Friday, May 27, 2011

Piecewise

A piecewise-defined function is a combination of at least two different functions at non-overlapping set of domain. The domain and range of the function is different for each one of them. For examples,

1. Let's say we have two equations $y_1=a$ for $x<0$ and $y_2=b$ for $x\geq 0$. Combine these two equations and we get one piecewise equation as below.

Now, try to drag point A across y-axis. You'll find that point A will 'jump' from the left line to the right when x = 0. Now, click on 'View Domain' and you'll find that a pink point appears on the x-axis. If you drag point A away from the previous position,  the pink points showing the domain on the x-axis. If you want to view the range on the y-axis, simply click on 'View Range'.
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3. For the second piecewise equation
perhaps you can move the values of 'a' and 'b' to view the changes of the graphs and also the domain and range. Likewise, click on 'View Domain', 'View Range' and drag point A to view the domain and range of the function.
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5. For the third piecewise equation
,
try to change the values of 'a' and 'b' to view the changes happen to graph. Then, click on 'View Domain', 'View Range' and drag point A to view the domain and range of the function.
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Power

A power function is defined as
$f(x) = x^{a}$

where a is any real number.

Slide the value of a in the graph below to obeserve the changes of graph for
a) a > 1
i) a is odd number
ii) a is even number
iii) a is other number
b) 0 < a < 1
c) a < 0
i) a is odd number
ii) a is even number
iii) a is other number

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Betty, Created with GeoGebra

Thursday, May 26, 2011

A rational function with quadratic as numerator and linear as denominator might be able to be simplified to

$f(x)=\frac{a(x-b)(x-c)}{d(x-e)}$.

Slide the values of a, b, c, d and e to view the changes of the graph and the asymptotes.

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Betty, Created with GeoGebra

A rational function with quadratic both for numerator and denominator can be written as

$f(x)=\frac{a(x-b)(x-c)}{d(x-e)(x-f)}$.

Slides the values of a, b, c, d, e and f to view the changes of the graph and the position of the asymptotes.

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A rational function with linear function as numerator and quadratic function as denominator might be able to be written as:
$f(x)=a\frac{x-b}{(x-c)(x-d)}$.

You may slide the values of a, b, c and d to see the changes of the graphs and asymptotes.

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Linear-Linear

A rational function with linear function for both denominator and numerator is in the form of

$f(x)=\frac{ax-b}{cx-d}$.

Slide the values of a, b, c and d to observe the changes of the graph and the position of the asymptotes.

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Constant-Linear

A rational function with constant function as numerator and linear function as denominator is in the form of

$f(x)=\frac{a}{x-b}$   or    $f(x)=\frac{a}{bx-c}$.

Slide the values of a, b and c to look at the changes of the graphs and also the position of the asymptotes. The red line is the vertical asymptote whereas the blue line is the horizontal asymptote.

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Betty, Created with GeoGebra

Tuesday, May 24, 2011

Rational Function

A rational function is a division between a polynomial with another polynomial:

$f(x)=\frac{P(x)}{Q(x)}$

If we want to sketch the graph of rational function, there are a few things you need to take note of:

1. Asymptotes:
■Vertical Asymptote(s): Obtained from taking Q(x) = 0.
■Horizontal Asymptote: Exists when the degree of P(x) is at most of the degree of Q(x).
■Oblique Asymptote: Exists when the degree of P(x) is a degree higher than the degree of Q(x).
■Curve Asymptote: Exists when the degree of P(x) is at least two degree higher than the degree of Q(x).

2. Roots/Zeros
Starts from taking P(x) = 0

3. Maximum/Minimum points
Starts from taking P’(x) = 0

You may click on some of the examples listed under Rational Function in the Content alongside.

