Crash course Unit 2 Unit 3 First semester Unit 1 Functions Trigonometry Function: An equation/set of points where each 1st

coordinate is assigned with only one 2nd coordinate. Domain: A collection of all the

1st coordinates in a function Range: A collection of all the 2nd

coordinates in a function Parent Functions Equation: f(x)=x

Continuity: All values of x

Extrema: None

End Behavior: x-> , y-> ; x-> - , y-> -

Symmetry: Odd about the origin

Asymptotes: None

Domain:

Range: Quadratic

Function Equation: f(x)=

Continuity: For all values of x

Extrema: Absolute Relative Min- (0,0)

End Behavior:

Symmetry: Even

Asymptotes: None

Domain:

Range: Cubic Function Equation: f(x=

Continuity: All values of x

Extrema: none

End behavior:

Symmetry: Odd

Asymptotes: none

Domain:

Range: Basic Characteristics Continuity: Yes No No When x=2, the graph is not continuous

Type of discontinuity: Point Discontinuity

or

Removable Discontinuity When x= 3, it is not continuous

Note: The dots can either be both empty or one filled, you cannot have both dots filled.

Type of discontinuity: Jump Discontinuity

or

Non- removable discontinuity Asymptotes: The line that governs where the function goes. Vertical Asymptotes cannot be crossed/touched but horizontal ones can be. Insight to Trig:

If f(x)=

m>n

if the degree 'm' is >, then there is no Horizontal Asymptote or hole.

m<n

if the degree of 'm' is <, then the horizontal asymptote is- y=0

m=n

if the degree of m is equal, then the horizontal asymptote is the ratio of the leading coefficients. y= Equation: f(x)= 1/x

Continuity: x 0

Extrema: None

End Behavior:

Symmetry: Odd

Asymptotes: H.A- y=0

V.A- x=0

Domain: x 0

Range: y 0 Rational

Function Square Root

Function Equation: f(x)=

Continuity:

Extrema: Absolute min- (0,0)

End Behavior:

Symmetry: None

Asymptotes: None

Domain: [0, )

Range: [0, ) Equation: f(x)= |x|

Continuity: All values of x

Extrema: Absolute/Relative Min- (0,0)

End Behavior:

Symmetry: Even

Asymptotes: None

Domain:

Range: [0, ) Absolute Value

Functoin Exponential

Function Equation: f(x)=

Continuity:All values of x

Extrema: None

End behavior:

Symmetry: None

Asymptotes:H.A- y=0

Domain:

Range: (0, ) Logarithmic

Function Equation: f(x)=

Continuity: (0, )

Extrema: None

End Behavior: x->0, y-> -

Symmetry: None

Asymptotes: V.A- x=0

Domain: (0, )

Range: Sine Function Equation: f(x)= sin(x)

Continuity: All values of x

Extrema: Absolute/Relative max- 1

Absolute/Relative Min- -1

End Behavior: Inconclusive

Symmetry: Odd

Asymptotes: None

Domain:

Range: [-1, 1] Cosine Function Equation: f(x)=cos(x)

Continuity: All values of x

Extrema:Absolute/Relative max- 1

Absolute/Relative Min- -1

End Behavior: Inconclusive

Symmetry: None

Asymptotes: None

Domain:

Range: [-1, 1] Parent Functions Cont. Transformations Lets take f(x) and transform it. The meaning is in light blue.

g(x)=f(x) + k

g(x)=f(x) - k

g(x)=f(x-h)

g(x)=f(x+h) Algebra: Meaning: Shift up 'k' units

Shift down 'k' units

Shift right 'h' units

Shift left 'h' units f(x)= vs g(x)= -

h(x)= The function is flipped over the x-axis

The function is flipped over the y-axis Determining Symmetry

with just the function Odd Functions: f(-x)= -f(x)

ex: f(x)= -3x ... Is f(x) odd?

f(-x)= (-x)^3 - 3(-x)

= - + 3x

f(x) is odd because - +3x is the

opposite of f(x) Even Functions: f(-x)= f(x)

ex: f(x)= |x| ... is f(x) even?

f(-x)= |-x|

= |x|

f(x) is even because |-x| is equal to |x| Algebra of Functions Algebra of functions is easy. Here are some basic

rules:

f(x) + g(x)= (f+g)(x)

f(x) + g(x)= (f-g)(x)

f(x)g(x)= (fg)(x)

f(x)/g(x)= (f/g)(x)

Composition of functions:

(f o g)(x)= f(g(x))

(f o g)(x)=f(g(x))

f(x)=3( -x+4)+1

f(x)=3 -3x+12+1

f(x)=3 -3x+13 Lets use this function pair as an example

f(x)= 3x+1

g(x)= -x+4

We are going to determine (f o g)(x) and (g o f)(x) (g o f)(x)= g(f(x))

g(x)= (3x+1)^2 -(3x+1)+4

g(x)= 9 +6x+1-3x-1+4

g(x)= 9 +3x+4 Ashley L.

