Precalculus
f(x) = ax² +bx + c If a<0 the range is
(-∞,k)
Domain of sin and cos is:
(-∞,∞)
Range of tanx
(-∞,∞)
The domain of any polynomial is
(-∞,∞)
The Remainder Theorem Continued:If a polynomial f(x) is divided by x-a, then the remainder R is given by R=f(x) Find the remainder of F(x) = x³-2x²+3 is divided by x-1
(1) = (1)³-2(1)²+3 =1-2+3 =-1+3 =2
Vertical Asymptote of h(x) = x²-9/x-3
(x-3)(x+3)/x-3 No vertical asymptote with no denominator
Find the standard form of the equation of the circle with center (7,-3) and passes through the point (5,-2)
(x-7)² +(y+3)² =5
Find the angle between 0° and 360° that is coterminal with the given angle a) -65°
-65° + 360° = 295°
Find sin(-3)
-sin(30) = -1/2
Odd functions: sine, cosecant, tangent, cotangent sin(-x) = tan(-x) = csc(-x) = cot(-x) =
-sin(x) -tan(x) -csc(x) -cot(x)
Loga1 =
0
ln 1 = 0
0
log 1 =
0
In e =
1
Logaa
1
Pythagorean identities cos²t + sin²t =
1
log 10 =
1
tan 45 = tan π/4 =
1
(1+√2)(3-√2)
1 + 2√2
θ = how many radians
1 radian
Steps to graphing a polynomial:
1) End behavior 2) Zeros 3) Multiplicity 4) Cross/Touch 5) Above/Below x-axis 6) Y-intercept
Finding the vertex of f(x) = ax² +bx +c
1) Find the x-coordinate h=-b/2a of the vertex. 2) Calculate k=f(-b/2a) to find its y-coordinate
To find the horizontal asymptote
1) If n<m, the x-axis, or y=0 is the horizontal asymptote graph of f. 2) If n=m, the line y= aₙ/bₙ is horizontal asymptote of the graph of f. 3) If n>m, the graph of f has no horizontal asymptote n = numerator m = denominator
Ways to test for symmetry
1) Symmetric about y-axis if replacing x with -x results in an equivalent equation 2) Symmetric about x-axis if replacing y with -y results in an equivalent equation 3) Symmetric about origin if replacing x with -x and y with -y results in an equivalent equation.
What values are allowed for x? f(x) = 1/-x²
1-x² = 0 x² =1 x=1, x=-1 Domain: (-∞,-1) ∪(-1,1) ∪(1,∞)
Trigonometric functions of real numbers Sec(t) =
1/x (x>0)
Note the reciprocals: Sec(t) =
1/x = 1/cos(t)
Trigonometric functions of real numbers csc(t) =
1/y (y>0)
Note the reciprocals: Csc(t) =
1/y = 1/Sin(t)
3√4 + 2√4
10
3√8 + 5√2
11√2
Domain of f(x) = log3(2-x)
2-x >0 -2 -2 ------------- -x >-2 x<2 (-∞,2)
Common acute angles: π/6 =
30°
Trigonometric Function values of Common angles Common acute angles: π/6 = π/4 = π/3 =
30° 45° 60°
√2(3+√3)
3√2 +√6
Common acute angles: π/4 =
45°
2√3 +3√3
5√3
√3 +4√3
5√3
√3(2√3 + √5)
6 + √15
Common acute angles: π/3 =
60°
Find the angle between 0° and 360° that is coterminal with the given angle b) 700°
700° -360° =340°
Trigonometric Functions of Angles Values of the trigonometric functions of an angle. Let p(x,y) be any point on the terminal ray of an angle θ is standard position (other than the origin) and let r=√x²+y² Then r>0, and Cos θ =
= x/r
Trigonometric Functions of Angles Values of the trigonometric functions of an angle. Let p(x,y) be any point on the terminal ray of an angle θ is standard position (other than the origin) and let r=√x²+y² Then r>0, and tan θ =
= y/x (x≠0)
Interest Earned =
A - P
Compound Interest Formula
A = P(1+ r/n)nt A = amount after t years p = principal r=rate n = number of times interest is compounded t = number of years
Continuous Compound Interest Formula
A = Pert A = amount after t years p = principal r= annual interest rate t= years
Area of a sector formula
A = ½r²θ
Periodic Functions
A function f is said to be periodic if there is a positive number p such that f(x+p) = f(x) for every x in the domain of f the smallest value of p (if there is one) for which f(x+p) = f(x) is called the period of the f. The graph of f over any interval of length p is called one cycle of the graph.
