Aleks Study Guide

Set builder/interval notation -if x>4

(4,∞), no bracket because there is no line under the > sign

Multiplication involving binomials and trinomials in 2 variables

(4v-2w+9)(7v+3w-4) 4v(7v+3w-4)-2w(7v+3w-4)+9(7v+3w-4) simplify

Circumference Ratios

-wheel is .5m in D and races for 61 m how many times does it turn equation is- # of times spinning x circumfrence =length

Graphing a cubic function

-y=ax³ -(x,y) or (-x,-y)

solving absolute value equation of the form ax+b=cx+d

-|5x-7|=|8-5x| 5x-7=8-5x and 5x-7=-8+5x(no solution)

Square Root Multiplication 2√96x √24

2x4√6x2√6 16√36 16(6)

reference angles trig

2π+fraction or if its an angle 360+angle so if -204 360-204

Solving multistep equations involving a log

4+log5(x+7)=5 log5(x+7)=1 5^1=x+7

Writing Equations of vertical and horizontal lines through a point

4,0 vertical x= 0,4 horizontal y= 2,-7 x=2 y=-7

rational exponents: powers of powers with exponents

multiply with parenthesis put negative value on the bottom

Perimeter involving rectangles and circles

square that has a half circle cut out of side 26x34 c=πd 26π/2 because it is only half of the circle so it would be 26+34+34+26π/2

solving a quadratic inequality written in factored formn

x(x-6)>0 -(x-6)(x+4) 6, -4

Polynomial Long Division

x+6 divided by 5x^2+37x+37 how many times does x go into 5x^2 then multiply 5x(x+6) and subtract from top -how many times does 7x going into x(7) mulitply 7(x+6) and substract from bop and go until you reach a value without x

solving a quad inequality in factored form

(x-3)(x+7)≥0 if it is less than 3 it is negative if it is less than 7 it is negative [ut on number line and write out where it would be positive and negative where the two values line up either(two positives or two negatives) and graph that make sure to keep in mind if circles are filled in our not

graphing a circle given its equation in standard form

(x-h)^2+(y-k)^2=r^2 r=radius h, k is point and graph points equally away using radius to create a circle

amp period and phase shift

-2sin(2x-π/3)-2 amp=2 always positive phase shift=c/b π/3/2=π/6 period 2π/b so 2π/2=π

solving a percent mixture problem using a system of linear equations

-70% fruit juice 95% wants 75% how many pints of existing will be used to make 140 pints x+y=140 .70x+.95y=140(.75) solve for it

Unions and Intersections of Intervals

-U includes both ∩ is where they intersect -the first place they intersect U is the one that includes both

writing expressions as single logs

-adding goes to multiply -substrate goes to divide -numbers in front become exponents -if there is a gcg of log pull it out front such as log7(x^5y^7/w^1/3)

Simplifying a higher root of a whole number

-cube root of 81 √3,√27 3√3

writing equation of a line through two points

-do slope formula y-y/x-x y=mx+b plug in point to y and x to get b

Converting between natural logs and exponential equations

-e^y=8 lne^y=ln8 y=ln8(e and ln cancel each other out) lnx=9 -e^lnx=e^9 x=e^9(ln and e cancel out) -ln5=y e^ln5=e^y 5=e^y

Write an equation for a function after vertical and horizontal shift

-f(x-a) shift right -f(x)+b up

Multiplying rational expressions with quadratics

-factor or take out a gcg to simplify and cross out stuff

solving a distance rate and time problems with linear equations

-flying against jet stream 4830/7 hours and other is 3150 with jet stream in 3 hours (x-y)7 (x+y)3 7(x-y)=4830 . x-y=690 3(x+y)=3150 x+y=1050 y=1050-x y=-x+690 se equal to each other x=870 y=180(jet stream) -motorboat 192km in 6 hours and upstream 348 downstream 6(x+y)=192 6(x-y)=348 set equal and y=stream x=speed

