ± √(b² - 4ac)/2a Use to find the zeros of a quadratic

45-45-90 triangle

x, x, x√2

30-60-90 triangle

x, x√3, 2x

Quartic function

x^4 4 lines on the graph= 4 zeros

Quintic function

x^5 5 lines on the graph= 5 zeros

Exponential growth

y= ab^x A is the initial amount B is the growth factor (over 1, decay is under 1) X is the time in years or amount of time that it occurs To find using points or table- 1. Enter in stat --> edit 2. Stat --> calc --> ExpReg(0) To graph 1. Put in Y= If you want to see the points do 2nd+Y= and turn on the stat plot. Press Zoom 9 (zoom stat) to see points, you can also see them in the table. 2nd+4 clears list without resetting There is one pregnant guppie. The number of guppies triples every 4 months. How many are there in 2 years? - Initial is one (A) - Growth factor is tripling, so 3 (B). - There are 24 months in 2 years, so 24/4=6 times in 2 years (X). 1(3)^6= 729 guppies If you look at the table for the graph then it will show how many guppies there are each time it triples, but remember that it triples 3 times a year.

Graphing quadratics

y=ax^2+bx+c 1. Put in y= and find the vertex (2nd+graph) - or press 2nd+trace and choose maximum or minimum 2. Plot vertex 3. Choose point from table to make line 4. Graph points and draw -b/2a= axis of symmetry

Writing a polynomial function from zeros

zeros--> expression --> polynomial Zeros are the opposite of the factors in a polynomial. So write an expression using the opposite signs and then use the box method. Ex. zeros= -2,3,3 1. make expression= (x+2)(x-3)(x-3) 2. use box method and get x^3-4x^2-3x+18 Zeros with √ must be written with positive and negative sign. Ex. √6 = (x-√6)(x+√6) Using a box for these will cancel out the middle terms anyway, so you can simplify to x^2-√36, same with imaginary zeros Ex. 3i=(x-3i)(x+3i)= x^2+9