Exponential Function
A function f of the form f(x) = ax , a>0 and a≠ is called an exponential function with base "a" and exponent "x". Its domain is (-∞,∞) Example: Evalulate Exponential Function a) f(x) = 3x-2 find f(4) f(4) = 34-2 =3² =9
The Factor Theorem
A polynomial f(x) has (x-a) as a factor if an only if f(a) = 0.
Function
A set x to a set y is a rule that assigns to each element of x exactly one element of y.
Intersection
A ∩ B All elements in a AND b. Where they overlap. Ex: A = {-2,-1,0,1,2,3) B = {-4.-2.0, 2,4} A∪B = (-4.-2,-1,0,1,2,4} A ∩B = {-2,0,2}
Union
A ∪ B (Looks like a U). All elements that are in A, B, or both. List numbers only once from smallest to largest.
How to perform vertical shifts?
Add or subtract on outside of function y =|x| What does y=|x| + 2 - shifts up What does y =|x| -3 - shifts down
How to perform a horizontal shift ?
Add/subtract on inside of function Ex: f(x) = x² g(x) = (x-2)² - shifts rights h(x) = (x+3)² - shifts left
If you are given a factored function you can find the end behavior by?
Adding together the exponents to find the degree.
If the answer to your problem looks like 6x=6x then the answer is?
All real numbers
Remember an Undefined slope and a zero slope will only touch?
An undefined slope only touches the x-axis while a zero slope only touch the y-axis.
Coterminal angles
Angles with the same initial and terminal sides.
Length of an Arc of a circle
Arc length formula The length s of an arc intercepted by a central angle with radian measure θ in a circle of radius r is given by S = rθ
Circumference of a circle
C = 2πr
Find the complement and supplement a) 73°
Complement: x + 73° = 90° x = 90° -73° = 17° Supplement: x + 73° = 180° x=180°-73° = 107°
Find the complement and supplement 110°
Complement: x+110° = 90° No complement 110°>90° Supplement: x+110°=180° x=180°-110° =70°
Four methods to solve quadratics
Complete square Quadratic formula Factor Square Root
Find the exact value of y: y = arcos(-½)
Cos y = -½ y= 2π/3
Even functions Cosine and secant Cos(-x) = Sec(-x) =
Cos(x) sec(x)
Absolute value
Distance (Positive numbers) |3| = 3 |-2| = 2
Domain and range of f(x) = ax
Domain: (-∞, ∞) Range: (0,∞)
Find the domain: f(x) = 3x²-12/x-1 x-1 =0 x=1
Domain: (-∞,1)∪(1,∞)
g(x) = x/x²-6x+8 x²-6x+8 (x-4)(x-2) =0 x=4, x=2
Domain: (-∞,2)∪(2,4)∪(4,∞)
Domains of Logarithmic Functions f(x) = ax y=logax
Domain: (-∞,∞),Range: (0,∞) Domain: (0,∞), Range: (-∞,∞)
Domain and range of y= logaX
Domain: (0,∞) Range: (-∞,∞)
y = aⁿ - C
Down shift
What does it mean for something to be Above/Below the x-axis
Draw a number line and place your multiplicities. Play random numbers that a inbetween the multiplicities. When you plug that
Even-Odd properties Is it even or odd and its symmetry cos(-t) = cos(t)
Even and symmetric y-axis
What is the Exponential function? What is the exponential function of growth? Decay?
