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!

Characteristics of a Polynomial Function

Did you know???

- Negative leading coefficients result in a negative finite difference. Similarly, a positive leading coefficient results in a positive finite difference.

   i.e. y = -2x³ Since -2 is the leading coefficient and is a negative number, hence, the constant finite 

         difference for this polynomial function will also be a negative constant

- The number of finite differences is the same as the degree of the polynomial function.

   i.e. y = -2x³ Since the degree of the polynomial function is 3, hence, this polynomial function has 3

         finite difference / hence, this polynomial function reaches its constant at the 3rd finite difference

- Simple??? I say very this is pretty straight forward. What makes it simpler is the existence of a formula for finding finite differences without using a graphing calculator!

                                      Constant Finite Difference = Leading Coefficient x Degree!

C.F.D = a x n!

For example,

A polynomial function has 4 finite difference and its constant is -48

Therefore, n = 4   >>>   Remember? The number of finite differences = The degree of a P. function

Using the formula,

C.F.D = a x n!

    -48 = a x (4 x 3 x 2 x 1)

       a = -2

Don't believe me? Try more questions Here

At first, I didn't quite understand how an even/odd degree polynomial function affected the maximum/minimum points of its graph. But after i read through this table, everything seemed to make MUCH more sense! Hoorayyy!!!

Description	Odd Degree (n)		Even Degree (2n)
Leading Coefficient	+	-	+	-
General Shape	Refer to picture A	A reflection in the x-axis of its original shape	A reflection in the x-axis of its original shape	Refer to picture B
Domain	{ x ϵ R }	{ x ϵ R }	{x ϵ R }	{ x ϵ R }
Range	{ y ϵ R }	{ y ϵ R }	{ y ϵ R | y has a restriction }	{ y ϵ R | y has a restriction }
End Behavior	Extends from quadrant 3 to quadrant 1	Extends from quadrant 2 to quadrant 4	Extends from quadrant 2 to quadrant 1	Extends from quadrant 3 to quadrant 4
Number of absolute maximum / minimum points	0	0	1	1
Total number of local maximum / minimum points	Only a maximum of n-1	Only a maximum of n-1	Only a maximum of n-1	Only a maximum of n-1
Total number of x-intercepts	Maximum n, minimum 1	Maximum n, minimum 1	Maximum n, minimum 0	Maximum n, minimum 0

Note :

1) Both ends of the graph with an odd degree faces an opposite direction (1 north, 1 south). Whereas, the ends of the graph with an even degree faces the same direction (both north / both south).

2) ONLY the even degree has :

   - restrictions for its range (y values)

   - ONE absolute maximum / minimum point (ABSOLUTE basically means MOST. So, absolute

     maximum/ minimum point is the point on the graph with even degree that has the highest or lowest

     peak)

3) ALL the types of polynomial graphs can have a maximum number of local maximum / minimum

    points of n-1. Nothing more than that!!!

*** By the way, an absolute maximum / minimum point need not be a turning point on a graph. It can also be an end of the graph which has not restrictions***

I hope this piece of information helped you well! It cleared my doubts, totally!

See you soon!

Power Functions



POWER functions



Power functions are basically the simplest of all polynomial functions. Very basic, but can be tricky at times. Just read the functions carefully, and be sure not to overlook any detail because every part of a power function is significant.

It's a lot easier than you think to imagine the graph of a power function just by STARING at a power function. Of courseeeeeeeeeeeeee, you need some basic knowledge to be able to do magic! No worries, the table below explains everything.

 

 

Key Features	y = xn ; where n is an odd number	y = xn ; where n is an even number
Domain	{ x ϵ R }	{ x ϵ R }
Range	{ y ϵ R }	{ y ϵ R | y ≥ 0 }
Symmetry	Point symmetry about (0,0)	Line symmetry at x = 0
End Behavior	As x → ∞ , y → ∞ As x → -∞ , y → -∞	As x → ∞ , y → ∞ As x → -∞ , y → ∞

Before further explanation, there are new terms that need to be remembered as it will help in understanding future polynomial questions.

          Degree = The highest power in a power / polynomial function

               e.g. y = 3x² + 2x + 1    this means, the degree of the given power function is 2 (since 2 > 1)

          Leading coefficient = The coefficient of the term with the highest power in a power/polynomial function

               e.g. y = 3x² + 2x + 1    this means, the leading coefficient is 3, since it is the coefficient of x²

Tip :

1) The range of a power function remains +∞ at all times ONLY for a power function with an even degree.

2) Only the range of a power function with an even degree has restrictions.

3) For a point symmetry, the coordinate of that point must be stated to show a clearer understanding of the question. Whereas for a line symmetry, the value of x at the symmetry must be stated. (note: x need not necessarily be 0. It can be a whole number depending on the power function)

To practise, try imagining the shapes of the graphs just by analyzing the power function given. And when you've become an expert, try vice versa. When you're an expert in both the ways, you deserve a pat on the shoulder! =)

Wait, wait, wait. It's not the end yet! Just a liiiiittle more...

End Behavior	Function
Extends from quadrant 3 to quadrant 1	y = xn where n is an odd number (if the coefficient of x is a positive)
Extends from quadrant 2 to quadrant 4	y = -xn where the coefficient of x is a negative number (if n is an odd number)
Extends from quadrant 2 to quadrant 1	y = xn where n is an even number (if the coefficient of x is a positive)
Extends from quadrant 3 to quadrant 4	y = -xn where the coefficient of x is a negative number (if n is an even number)

Note that polynomial graphs vary by THEIR DEGREE (odd/even) & THE SIGN OF THEIR LEADING COEFFICIENT (positive/negative)

If you intend to use a graphing calculator, refer to this link Click Here for a step-by-step guideline on how you can plot a graph by just keying in a function in your graphing calculator (I am referring to Texas Instrument TI-84 which looks like below)

 

Happy Math! Until next time...