Word Problems with Exponential Growth

-forest is 3900km2 and de3crases by 65 every 8 years -3900(1-.65)^x -population is 290000 and grows by 4.25 8 years 29000(1+.0425)^x

finding values of trig functions given info about an angle

-graph it in the quadrant it is suppossed to be in

graphing a linear inequality in a plane

-if there is no line under > then it is a dashed line

Adding Rational Expressions with a common denominator and binomial numerators

-if they have a common denominator it can all be over one

radical expressions: products/quotients with negative exponents)

-if they have the same base add exponents together

Complex Fractions without variables

-if you have two fractions on top of each other x/y÷g/p x/y × p/g

graph/domain/range of a log

-log4x 4^y=x find points and the negative affects the y -log3(x)-1 -3^y=x shift y down 1 -always write down points first and then apply -logs only have vertical asymptotes get from inside parenthesis log3(x-2) it is -2 -range always -infinity to infinity -inside of parenthesis affect it

using trig to find angles of elevation and depression

-make sure to do arc sin cos and tangent or -1 in calculator

Finding values of trig functions

-remember that quadrants matter to find negatives -sec=1/cos h/a -csc=1/sin h/o -cot=1/tan(cos/sin) a/0

applying quad formula

-remember there are 2 values -don't simplify all the way

Inverse functions

-switch x and y and solve for y again

Volume of a cone

1/3bh b=πr²

coterminal angles

13π/3 divide to get between 4π and 5π so 13π/3-4π=π/3 if negative (-25π/10) between -3π and -2π so -25π/10-(-3π)

Solving a value mixture problem using a linear equation

2 coffees. Type a costs 5.95/lb and b is 4.60 /lb this month uses 4x as many pounds of type b than a with a total of 633. How many pounds of type a 5.95a+4.60b=633 4a=b 5.95a+4.60(4a)=633 at museum child costs 5.6 and adult is 9.3 sold 154 for a total of 1095.5. How many child tickets 5.6c+9.3a=1095.5 c+a=154 a=154-c

Exponentials and log word problems

2600 is compounded annually at 5.25% interest how many years till 5000 in account 2600x1.0525^x=5000 1.0525^x=5000/2600 log1.0525^x=log 5000-log2600/log1.0525 *if you are losing money it is 1-%

Factoring a polynomial by grouping

4w^6+7w^4-24w^2-42 do half of the equation w^4(4w^2+7)-6(4w^2+7) (w^4-6)(4w^2+7)

Least Common Multiple

4x^4 and 6n^3 both going into 12 so 12x^4n^3

exponential equations

9^x+1=27 3^2(x+1)=3^3 2(x+1)=3 and solve for x

Union and Intersection of finite sets

Intersection ∩ Union U an intersection has to be a member of both groups a union has to be a member of either group

union and intersection of intervals

U not overlap ∩ overlap(intersection) -possibility of no intersection if it is filled in it is a bracket

graphing a parabola of form ax^2+bx+c

a(x-h)2+k h,k is vertex y=-x^2+4x-5 y=-(x^2-4x)-5 complete square b/2^2 -4/2^2=4 y=-(x^2-4x+4)-5+4 (4 x -1=-4 so add to end) y=-(x-2)^2-1 so vertex is 2,-1 and plug in other points to get a graph

Finding local max and min of a function/graph

all values at which f has a local min: x values all local min values of f=y values y=shorter expression

sketch graph with addition

always 0 π/2 π 3π/2 2π sin starts at origin so y is 0, 1,0,-1,1 cos 1,0,-1,0,1 3/2(cos(x-π/3) y by 3/2 x set x-π/3=x value and solve for x

sketch graph of sink or cos with multiplication

always 0 π/2 π 3π/2 2π sin starts at origin so y is 0, 1,0,-1,1 cos 1,0,-1,0,1 y=2sin(2/3x) multiply y by 2 multiply x by 3/2