Exponential function: f(x) =ex Growth: A(t) = A0ekt Decay: A(t) = A0e-kt
Period of the sine and cosine functions
For every real number t, sin(t+2π) = sin(t) and cos(t+2π) = cos(t) The sin and cos functions are periodic with period 2π
To find the domain you need to look
From left to right
Stretching/Compressing
Given g(x) =af(x) 1) a>1, vertical stretch 2) 0<A<1, vertical compression Example: f(x) = x² g(x) = 3x² (stretch) h(x) = ½x² (compression) Given g(x) = f(bx) 1) 0<b<1, horizontal shift 2) b>1, horizontal compression
Reflections over x-axis
Given y=f(x), then g(x)=-f(x) is the reflection over the x-axis.
Reflections over y-axis
Given y=f(x), then g(x)=f(-x) is the reflection over the y-axis. Example: f(x) = √x g(x) = √-x
Phase Shift
Graphing y = sin(x-c) Example: y = sin(x-π/2) shifted right π/2 Graphing y=asin[b(x-c)] and y=acos[b(x-c)] with b>0 Step 1: amplitude = |a| period = 2π/b phase shift = c Step 2: cycle begins at c one cycle is [c, c+2π/b] Step 3: Divide into 4 equals parts to get key points Step 4 sketch
Finding the vertex of f(x) = ax² +bx +c with an equaion
H = (-b/2a) k = f(-b/2a)
y = aⁿⁿ
Horizontal compression/expansion
Simple Interest Formula
I = Prt Principal: amount borrowed time: how long you borrow (in years) Interest rate: percent charged for the use of the principal for given time Interest: fee for borrowing money
Symmetry on a Unit Circle
If P = (a,b), then following symmetric images of p are also on the unit circle. a) Q = (a,-b) b) R = (-a,b) C) S =(-a,-b) D) T = (b,a)
The Remainder Theorem
If a polynomial F(x) is divided by x-a, then the remainder, R, is given by R= F(a) If the remainder is zero then it is a factor of it.
Zero product rule
If a × b = 0 a, b , or both must be zero
Real Zeros of Polynomial Functions
If f is a polynomial function an C is a real number, then the following statements are equivalent. 1. C is a zero of f. 2. C is a solution (or root) of the equation f(x) = c 3. C is an x-intercept of the graph of f 4. (x-c) is a factor of f
Vertical line test:
If no vertical line intersects the graph of a relation at more than one point then the graph is a function.
Oblique or Slant Asymptote Example: Find Slant Asymptote f(x) = x²+x/x-1
If the degree of N(x) is exactly one more than the degree of D(x), then there will be a slant asymptote in the form mx+b x-1/x²+x = y = x+2
Trigonometric Function of an angle
If θ is an angle with radian measure t, then Sin θ = Sin t Csc θ = csc t Cos θ = cos t Sec θ = sec t tan θ = tan t cot θ =cot t If θ is given in degrees convert θ to radians before using these equations.
Find the domain: P(x) = 2x -1
In the function no value of will result in a denominator of 0 or cause you to take an even root of a number. (-∞,∞)
Vertical Asymptote
Is a vertical line that the graph of a function approaches but does not touch. The graph of a rational function will never intercept a vertical asymptote
A angle
Is formed by rotating a ray about its endpoint
When will a line cross/touch a graph?
It will cross a graph when the multiplicity is odd and it will touch it when the multiplicity is even.
An angle in a rectangular coordinate system is in standard position if
Its vertex is at the origin and its initial side lies on the positive x-axis.
y = aⁿ⁺ⁿ
Left shift
An intermediate value Theorem
Let a and b be two numbers with a<b. If f is a polynomial function such that f(a) and f(b) have opposite signs then there is at least one number c, with a<c<b for which f(c) is 0. A polynomial function of degree n with real coefficients has, at most n real zeroes. If c is a zero of multiplicity m for a polynomial function f(x) with corresponding factorization f(x) = (x-c)ⁿ q(x), q(x) ≠0 then, near x=c, f(x) looks very much like the graph of A(x-c)ⁿ, where A is the constant A=q(c) Basically plug in the values and check to see if there is a difference in signs.