Christine M.

Connie H. Rational Fin Inverses of Functions Inverse Function: The relation formed when the independent variable is exchanged with the dependent variable (x and y are switched) Steps to Finding the inverse of a function 1: State the original Function

2: Switch x and y

3: solve for x EX: y=3x+2

x=3y+2

y=(x+2)/3 Green: y=3x+2 Red: y=(x+2)/3 Characteristics of inverse functions y=x^2 x=y^2 Notice in the example above the domain of the original equation is the same as the rang of the inverse. And the range of the original graph is the same as the Domain of the inverse. This is true of all functions and their inverses. Synthetic Division What is it? Synthetic division is a method of factoring of a polynomial by another polynomial of the first degree. Why do we use it? We use synthetic division because it typically reduces the amount of time needed to factor a polynomial Ex: How did you get there?? 2 1: you take the constant of the polynomial with the degree of one, and reverse the sign

2: Place the coefficients of the variables of the polynomial with the higher degree in a line 4: Carry down the first number ( in this case 1)

5: multiply the outside number (in this case the 3) by the number you just carried down

6: Take the product of those two numbers and place it under the second coefficient

7: Take the sum of the coefficient and the new product and carry it down

Repeat steps 5,6,and 7 until you can go no further.

8. To find the answer take all the numbers at the bottom and starting from the rightmost number that is not 0, use those numbers as coefficients for variables going up a degree each number starting with the constant Answer What if there's a remainder?? If there is a remainder, you simply divide the remainder by the first degree polynomial. It can stay in fraction form. Answer: Inequalities How to solve Polynomial inequalities 1: Factor the polynomial 2: Take the constant of each individual first degree polynomial and reverse the sign. These will be the critical values (zeroes on the graph). -NOTE- If there is a lone x, it counts as a c.v. (0) 3: Place the critical values on a number line. Check the numbers located between the c.v.'s to see if the inequality is valid through those x-values If the inequality is greater than or EQUAL TO of less than or EQUAL TO. The c.v.'s are included in the valid x-values. Use solid points on the number line to show this and hollow points if it is not greater than or EQUAL TO of less than or EQUAL TO. Critical values: (0,-2,1) Graphing Polynomial Inequalities Place the critical values as zeroes of the function and sketch the graph based on your knowledge of parent graphs. Unit Circle In this case the equation was a cubic function and you can clearly tell that the inequality is only valid in the shaded regions of the graph A circle that has a radius of 1 is known as the Unit Circle, and it can be used to learn trigonometric values and angles. Each point on the unit circle also corresponds with a sine, cosine, and tangent value. The points on the circle correspond with certain angles, which can be measured in degrees or radians. Angles Each angle on the unit circle can be measured in degrees or radians, and the measurement can be converted from one form to the other.

By multiplying the measure of the angle in degrees by pi and then dividing by 180, you get the measurement in radians.

By multiplying the angle measurement in radians by 180 and dividing by pi, you get the measurement in degrees. Cosine, Sine, and Tangent Values of the Unit Circle Answer: Rational inequalities Each point has an ordered pair.

The x value of the ordered pair is the cosine value.

The y value of the ordered pair is the sine value.

By dividing the sine value by the cosine value, you get the tangent value. 1: Set one side equal to zero

2:collect all terms under 1 denominator

3:Determine the critical values

4:Place the critical values on a number line and check regions

5:Solve C.V.'s: 7,-2,1 = The unit circle also consists of 4 quadrants, shown using Roman numerals. 1: 2: 3: 4: 5: Note: because the c.v.'s are in the denominator, the inequality won't work on those values All Sine Tangent Cosine In the first quadrant, all values are positive.

In the second, all sine values are positive.

In the third, all tangent values are positive.