Area involving inscribed figures

area of circle πr^2 area of triangle 1/2bh find areas of circles or triangles and subtract

range of a quadratic function

completing the square yay -3x^2-6x+1 make in form of a(x-h)^2+k vertex is h,k so 3(x^2-2x)+1 (h/2)^2 2/2^2=1 3(x^2-2x+1)+1-3 (1x3) and substract 3(x-1)^2-2 (1,-2) plug in points to graph a is positive so graph opens up range is (-2,∞) if graph opens down it would be (-∞,-2)

trig functions with special angles

cos=x value sin=y value -90 3π/2 0, -1 cosx=0

trig functions and special angles

cot 5π/3 cos/sin so x/y value -if you get a negative degree substract from 360 to find reference angle and solve from there

finding trig ratios given a right triangle

csc=h/o (sin) sec=h/a(cos) cot=a/o

inverse functions rational

domain of the inverse is the range of f find range by doing inverse of f find domain by using f

composition of two functions

domain under a square root is that it has to be greater than 0 domain for a fraction is that the denominator cannot equal 0 -remember brackets and parenthesis

determining end behavior of polynomial function

exponent odd and n>0 falls left rises right e odd and n<0 rises left falls right e even and n>0 rise right and left e even and n<0 fall left and fall right

word problems involving area between two circles

find area of bigger and then smaller (πr₂) so if 1 gallon covers 5 yards how many do we need divide the answer by 5!! Do not forget this part

Writing Equation of lines given y-intercept and another point

find slope by doing y2-y1/x2-x1

complex fractions

if dividing fractions multiply by opposite and simplify

reference angles

if in quad 1 x=x if in quad 2 180-x if in quad 3 x-180 if in quad 4 360-x -255 (quad 3) 255-180 (make positive) -7π/9 (quad 2) 180-7π/9 *if it is negative keep x positive when using equations

domain and range of a graph

if it is an open circle it is a parenthesis ()

Sketching an angle in Standard Position

if it is positive it goes clockwise if it is negative it goes counterclockwise 1/4 revolution would be 2π x 1/4=π/2 -if it is -5π/6 use the opposite side so it would be 7π/6

solving equations with 0 one or infinite solutions

if its infinite its all real numbers

Vertical Line Test

if there is a hole and a solid hole on top of each other it does past vertical line test

Expanding a log

if they are multiplied together they can be added if they are divided they can be subtracted bring exponents to the front or numbers in front as an exponent log(xy^6) logx+6logy logz^3/x 3logz-logx

solving trig equations

if you get a tan value of 1/√3 find where on unit circle that sin/cos=1/√3

Solving an absolute value inequality

if |A|<c -c<a<c if |A|≤c -c≤a≤c if |A|>c then a<-c or a>c if |A|≥c then a≤-c or a≥c |w+8|>2 w+8<-2 or w+8>2 |y+3|≤4 -4≤y+3 or 4≥y+3 so if < then set -c<a<c if > then a<-c or a>c

Solving Word Problems with quadratic equation

length of a rectangle is 10m less than 3 times the width and area is 77 -L=3W-10, LW=77 (3W-10)(W)=77

graphing a square root function

look at square root and find when it is equal to 0 and that is your left most point and then find values to the right of that √x+1 -1,0 and find points to right of that

finding the x intercepts of a parabola

mp formula -x1+x2/2, y1+y2/2 y=-x^2-14x-49 -1(x+7)(x+7) -7,0 y=-x^2-2x+24 (x+6)(x-4) -6,4 -6+4/2=-1 plug in -1 into equation and get point

rationalizing the denominator of a rational function

multiply by opposite reciprocal and simplify -7/3√2-1 multiply by 3√2+1/3√2+1

Evaluating Functions: rational and radical

plug in number given into equation

Graphing a parabola

plug in points

composition of 2 functions

q(P)(2) plug p into q then plug 2 in

solving the absolute value in form |ax+b|=c

rules: if c≥0 than a=c or a=-c if c<0 than |A|=b has no solution ex. |2u-10|=4 4≥0 2u-10=4 or 2u-10=-4 |2u+4|=-10 -10<0 NO SOLUTION

domain of a square root function: advanced

set it equal to or greater than 0 and solve and graph on a number line to find the domain (has to be greater than 0) y=√-x √-x≥0 x≤0 (-infinity, 0) since 9 wasn't under square root you don't have to put it