Graphs of the Reciprocal functions
Let f(x) be the reciprocal of g(x): f(x) = 1/g(x), where g is any trigonometric function periodicity: if g(x) has period p, then f(x) also have period p Zero: If g(c) =0, then f(c) is undefined, so if c is an x-intercept of the graph g, then the line x=c is a vertical asymptote of the graph Even-odd a) If g(x) is odd, then f(x) is odd b) If g(x) is even, then f(x) is even Special values: a) if g(x₁) = 1 then f(x₁) =1 Both pass through (x₁,1) b) if g(x₂) =-1 then f(x₂) = -1 both pass through (x₂,-1) Sign a) If g(x) >0 on an interval (a,b) then f(x) >0 on the interval (a,b) b) if g(x) <0 on an interval (c,d), then f(x) <0 on the interval (c,d) Increasing-Decreasing a) If g(x) is increasing, then f(x) is decreasing b) f g(x) is decreasing then f(x) is increasing Magnitude: a) if |g(x)| is small, then |f(x)| is large b) if |g(x)| is large then |f(x)| is small
To find the rational zeros of a polynomial function
List all of the factors of the constant term. (p) List all of the factors of the leading coefficient. (q) Then form all possible ratios with p/q
What does (D∪E)∩F mean?
List all of the points in D and E then only list the points from that list that are in F.
Quotient Rule of Logarithms
Loga (M/N) = Loga M - Loga N
Product Rule of Logarithms
Loga (MN) = Loga M + Loga N
Power Rule of Logarithms
Loga Mr= rloga M
What is logarithmic and exponential form?
Logarithmic: y = Loga X Exponential: X = ay
Mid point formula
M = (x₁+x₂/2, y₁+y₂/2)
To write a complex number in standard form you need to?
Multiple numerator and denominator by conjugate of denominator.
To divide complex numbers or write in standard form you need to?
Multiply denominator and numerator by conjugate of denominator.
Can an X-value have more than one y-value?
No One number can't be equal to two different y-values but two different x-values can be equal to one y-value.
Find the zero of x²+1=0
No real solution
Solving a Quadratic equation by factoring
Note: a × c = 2(-3) =-6 ---> +6 and -1 Ex) 2x² + 5x = 3 2x² + 5x -3 = 0 2x² + 6x -1x - 3 = 0 2x(x+3) -1 (x+3) = 0 (x+3)(2x-1) =0 x+ 3 = 0 x = -3 2x -1 = 0 x = ½
Even-Odd properties Is it even or odd and its symmetry sin(-t) = -sin(t)
Odd and symmetric to origin
Intervals
Open circle = ( , ) , > , < Closed circle = ] , [ , ≤ , ≥
Relative maximum points correspond to the?
Peaks on a graph
Example: Graph y=3cos1/2x
Period = 2π/1/2 = 4π One cycle from [0,4π] key points: 0, π, 2π, 3π, 4π (0,3), (π,0),(2π,-3),(3π,0),(4π,3)
How to find y-intercepts
Plug in x=0 and solve for y
How to find x- intercepts
Plug in y=0 and solve for x
Range of a function: Given f(x) = x² with domain x =[3,5], what is the range?
