Precalc Final

convert (8√3, -8) to polar coordinates

(x, y)= (rcosθ, rsinθ) 16cos11π/6, 16sin11π/6 r, θ (16, 11π/6)

log(base3)(1/9)³

-6

(1+i) is a root of x³-8x²+14x-12=0

1-i is also a root as for the other roots, factors of constant: ±12 ±1 ±4 6 ±3 ±2 factors of coefficient: ±1 constant factors/coefficient factors are the potential zeroes literally have to do synthetic division w all of them The other root is 6

How do you get the partial fraction decomposition of 7/ (x²+13x+40)

1. factor the bottom 2. write a partial fraction like A1/(x+8) + A2(x+5) = 7/ (x+8)(x+5) 3. multiply everything by (x+8)(x+5) and simplify 4. You have 7=A1(x+5)+ A2(x+8) plug in a number like -5 that will make one of the variables 0 5. Solve+simplify A1=-7/3 and A2=7/3 6. Final equation: (-7/3)/(x+8) + (7/3)(x+5)

solve 2(3^(2t-5))-4=11

3 ^(2t-5)= 7.5 log (base3)7.5= 2t-5 t= [(log (base3)7.5)+5]/2

Whats the equation of a parabola? What is p? If the vertex was at the origin and the focus was (0, 6) what would be the equation?

4p(x-h)=(y-k)² if it's sideways 4p(y-k)=(x-h)² if it's regular p=the distance between the vertex and the focus So if the focus is on the positive y axis that tells you it's a normal, upward-facing parabola Then focus to vertex=6 So 4(6)y=x² x²=24y

component form

<x, y, z> vector form

Half life equation

A=A₀ (1/2)^(t/h) if h is half-life

continually compounded interest formula

A=Pe^(rt)

How can you tell how many extrema (relative min and max) a polynomial graph will have (FREE RESPONSE)

At most (highest degree)-1

Find the center and focus of the ellipse (x+5)²/5 + (y+9)²/9 What is the equation of an elipse?

Center at (-5, -9) remember that the form is (x-h)²/b² + (y-k)²/a² =1 for a skinny elipse with the bigger number under y The foci are c units away c²=a²-b² c²=9-5 c=2 So foci at (-5, -11) and (-5, -7)

What do you do if youre asked to graph something like f(x)= (x²-5x-3)/(x-3)

Degree of num>degree of denominator It has a slant asymptote Need to do (x²-5x-3)÷(x-3) Use synthetic division, and ignore the remainder You get y=x-2 That's the equation of the slant asymptote Make sure to also find zeroes and holes

How are a, b, and c related in an ellipse? In a hyperbola?

Ellipse: a²-b²=c² Hyperbola: a²+b²=c²

f(x)=x²-5 is one to one on domain x≤0. Find f⁻¹(x)

Find the inverse by switching x and y and solving for y You get f(x)=±√(x+5) but the only real solution is -√x+5 because it's one to one for negative numbers only. Does not exist for +√(x+5)

Sketch the graph of 3x⁴-4x³ (FREE RESPONSE)

If you factor you get x³(3x-4) So 0 is a zero, it's odd degree so the line crosses at 0 x=4/3 is also a zero, its odd degree (power of 1) so crosses there too To fourth power and pos so rises in both directions, max of 3 extrema Then plot some additional points and draw graph

How do you find the inverse of a matrix w/ a calculator

Just put the matrix into [A] and then do [A]⁻¹

How many positive, negative, and nonreal roots are there of 4x¹¹+7x²+7x+3=0

No positive roots bc no sign change Plug in f(-x) and you get 3 sign changes. Either 3 or 1 negative roots Then the highest degree here is 11. That means theres 11 total solutions, real and nonreal combined. Since only 3 or 1 real solutions, theres either 10 (11-1) or 8 (11-3) nonreal solutions

Rabbit population is (25+3000t)/(.9t+4) What is eventual pop of rabbits

Num and denom are same degree- asymptote is a ratio of highest-degree terms 3000/0.9 = 3333

When the leading coefficient is odd (for example, 3 in x³+x²+7), then the graph If the leading coefficient is even, the graph, If its positive, If its negative (FREE RESPONSE)

Odd and pos = graphs falls to left, rises to right Odd and negative= graph rises to left, falls to right Even and pos= rises in both directions Even and neg= falls in both directions

how do you find a vector orthogonal to PQ and PR

PQ×PR

How do you graph something (for example r=2/(3+cosθ) ) in polar coordinates? How do you know if it's an ellipse, hyperbola, or parabola?

So you graph by going out r units and rotating θ degrees If e<1, it's an ellipse If e=1, parabola If e>1, hyperbola

How do you multiply matrices

Take the dot product of the row of the first matrix and the column of the second matrix

What is p

The absolute value of the distance between the focus to the vertex Or the distance from the directrix to the vertex It's the same thing as c in normal form

Find a polar equation for parabola with focus at pole and directrix at x=-8

This is a sideways-facing parabola that opens to the right- want to use ep/1-cosθ Just like with hyperbolas, the sideways means cos, and the negative directrix means -cos r=8/(1-cosθ)

What are the vertical and horizontal asymptotes of (5x-5)/(x²-4x-5)

Vertical: x=5 and x=-1 Horizontal is 0 because deg of denominator> degree of numerator