In the fourth, all cosine values are positive. Decomposition Breaking down Fractions 1: Factor the denominator of the function

2: Set the original fraction equal to A divided by one factor of the denominator PLUS B over the other factor of the denominator

-combine the fractions under a common denominator (cross multiply)

4: Set the x values equal to each other and the A+B values equal to the constant. Solve for A and B the plug the values into the original equation A+B=2 4A-2B=26 Secant, Cosecant, and Cotangent 6A=30 A=5 20-2B=26 -2B=6 B= -3 Sine, cosine, and tangent each have a related function, which can be found by using the reciprocal of the original functions or values.

The reciprocal for sine is cosecant. Answer: sine 1 The reciprocal for cosine is secant. 1 cosine The reciprocal for tangent is cotangent 1 tangent Green highlighter means cross multiply Arcs and Area An arc is a segment of a circles circumference. To find an arc length on the unit circle, multiply the radius by the angle measure in radians. To find the area of a part of the unit circle, multiply theta by 1/2 and multiply that by r^2, if using radians. For degrees, use this: Inequalities: The relationship between two expressions that don't equal each other; using greater than, greater than or equal to, less than, and less than or equal to. Decomposition: The breaking down of a whole into separate parts. Reference Angles The angle can be found by subtracting theta from 180 (if using degrees), or from pi (if using radians). Reference triangles A reference triangle is the acute angle formed when the point of an angle is connected to the x axis with a perpendicular line, which forms a triangle. A reference angle is the acute angle measure from the terminal side of an angle to the x axis. Using the reference triangle, you can find the trigonometric ratios by using Soh, Cah, Toa. And you can find the ratios for the inverse functions by finding the reciprocal of these ratios. Co-terminal Angles Co-terminal angles are angles that have the same terminal side. To find a co-terminal angle, add or subtract 360 degrees to an angles measure. If using radians, add or subtract 2pi. Co-functions Co-functions are the functions related to each of the trigonometric function. NOTE: in this image, B is used instead of theta If using radians, subtract theta from pi/2 instead of 90. Turns To find the measure of an angle that was rotated, first you need to know that clockwise rotations are negative, whereas counterclockwise rotations are positive.

Multiply the turn by 360 degrees, or by 2pi.

For example: two and one half of a rotation counterclockwise

(5/2)(360)=900 degrees.

or

(5/2)(2pi)=5pi/2 Graphing Sine Function Maximum= 1

Minimum=-1

Range is y= [-1,1]

Domain is all real numbers

Amplitude=1

Equilibrium value=0

Period=2pi

Sine is an odd function

Frequency is 1/(2pi) Cosine Function Maximum= 1

Minimum=-1

Range is y= [-1,1]

Domain is all real numbers

Amplitude=1

Equilibrium value=0

Period=2pi

Cosine is an even function

Frequency is 1/(2pi) Secant Function Maximum= infinity

Minimum=negative infinity

Range is y= (-infinity, -1] U [1, infinity)

Domain= x cannot equal (pi/2)+Kpi

Equilibrium value=0

Period=2pi

Secant is an even function

Asymptotes x=(pi/2)+Kpi

Frequency= 1/(2pi) Cosecant Function Maximum= infinity

Minimum=negative infinity

Range is y= (-infinity, -1] U [1, infinity)

Domain= x cannot equal (pi)+Kpi

Equilibrium value=0

Period=2pi

Cosecant is an odd function

Asymptotes x=pi+Kpi

Frequency= 1/(2pi) Tangent Function Maximum= infinity

Minimum=negative infinity

Range is y= all real numbers

Domain= x cannot equal (pi/2)+Kpi

Equilibrium value=0

Period=pi

Tangent is an odd function

Asymptotes x=(pi/2)+Kpi

Frequency= 1/(pi) Cotangent Function Maximum= infinity

Minimum=negative infinity

Range is y= all real numbers

Domain= x cannot equal (pi)+Kpi

Equilibrium value=0

Period=pi

Cotangent is an odd function

Asymptotes x=pi+Kpi

Frequency= 1/(pi) NOTE: Cotangent is always decreasing NOTE: Tangent is always increasing All transformations for trigonometric functions follow the same rules as regular functions Trigonometry Graphing Terms Equilibrium- the point that is halfway between the maximum and minimum values

Amplitude- the vertical distance between the equilibrium value and the maximum value or the minimum value

Period- The horizontal distance needed to complete one cycle of a function

Frequency- The reciprocal of the period

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# Math Project 1st sem

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