Range: [9,25]
y = -aⁿ
Reflect over x-axis
Describe the transformation: y= a⁻ⁿ
Reflect over y-axis
Arc length formula
S = rθ
What is happening y= -½(x+4)² -2
Shift down 2 Shift left 4 Reflect over x-axis Vertical compression by 1/2
y = aⁿ⁻ⁿ
Shift right
What is happening? y = -|x-2| + 3
Shifts up 3 Shift left 4 Reflect over x-axis
Finding Exact Trigonometric Function Values Find the values (if any) of the six trigonometric functions at each value t a) t=0
Sin 0 = y = 0 cos 0 = x =1 tan 0 = y/x = 0 csc 0 = 1/y = 1/0 = 0 sec 0 = 1/x = 1 cot 0 = x/y = 1/0
Trigonometric Function of an angle Sin θ =
Sin t
Trigonometric functions of real numbers Let, t, be any real number and let p(t) = (x,y) be the terminal point on the unit circle associated with t. Then
Sin t = y Cost t =x tan t = y/x (x>0) csc t = 1/y (y>0) Sec t = 1/x (x>) Cot t = x/y (y≠0)
Trigonometric Functions of Angles Values of the trigonometric functions of an angle. Let p(x,y) be any point on the terminal ray of an angle θ is standard position (other than the origin) and let r=√x²+y² Then r>0, and
Sin θ = y/r Cos θ = x/r tan θ = y/x (x≠0) csc θ = r/y (y≠0) sec θ = r/x (x≠0) cot θ = x/y (y≠0)
Amplitude and Period
Sinusoidal graphs or sinusoidal curves in the form y= af[b(x-c)] +d Amplitude: Let f be a periodic function. Suppose M is the maximum value of f and m is the minimum value of f the amplitude of f is defined by; Amplitude = 1/2(M-m)
Find the exact value of y: a) y= arcsin√3/2
Siny = √3/2 y=π/3
Standard form of equation: Ax + by =C Slope? Y-intercept?
Slope: m = -A/B Y-intercept: b= c/B
To Find the vertical asymptotes
Solve the denominator for zero
Horizontal line test
States that if every horizontal line intersects the graph of a function "f" in at most one point, then "f" is one to one.
Graphing Rational Functions Example: Sketch f(x) = 2x²-2/x²-9
Step 1: Find intercepts 2x²-2=0 2(x²-1)=0 2(x-1)(x+1)=0 x=1 x=-1 y = 2(0)²-2/0²-9 = 2/9 y=2/9 Step 2: Find asymptotes Vertical Asymptote: y= 2x²-2/x²-9 = 2(x-1)(x+1)/(x-3)(x+3) x=3 x=-3 Horizontal Asymptote: y=2x²-2/x²-9 y=2 Step 3: Locate the graph relative to the horizontal asymptote If y=l is a horizontal asymptote, divide N(x) by D(x) and write f(x) = N(x)/D(x) = k+R(x)/D(x) Use a sign graph for f(x)-k = R(x)/D(x) and "test" numbers associated with the zeros of R(x) and D(x) to determine intevals where the graph of f is above or below y=k. f(x) = 2x²-2/x²-9 =2 + 16/x²-9 x²-9/2x²-2 = 2 Remainder 16 R(x) has no zeros and D(x) has zeros -3 and 3
Example: Graph y=3sin[2(x-π/4)]
Step 1: amplitude = |3| =3 period = 2π/b = 2π/2 = π phase shift = π/4 shift to right Step 2: cycle begins at π/4 one cycle is [π/4,π/4+π] = [π/4,5π/4] Step 3: Divide cycle into 4 equals parts 1/4(period) = 1/4 = π/4 π/4 π/4 + π/4 = 2π/4 = π/2 π/2 + π/4 = 3π/4 3π/4 + π/4 = π π + π/4 = 5π/4 Step 4: (π/4,0), (π/2,3), (3π/4,0),(π,-3), (5π/4,0)
What is a complement
Sum of angles is 90°
Perpendicular lines have slopes
That multiple to -1
Area of a sector
The area A of a sector of a circle of radius r formed by a central angle with radius measure θ is A = ½r²θ
The Unit Circle
The circle with radius 1 with its center at the origin in the xy-plane, it is the set of points (x,y) that satisfy the equation x² + y² =1
Circumference of a Unit circle
The circumference of a unit circle is 2π. This means the arc length for one revolution around the unit circle is 2π units
Standard form of a circle
The equation of a circle with center (h,k) and radius,r, is: (x-h)² + (y-k)² = r²
Complex Numbers
The equation x² = 1 has no solution in the set of real numbers to solve. We extend the real number system to the complex number system. We define a new number i with the following rules i = √-1, i² = -1 We will use the form: a + bi a is the real part whereas b is the imaginary part.
