Graphs of Reciprocal Trigonometric Functions

Now, the R-E-C-I-P-R-O-C-A-L!!! you really cant deny that mathematicians from the past have genuinely creative and curious minds. =]

This part of math has nothing new to learn, but only new things to remember...
Since trigonometric function graphs are periodic, they are continuous and will definitely have x-intercepts.

To remember:

1. The x-intercepts of a trigonometric function become the vertical asymptotes of the respective trigonnometric function's reciprocal.

2. The reciprocal of a trigonometric function IS NOT THE SAME as the inverse of the trigonometric function.

3. Practise, practise, practise with the graphing calculator!

Examples:

If y = sin x is graphed, its x-intercept lies at π.
If y = 1/sin x is graphed, its x-intercept of  π becomes its vertical asymptote. Hence, y = 1/sin x has a vertical asymptote at x =  π

The reciprocal of a trigonometric function is 1 divided by the trigonometric function. However, Trigo^-1  is asking for the angle that has the trigonometric ratio equal to x (the inverse).

For further help of this, this video can probably help you understand a little better as it provides detailed examples. Good luckie!!!~

Graphs of Sine, Cosine and Tangent Functions

FIRST OF ALL, you must remember that trigonometric functions like sine, cosine and tangent are periodic. In other words, they are continuous with the same pattern  =D  think of eternal~
I would say this part of math is fairly easy to understand. But it's just the matter of understanding and knowing it well without memorizing.

Remember how polynomial functions could have their transformations? Like the vertical stretch, horizontal shift, etc? Trigonometric functions can too. But there are 'special ways' to understand their transformation. Not that much of a difference, its just the way to find the value of each transformation.

Before that, note there are 3 graphs at the top. Now, these 3 graphs are all trigonometric graphs over ONE CYCLE.

What is a PERIOD?
A period is basically the x value at the end of a cycle (from the beginning of the first cycle to the end of the first cycle). In this case, the period for sinx graph is 2pi, for cosx graph is 2pi, and pi for the tanx graph. It means how 'long' it takes for one cycle to be completed.

What is an AMPLITUDE?
An amplitude is like the 'height' of the graph from the x axis. Ask yourself "How high and how low does the graph go, in terms of the y values?" and you will get the amplitude of the graphs. In this case, the amplitude for the sinx graph is 1unit, for the cosx graph is 1unit, and 1 unit as well for the tanx graph.
To refresh,

y = a sin k (x - d) + c, where

a = vertical stretch / vertical compression
k = horizontal stretch / horizontal compression
d = horizontal shift (left if +, and right if -)
c = vertical shift (upwards or downwards)

Normally, the questions asked in this topic will ask for a new equation based on the information they provide you with, such as the vertical shift, horizontal compression and more. All you need to do is just to fill in the value they provide at the correct space in the general equation. For example,

A sine function has a vertical stretch of 2.
Hence, the equation of the function is  y = 2sinx

However, sometimes they give you the period of the graph and expect you to find its horizontal stretch/compression. Then comes the formula  

Period = 2pi / k, where k is the horizontal stretch/compression.

For example,
Find the value of k if the period is 2pi
Hence, 2pi = 2pi / k   >>>   k = 4.
y = sin4x

Simple as that! If you need more help, Click Here for a YouTube Tutorial

Bye!

By the way. . .

After learning about speedometers and secants yesterday, I became mathcrazee while driving my way home.

I noted the mileage on my dashboard and the time I left college. Drove home as usual and noted my end mileage and the time I arrived home. Just for the fun of it, I calculated my average rate of speed throughout my journey. 

I subtracted the difference in my beginning and end mileage and divided it by the time I reached home. As a result, I found out that I was driving at 80km/h. 

Ok I know...sounds crazy. But why not apply these stuffs to real life?? Wouldn't it be much easier to remember or understand it this way, instead of blindly memorizing from the book?

Try it yourself. It really works! 

Now...it's the Slope of Tangents and Instantaneous Rate of Change

In the previous post, I explained all about slopes of a secant on a graph. Just refreshing, secant line is a line connecting any 2 points on a graph. Whereas, a tangent line is a line passing through only 1 point on a graph.

The point on the graph is where the instantaneous rate of change takes place ; that is, the
change in y ÷change in x without an interval
Unlike the slope of a secant line, we have 2 reference points to determine its slope. How about a tangent line where there is only ONE point to refer to?

Steps:
1. Draw a tangent line through the point of reference ; a line cutting through only 1 point on the graph. NOTHING MORE THAN THAT ONE POINT!

2.  As you can see, the tangent cuts 1 point on the graph. Now, take any 2 points from the yellow straight line, a.k.a the tangent line, and find its slope, just the way you found the slope for a secant line.
3. When you have found your slope of the tangent line, this is called the instantaneous rate of change at the red point when x = ...

Note : You have to know the secant line well enough first to be able to master the tangent line questions.

See you!

