5.2 Graphs of Reciprocal Trigonometric Functions

CHAPTER 5 - Trigonometric function

5.1 Graphs of Sine, Cosine, and Tangent Functions

here's an interesting song! haha:) it's quite entertaining! Well, that is the primary trigonometry identities: [SOH,CAH,TOA]
Sin x = Opposite / Hypotenuse
Cos x = Adjacent / Hypotenuse
Tan x = Opposite / Adjacent

• Basic graphs of trigonometric functions are:

Sine Graph

Cosine Graph

Tangent Graph

• Graphs y = sin x, y = cos x, y = tan x are periodic.

 

Original Graph	y = sin x	y = cos x	y = tan x
+ vertical translation v  Moves upward/downwards by c units	y = sin x + c	y = cos x + c
+ amplitude v  Amplitude increases/decreases by a factor	y = a sin x	y = a cos x	No amplitude because there is no max/min value
+ phase shift v  Shifts by d units to the left/right	y = sin (x-d)	y = cos (x-d)
+ period (k-value) v  Number of cycles in one period	y = sin kx	y = cos kx	π

5.2 Graphs of Reciprocal Trigonometric Functions

Domain = {x Є R | x ≠ kx } , whereby k is an integer.

Range = { y > 1 or y < -1 }

Period = 2π

Vertical Asymptotes = x = kx , whereby k is an integer

Domain = {x Є R | x ≠ (2kx+1)(π/2) } , whereby k is an integer.

Range = { y > 1 or y < -1 }

Period = 2π

Vertical Asymptotes = x = (2kx+1)(π/2) ‘ whereby k is an integer

Domain = {x Є R | x ≠ kπ } , whereby k is an integer.

Range = { -1< y < 1 }

Period = π

Vertical Asymptotes 

5.3 Sinusoidal Functions of the form f(x) = a sin [ k(x-d) ] + c & f(x) = a cos [ k(x-d) ] + c

  f(x) = a sin [ k(x-d) ] + c 
  f(x) = a cos [ k(x-d) ] + c

a = amplitude
k-value = 2 / Period
d = phase shift to left / right
c = vertical translation upwards / downwards

EG:

y = sin x

y = 5 sin x ( where An =5)

y = sin x + 1

= x = kx , whereby k is an integer

5.4 Solve Trigonometric Equations

1.Trigonometric equations can be solved in two ways:

• algebraically by hand
• graphically with technology

2. There are often multiple solutions. So, be sure to find all solutions that lie in the domain of interest.
3. Quadratic trigonometric ewuations can usually be solved by factoring.

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Solving Algebraically

Solving Graphically

Trigonometry!

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Unit 3: Rational Equations

3.1 Reciprocal of a Linear Function

• The reciprocal of a linear function has the form:

                     f(x) = 1 / kx – c

• The restriction on a domain of a reciprocal linear function can be determined by finding the value of x that makes the denominator equal to zero, that is x = c / k.
• The Vertical Asymptote of a reciprocal linear function has an equation of the form x = k / c. 
• The horizontal asymptote of a reciprocal linear function has equation y = 0.
• If k > 0, the left branch of a reciprocal linear function has a negative, decreasing slope, and the right branch has a negative, increasing slope.
• If k < 0, the left branch of a reciprocal linear function has a positive, increasing slow, and the right branch has a positive, decreasing slope.

 3.2 Reciprocal of a Quadratic Function

• Rational quadratic functions can be analyzed using key features:asymptotes, intercepts, slope (positive or negative, increasing or decreasing), domain, range, and positive and negative intervals.
• Reciprocal of quadratic functions with two zeros have three parts, with the middle one reaching a maximum or minimum points. This point is equidistant from the two vertical asymptotes.
• The behavior near asymptotes is similar to that of reciprocals of linear functions.
• All of the behaviors listed above can be predicted by analyzing the roots of the quadratic relation to the denominator.

 3.3 Rational Functions of the Form  f(x) = (ax + b) / (cx + d) 

• A rational function of the form f(x) = (ax + b) / (cx + d) has the following key features:
• The vertical asymptote can be found by setting the denominator equal to zero and solving for x, provided the numerator does not have the same zero.
• The horizontal asymptote can be found by dividing each term in both the numerator and the denominator by x and investigating the behavior of the function as x -> positive or negative infinity.
• The coefficient b acts to stretch the curve but has no effect on the asymptotes, domain, or range.
• The coefficient d shifts the vertical asymptote.
• The two branches of the graph of the function are equidistant from the point of intersection of the vertical and horizontal asymptotes.
• Analysis of End Behavior
• For vertical asymptote
• Substitute a number very close to the VA from the right, and a number from the left
• Analyze the result of that number and express the end behavior
• Whether As x -> VA +/- , y -> +/- infinity
• For horizontal asymptote
• Substitute a very large negative and positive number for x and analyze the behavior of y.
• Express the end behavior with the results from that substitution
• As x -> +/- Infinity, y -> HA from above/below

 3.4 Solve Rational Equations and Inequalities

• To solve rational equations algebraically, start by factoring the expressions in the numerator and denominator to find asymptotes and restrictions.
• Next, multiply both sides by the factored denominators, and simplify to obtain a polynomial equation. Then solve.

3.5 Making Connections With Rational Functions and Equations 

• For Rational inequalities
• Set the right side of the equation zero.
• Factor the expression to find restrictions
• Based on the assumption that x = a / b is true if and only if a * b = x.
• On the left side of the equation, take the denominator and multiply it by the numerator.
• Since the equation is already factored, the roots are clearly shown. Graph or use Interval Table method to find the intervals x which satisfy the equation