Inverse Tangent function
The equation y= tan⁻¹x means tany = x where -∞<x<∞ and -π/2 <y<π/2. Read tan⁻¹ x as "the inverse tangent at x"
Amplitude of Sine and cosine
The functions y= asinx and y =acosx have amplitude = |a| and range = [-|a|, |a|] Example: Graph: y=sinx y=3sinx y=1/3sinx Vertical stretch/compression
The multiplicities of a function are equal to
The highest exponent of the zero
Radian measure of a central angle
The radian measure θ of a central angle intercepts an arc of length s on a circle of radius r is given by θ = S/r radian
Cosine and Sine
The terminal point p(t) = (x,y) on the unit circle associated with a real number t has coordinates (cos t, sin t) because x = cos t and y = sin t
If your zero is the square root of a negative number then
There are no real solutions
Relative minimum refer to?
Troughs
If there are no relative extreme points then there are
Turning points
A vertical line has a slope of?
Undefined
y = aⁿ + c
Up shift
y= caⁿ
Vertical stretch/compression
The factor theorem states that to determine if a polynomial is a factor of another polynomial you have to?
Write the divisor in the form x-k, so if problem states x+6 then it will be x-6. Then use synthetic division to divide the polynomial. If the remainder is zero then x+6 must able to divide evenly into the polynomial so it is a factor of it .
Y = logaX in exponential form
X=ay
Vertical Shifts
Y = acos[b(x-c)] + d +d shift up -d shift down
Inverse Sine Function
Y = sin⁻¹x means siny=x, where -1≤x≤1 and -π/2≤y≤π/2 Read sin⁻¹x as "inverse sine at x"
Does the function define y as a function of x? 4-y=9x
Yes, you can solve for "y"
A horizontal line has a slope of?
Zero
Evaluate expression
[ (9+5) ÷ ] ×35 = [14÷17] ×35 = [2] x 3-5 = 6 -5 1
Range of sin and cos is
[-1,1]
f(x) = ax² +bx + c If a>0 the range is
[k,∞)
If a>0 then the range is ... If a<0 then the range is ....
[k,∞) (-∞,k)
Converting radians to degrees
______ (180/π)
Converting degrees to radians
_____° (π/180)
Distributive propert
a (b+c) = ab + ac
Symmetric coordinates on a unit circle
a) Q = (a,-b) b) R = (-a,b) C) S =(-a,-b) D) T = (b,a)
Factoring
a. x² +4x + 4 = (x+2) (x+2) = (x+2)² b. x³ + 2x² + 3x + 6 = x²(x+2) + 3(x+2) = (x+2) (x²+3) c. 6x² + 17x +7 3x(2x+1) + 7(2x+1) = (2x+1)(3x+7) d. 25x² -49 =(5x-7)(5x+7) e. x² +3x -10 = (x-2)(x+5) f. x³ -3x² -x + 3 = x²(x-3) - 1(x-3) = (x-3) (x²-1) = (x-3)(x-1)(x+1)
General form of the equation of a line
ax + by + c = 0
Power function
axⁿ
Exponent rules a⁰ = a⁻ⁿ = aⁿ x aⁿ = aⁿ/aⁿ = (aⁿ)ⁿ = (ab)ⁿ = aⁿbⁿ (a/b)ⁿ = aⁿ/bⁿ
a⁰ = 1 a⁻ⁿ = 1/aⁿ aⁿ x aⁿ = aⁿ⁺ⁿ aⁿ/aⁿ = aⁿ⁻ⁿ (aⁿ)ⁿ = aⁿⁿ (ab)ⁿ = aⁿbⁿ (a/b)ⁿ = aⁿ/bⁿ
A negative angle has what kind of rotation
clockwise
set
collection of objects
A quadratic function f(x) = ax² + bx +c can be changed to the standard form f(x) = a(x-h)² + k by
completing the square
Trigonometric Function of an angle cos θ =
cos t
Find cos(-2π/3) =
cos(2π/3) = -1/2
Find the exact value of y: y= cos⁻¹(√2/2)
cosy = √2/2 y=π/4
Trigonometric Function of an angle cot θ
cot t
A positive angle has what kind of rotation
counterclockwise
Trigonometric Function of an angle Csc θ =
csc t
Find the exact values y = csc⁻¹(2)
cscy = 2 sin(π/6) = 1/2 csc(π/6) = 1/sin(π/6) = 1/1/2 =2 y = π/6
Pythagorean identities 1 + cot²t =
csc²t
Distance formula
d = √(x₂ -x₁)² + (y₂-y₁)²
Angle measure? What are the two units of it?