Solving equations with zero, one or infinite solutions

set the equations equal to each other x= one solution 4=6 no solution 4=4 infinite solutions

finding solutions in an interval for sin/cos

sin is y cos is x 2cos-3=-2 cos=1/2 π/3,5π/3

Pythagreon identities

sin2u+cos2u=1 tan2u+1=sec2u cot2u+1=csc2u csc2-cot=1 cot2u+1-cot=1 cot(cot+1)=1 1 or 0 so when cos/sin =1, 0 -5sin=-2cos2+4 -5sin=-2(cos2)+4 -5sin=-2(1-sin2)+4 -5sin=-2+2sin2+4 2sin2+5sin+2 annd factor to get =1/2, -2 since sin never equals -2 you don't include to find value

double angle identities

sin2x=2sinxcosx cos2x=2cos2x-1 1-2sin2x if given a angle draw triangle and plug in values to double angle to find it

Using trig function ratio to find an angle in a right triangle

sohcahtoa use -1 to find an angle

Factoring when exponent is more than 2

substitute u in for x to get to x^2 so it can be factored x^4-5x^2+4 u=x^2 u^2-5u+4 u=4,1 take the square root of u so it is +/-(2,1)

Equations

the volume of a cone: 1/3bh (b(πr^2) the volume of a cylinder v=πr²h area of circle πr² area of triangle 1/2bh SA of a cylinder πr^2+πr^2+2πrh -csc=1/sin -sec=1/cos -cot=1/tan

finding x and y intercepts

to find x intercept: plug in 0 for y to find y intercept plug in 0 for x

Inferring properties of a polynomial from graph

use rules from above to find the degree it is one more than amount of extremas say there are three minimas and 1 max degree would have to be at least 5 but look at rules to determine if even or odd

Volume of a Cylinder

v=πr²h - volume is to the ³

Sketching graph of rational function

vertical asymptote is when x=0 in denominator horizontal is when y=0 rules for ha if numerator is greater than denominator no ha if numerator is less than denominator y=o if they are equal divide leading coefficients ^referring to exponents

Complex fractions involving multivariate monomials

when dividing fractions multiply by opposite and simplify -always simplify by subtracting or whichever one has more s4/s2 s2/1

Power product and quotient rules

when it is in a parenthesis()^x multiply it and if it is 2 variables add them together if it is a negative value bring it up to the top or down to bottom

Factoring a product of a quadratic trinomial and monomial

when you factor an amount out keep that value in your answer such as 4u^5(u+3)(u-3)

simplifying a product of rational expressions multivariate

when you multiply to exponents under square roots you add them when taking a square root just think of multiplying them by 1/2 and leave excess under square root

graph and domain and range of a logarithmic function

y=3^x+2 -this moves it two the left find points of just 3^x and then apply the x+2 domain is - infinity to infinity range is 0 to infinity because you can't have a negative exponent h.a=0 y=1/4x+1 move up 1 ha asymptote is 1 so range is 1, infinity) domain -infinity to infinity (always for these problems) y=-(1/2)x-2 find points of 1/2^x and then apply rules down 2 since ha is -2 range is -2, to infinity

transforming graphs by stretching or shrinking

y=a(fx) affects y coordinates y=f(xa) affects x coordinates and is opposite y=2f(x) find values of f(x) and multiply by 2 y=f(2x) find values of f(x) and multiply by 1/2

S.A area of a cylinder

πr^2+πr^2+2πrh