What is the difference between a zero, a hole, and an asymptote

When just the denominator is equal to zero at a value, there's an asymptote there When a value makes both the numerator and denominator equal to zero, there's a hole- a discontinuity at a single point A zero is simply where the value of the function is equal to zero. The numerator is 0. It's still on the line, and isnt undefined in any way

a. How is 2^(-x) transformed compared to 2^(x) b. How is -2^(x) transformed compared to 2^(x) c. How is 2^(x+1) transformed compared to 2^(x) d. How is 2^(x)+1 transformed compared to 2^(x) (FREE RESPONSE)

a. reflected across y axis b. reflected across x axis c. shifted left by 1 d. shifted up 1

a. how is log(x-1) transformed compared to log(x) b. how is log(x)+2 transformed compared to log(x) c. how is -log(x) transformed compared to log(x) d. how is log(-x) transformed compared to log(x) (FREE RESPONSE)

a. shifted right 1 b. shifted up 2 c. flipped across x-axis d. flipped across y-axis

Equation of a hyperbola- what is a, b, c

a=center to vertex b=center to less big part of box that you draw to get the asymptote c=center to focus a²+b²=c²

Find x intercept of f(x)=2ln(x-3)

can move 2 to the exponent f(x)=ln(x-3)² and then the x-intercept is where y is 0 0=ln(x-3)² Then ln(1)=0 So here, x must be 4

angle between vectors u and v

cosθ= u⁰v / ‖u‖‖v‖

Sketch the graph of (x²-9)/(x²-2x-3) (FREE RESPONSE)

factor zero: x=-3 (plug that in to get y-coordinate) Asymptote: x=-1 hole: x=3 y-int at (0, 3) x-int at (-3, 0)

2x²(e^2x) + 2x(e^2x)=0

factor out e^(2x) e^(2x)=0 and 2x²+2x=0 x=0 or -1

Solve 2s-5t=7 and -2s+4t=-6 using matrices

https://www.khanacademy.org/math/precalculus/precalc-matrices/solving-equations-with-inverse-matrices/v/solving-matrix-equation

Rewrite Log(base3)21 in terms of the natural logarithm

ln21/ln3

simplify 1/5[logx+log6]-[logy]

log (⁵√6x / y) see prob 14, ch 3 test

(e^x)=7

log(base e)7=x ln7=x

solve log(base3)x + log(base3)x-8 =2

log(base3)x²-8x=3² 3²=x²-8x 0=x²-8x-9 (x-9)(x+1) But be careful- -1 is NOT a solution because you cant take the log of a negative number

Simplify log₅150

log₅(25*6) log₅25+log₅6 2+log₅6

convert x²+y²=64 to polar form

polar form is r= form x²+y²=r² r=8

What is the polar form of the equation of a hyperbola, ellipse, or parabola? When do you use sin, and when do you use cos?

r= (ep) / (1±cosθ) or r= (ep) / (1±sinθ) Use sin if directrix is on y Cos if directrix on x When θ=0, π/2, π, 3π/2 that can tell you what the points are on the x and y axis. 0 is the right intersection of the ellipse with the x-axis, pi is the left intersection of the ellipse

Graph -1/(x+2) and compare it to the parent graph 1/x. How has it been transformed (FREE RESPONSE)

reflection in y-axis, horizontal shift 2 units to left

Convert r=12sinθ to rectangular

r²=12rsinθ x²+y²=12y x²+y²-12y=0 x²+(y-6)²=36

Standard form of a parabola Write the standard form of parabola w vertex at (4, 3) and passes through (-4, -2)

see ch 2 test prob 2 y= -5/64x ²+ 5/8x + 7/4

How do you put 3 equations into a matrix? How can you solve a matrix

see prob 5 To solve a matrix, you can use row operations (basically like solving algebraically but in a matrix) or you can do A⁻¹×B (see khan academy vid)

Asymptote of a hyperbola

see problem 51 in review. remember that the asymptotes go diagonally across the vertices. Calculate slope of the line of the asymptote

Lim(x→∞) [x/ (3+x)²]-5

the limit is -5 (NOT 0) degree of denom> degree of numerator

How do you find two pos real numbers whose product is a maximum and whose sum is 130

x+y=130 Set equal to x, x=130-y A=xy substitute A=(130-y)y A=130y-y² Max at x=-b/2a x=-130/(-1)(2)= 65 x+y=130 65+y=130 y=65

convert x=5 to polar form

x=rcosθ 5=rcosθ r= 5 /(cosθ)

What is x in polar form? What is y? What is r in rectangular form?

x=rcosθ y=rsinθ x²+y²=r²

Find a set of polar coordinates for (12, 12)

x²+y²=r² 12²+12²=r² r=√288 (√288, π/4)

Write the corresponding rectangular equation for x=6cosθ y=6sinθ by eliminating the parameter

x²+y²=r² 6cos²θ+6sin²θ=r² x²+y²=36

put r=4 in rectangular form

x²+y²=r² x²+y²=16

asymptote of a hyperbola

y=b/a (x-h) +k

Find the zeros of (x³-64)/(x²+5)

±√5 ARE NOT ZEROES if you plugged those in, they'd give you asymptotes They would NOT make the function equal to zero. The only zero is 4 The function still EXISTS at 4, it's just y=0 there

area of a parallelogram w/ 2 adjacent vectors

‖PQ×PR‖