degrees and radians
How to tell if a function is odd?
f(-x) = -f(x)
How to tell if a function is even?
f(-x) = f(x)
Standard form of a Quadratic Function
f(x) = a(x-h)² +k, a≠0 Vertex:(h,k) Symmetric with respect: x=h k, shift up/down h, shift left/right a, stretch
Quadratic Function
f(x) = ax² +bx + c, a≠0
Write an equation for a function that fits. The graph of f(x) =√x reflected over the y-axis and shifted two units down.
f(x) = √-x -2
Find the exact value of sin⁻¹(sin(-π/8))
in domain so = -π/8
A ray
is part of a line made up at a point, called the endpoint, and all of the points on one side of the endpoint
Powers of i i¹ = i² = i³ = i⁴ = i⁵ = i⁶ = i₇ = i₈ =
i¹ = i i² = -1 i³ = i² x i = -i i⁴ = i² x i² = 1 i⁵ = i⁴ x i = i i⁶ = i⁴ x i² = -1 i₇ = i⁴ + i³ = -i i₈ = i⁴ x i⁴ = 1
Natural Logarithm
ln x = loge x y = ln(X) X>0 if and only if x=ey In e =1 ln 1 = 0 ln ex=x elnx = x
Common logarithm
log x = log10 X y = logx (x>0) if and only if x=10y
ln x =
logex
Describe the slope of perpendicular lines
m = -1/m negative reciprocal
Describe the slope of parallel lines
m=m , same slope
Find the angle between θ and 2π radians that is coterminal with a) 19π/4 =
more than two revolutions, so we subtract the equivalent of two revolutions ( 2 × 2π = 4π) to obtain 19π/4 - 4π = 3π/4
irrational numbers
neither terminating or repeating π = 3.14....
Find the exact value of cos⁻¹(cos(5π/4)
not in domain/ can't use property cos(5π/4) = -√2/2 = cos(3π/4) cos⁻¹(cos(3π/4)) = 3π/4
elements
objects in a set
period of tanx:
period π
Trigonometric Functions of Angles Values of the trigonometric functions of an angle. Let p(x,y) be any point on the terminal ray of an angle θ is standard position (other than the origin) and let r=√x²+y² Then r>0, and sec θ =
r/x (x≠0)
Trigonometric Functions of Angles Values of the trigonometric functions of an angle. Let p(x,y) be any point on the terminal ray of an angle θ is standard position (other than the origin) and let r=√x²+y² Then r>0, and csc θ =
r/y (y≠0)
real numbers
rational and irrational numbers
Sets
roster method {1,2,3} Set builder notation {x|x is a natural number less than 63}
Trigonometric Function of an angle sec θ =
sec t
Find the exact value y = sec⁻¹ (½)
secy = 1/2 1/2 is not in domain (-∞,-1]∪(1,∞)
Pythagorean identities 1 + tan²t =
sec²t
Find the exact value of y: y = sin⁻¹(-3)
siny = -3 -3 is not in domain: undefined
Find the exact value of y: y=sin⁻¹(-½)
siny = -½ y=-π/6
What is a supplement
sum of angles is 180°
Trigonometric Function of an angle tan θ =
tan t
Find the exact value of y y = arctan(-√3)
tany = -√3 y= -π/3
Find the exact value of y Y = tan⁻¹(0)
tany =0 y =0
10logx =
x
10logx = x
x
Trigonometric functions of real numbers Cos(t) =
x
elnx =
x
ln ex=
x
Inverse function properties cos(cos⁻¹x)
x -1≤x≤1
Inverse function properties sin(sin⁻¹x) =
x -1≤x≤1
Inverse function properties sin⁻¹(sinx) =
x -π/2 ≤x≤π/2
Inverse function properties tan(tan⁻¹x) =
x -∞<x<∞
Inverse function properties cos⁻¹(cosx) =
x 0≤x≤π