Slopes of Secants and Average Rate of Change

Average rates of change are like a mixture of Physics and a little basic maths. It's basically the y-values divided by its x-values.  In other words, it is like the gradient/slope of any graph with respect to its x-values.

Let's just say...

A circular wave is formed on the surface of a basin filled with water. The table below shows the radius of the circular wave during the first 10s.

Time, t (s)	Radius, r (m)
0	0
1	2
2	4
3	6
4	8
5	10
6	12
7	14
8	16
9	18
10	20

If a question asks, determine the average rate of change of the radius, this means,

∆ y ÷ ∆ x = yx^-1

in this case, the specific formula would be... 

∆ r ÷ ∆ t = rt^-1

then,

(20 - 0) m ÷ (10 - 0) s = 2ms^-1

Note:

- Secant is the name for the line on a graph which connects ANY TWO points on a graph ; be it a quadratic graph, cubic graph or what not.

- Remember to have the x value written with a power of negative 1 (the small -1 on the top of it) or the answer will have no meaning

This part of math is just about UNDERSTANDING its formula, NOT memorizing it, but spending time to PRACTISE on it. I've found a pretty useful website that gives you examples and its solutions as well as some necessary explanation. Check it out Here!

Until next time!~

From the Unsensible to the "Ohhhh this makes sense!!!"

It's finally clearer now. Well, "Equations and Graphs of Polynomial Functions" isn't as difficult after all. Not until you differentiatet the terms properly. Phew, now I don't need to look like a blur girl who never attended math class before. =D

So! Two terms, that both make the world turn differently are...

1. Even Degree

As I've mentioned in my previous posts, even degree is a degree with an even number (2,4,6,8,...). Thus, a function with an even degree can be x to the power of 'something'.

WHEREAS *drum rolls*

2. Even Function (the graph looks like a mirror reflection across the y-axis)

- has a LINE SYMMETRY at x = 0

- ALL the powers of x's in the function are EVEN powers

- f(x) = f(-x)

  since any negative number to the power of an even power results in its original function, therefore,    

  when the function is EQUAL to the negative of its function, the function is even.

AND...

3. Odd Function (the graph looks like a mirror reflection across the y-axis)

- has a POINT SYMMETRY at (0,0)

- ALL the powers of x's in the function are ODD powers

- f(-x) ≠ - f(x)

   since any negative number to the power of an odd power results in a totally different function,

   therefore, when the function is NOT EQUAL to the negative of its function, the function is odd.

Warm Note : A function with a mix of odd and even powers is a 'neither odd nor even function'

Interesting way to remember:

Think of the white as the even function, the black as the odd function. Now these 2 functions are distinctly different from 1 another. A neither odd nor even function would be, the grey line seperating the black and white. =) simply simple!

Happy Maths!

Equations & Graphs of a Polynomial Function (going mumbo jumbo)

Oh my goddddddddddddddddddddddddddddddddddddddd!!! This isn't as easy as I expected! Especially with the textbook answers all different from Darren's. Sh*t.....what's going onnn...?? The more I look at those unanswered questions, the more I go mumbo jumbo up in my head. I REALLY do get the concept. But what is up with those weird questions answers? =(

Nevermind. Let's get the concet straight first!

Given the polynomial function

y = x² + 1

   = (x + 1)(x – 1)

Solely from an equation, this is the information you need to spot in order to sketch the function's graph without using a graphing calculator.

i) The sum of exponents of all factors (the equation has 2 factors with an exponent of 1. Thus, The sum of exponents of the equation is 2. note: each bracket is a factor.)

ii) The sign in front of the product of all x's (x x x = x² . Therefore, we can see that the sign in front of the product of all x's in the function is a positive sign)

iii) The x-intercept and y-intercept of the function (substitute y = 0 to get x-intercept and x = 0 to get y-intercept)

iv) The positive/negative sign of f(x) (substitute an x value to determine if f(x) is positive or negative at that particular x point)

 

Similarly, from a given graph, you will be able to identify and state the

i) Degree = 2 ; because it has n x-intercept, n - 1 local maximum/minimum point and its graph extends from quadrant 2 to quadrant 1. This information is enough to show that the graph has an even degree of 2 =D

ii) Sign of leading coefficient = Positive ; because the graph extends from quadrant 2 to quadrant 1

iii) x-intercept = -1, 1

      y-intercept = -1

iv)

Intervals	x < -1	-1 < x < 1	x > 1
Sign of f(x)	+	-	+

The new phrase of the day!

...has a zero at x = ? with order of ?...

Ok Ok...calm down. Read more, you'll understand better.

Let's say... y = -2 (x – 3)²

Expand it,    x - 3 = 0

                    x = 3

Hence, the function has a zero at x = 3 with the order of 2

Not too difficult right? The term "with the order of" is basically just the highest power of the function

Ps: I found this site where this person posted a quartic function question and there were many comments posted there. I think the comments, explanations and answers really help. Click Here

Alright then...not-too-happy-math! Till I solve more questions!