Inverse function properties tan⁻¹(tanx) =
x -π/2≤x≤π/2
What values are allowed for x g(x) = √x+1
x + 1≥ 0 x≥-1 [-1,∞)
y = ln (x) x>0 if and only if
x = ey
Find an equation of the vertical line through (4,8)
x =c x=4
logaax =
x for any real number
alogax =
x for any x>0
Vertical Asymptote of g(x) = x+2/x²-4
x+2/(x+2)(x-2) VA: x=2
Trigonometric Functions of Angles Values of the trigonometric functions of an angle. Let p(x,y) be any point on the terminal ray of an angle θ is standard position (other than the origin) and let r=√x²+y² Then r>0, and cot θ =
x/y (y≠0)
Trigonometric functions of real numbers Cot(t) =
x/y (y≠0)
Note the reciprocals: Cot(t) =
x/y = 1/tan(t)
Trigonometric functions of real numbers cot(t) =
x/y = cos(t)/sin(t)
A vertical line through (h,k) is
x=h
How to tell if a point is on the unit circle
x² + y² =1 Plug into the equation and see if it is true.
General form of a circle is
x² +y² +ax + by +c = 0
What are the zeros of 0=x²(x-3)(x+1)
x² =0 x= 3 x=-1
Trigonometric functions of real numbers Sin(t) =
y
Graph y=3tan(2x-π/2)
y = 3tan[2(x-π/4)] 2x =π/2 x= π/4 Step 1: vertical stretch =3 period: =π/2 phase shift π/4 to right Step 2: 2(x-π/4) =-π/2 x-π/4 = -π/4 x=0 and 2(x-π/4) = π/4 x=π/2
Inverse cosine function
y = cos⁻¹x means cosy=x where -1≤x≤1 and 0≤y≤π. Read cos⁻¹x as "inverse cosine at x"
Period
y = f(bx) period = 2π/b Example: y=sinx, sin(3x), y=sin(1/3x) y=sin(3x) period 2π/3 one cycle is completed [0,2π/3] Divide : 0≤x≤2π/3 into 4 equal parts key points: 0, π/6, π/3, π/2, 2π/3 y = sin(1/3x) period = 2π/1/3 = 6π one cycle is completed from [0,6π] Divide 0≤x≤2π/3 into four equal parts key parts: 0,π/6, π/3,π/2,2π/3 y=sin(1/3x) period = 2π/1/3 = 6π into 4 equal parts key points: 0, 3π/2, 3π, aπ/3, 6π
Transformations of y =logax when c>0
y = logax +C up y = logax -C down y = loga(x-c) right y = loga(x+c) left y= -logax reflect over x-axis loga(-x)reflect over y-axis y=Clogax vertical stretch/compression y= loga(cx) horizontal stretch/compression
Write the equation of the horizontal line through the point (5,1)
y =1
Trigonometric Functions of Angles Values of the trigonometric functions of an angle. Let p(x,y) be any point on the terminal ray of an angle θ is standard position (other than the origin) and let r=√x²+y² Then r>0, and Sin θ =
y/r
Trigonometric functions of real numbers tan(t) =
y/x (x>0)
Trigonometric functions of real number tan(t) =
y/x = sin(t)/cos(t)
A horizontal line through (h,k) is
y= k
Even-odd propery: tanx
y=tan(x) is an odd function symmetric about the origin Graph of y=tanx
Empty Set or null set
{ } or ∅
integers
{-3,-2,-1,0,1,2}
whole numbers
{0,1,2,3}
natural numbers (counting numbers)
{1,2,3}
Rational numbers
{anything as a fraction} {a/b, b≠0) terminating decimal = 0.5 = 1/2 non-terminating decimal = 0.666 = 2/3
The distance between a and b is
|b-a| = |a-b|
An angle θ has infinitely many coterminal angles:
θ + 2nπ, θ measured in radians θ + n ×360°, θ measured in degrees where n is an integer
180° = how many radians
π radians