Sunday, May 22, 2011

Area Between Two Curves

Let say we have 2 functions as given in the graph below:

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If we want to find the area between the two functions from x = a to x = b, we integrate
a) f(x) - g(x) from x = a to x = b if f(x) > g(x)
$\inline \int_{a}^{b}f(x)-g(x)\; dx$;

b) g(x) - f(x) from x = a to x = b if g(x) > f(x)
$\inline \int_{a}^{b}g(x)-f(x)\; dx$.

e.g. Slide the values of a and b to a = -5 & b = 6. The area found is 125.59 (positive). Take note that I take the integration of f(x) - g(x) to find the area falls between the two functions. Now, if we change the values of a and b to a = 9 & b = 12, we get -26.15 for the area. This is because we didn't change the integral to g(x) - f(x), and g(x) > f(x) for 9 < x < 12.

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Area

When we wish to find the area falls between function f(x) with x-axis for a < x < b, we integrate f(x) with respect to x from x = a till x = b, and we write it as $\inline \int_{a}^{b}f(x)\; dx$. Let's take a look at the graph below:

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Try to slide the values of a and b according to the suggested values below and look at the area found.

a) If a = 0, b = 7, f(x) is positive for 0 < x < 7, the area is 23.57 (positive).
b) If a = 7, b = 10, f(x) is negative for 7 < x < 10, the area is -6.9 (negative).
c) If a = 0, b = 10, f(x) is positive for 0 < x < 7 and is negative for 7 < x < 10, the "area" given is 16.67, which is the result of 23.57 + (-6.9), which means the "area" here is a SIGNED AREA.

What do you suggest if you wish to find the total area falls between f(x) and x-axis for 0 < x < 10?

Betty, Created with GeoGebra

Lower Sum

Another of the way to estimate the area below a curve y = f(x) for a < x < b is using the method of Lower Sum. In this method, we decide the number of rectangles to be built between x = a to x = b, which is denoted as n.

Look at the graph below, let's consider when n = 5 (5 rectangles are built). Do take note that for each rectangle, its height is taken from the y-value of the lowest point of the curve; while the width of the rectangle is $\inline \frac{b-a}{n}=\frac{5-0}{5}=1$. The summation of area of all the rectangles, b, gives to 19.5.

If you slide the value of n to higher value, you may see that the rectangles are narrowed and the estimation of area below the curve is getting nearer to the actual area below the curve.

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Upper Sum

One of the way to estimate the area below a curve y = f(x) for a < x < b is using the method of Upper Sum. In this method, we decide the number of rectangles to be built between x = a to x = b, which is denoted as n.

Look at the graph below, let's consider when n = 5 (5 rectangles are built). Do take note that for each rectangle, its height is taken from the y-value of the highest point of the curve; while the width of the rectangle is $\inline \frac{b-a}{n}=\frac{5-0}{5}=1$. The summation of area of all the rectangles, a, gives to 22.

If you slide the value of n to higher value, you may see that the rectangles are narrowed and the estimation of area below the curve is getting nearer to the actual area below the curve.

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Graph of f '(x)

Ever think of drawing the graph of y = f '(x)? First, let's take a look at the graph below:

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Recall the concept f '(a) is the slope of f(x) at x = a:
a) if f(x) is increasing, f '(x) > 0, which means the graph of y = f '(x) falls above the x-axis;
b) if f(x) is constant or reach to an extremum, f '(x) = 0, which means the graph of y = f '(x) falls on the x-axis;
c) if f(x) is decreasing, f '(x) < 0, which means the graph of y = f '(x) falls below the x-axis.

If you wish try out your understanding towards the concept above, perhaps you can ask your friend's help to give you a graph of a function (remember, a function is a one-to-one or many-to-one relationship) and you may try to sketch its derivative graph.

Betty, Created with GeoGebra

Saturday, May 21, 2011

Logarithmic Function

An basic logarithmic function has a form of f(x) = logbase (x), where x > 0 and the base is positive. Usually, two types of logarithmic functions are used:

a) logarithm of base 10 : log (x) or lg (x)
b) logarithm of base e (natural logarithm): ln (x)

The red curve given below is of base 10 by default and it is extended to the form of f(x) = a logb (cx + d) + e in order for you to shift, scale or reflect the graph; while the blue curve is of base e (the natural logarithm), which the function is g(x) = a ln (cx + d) + e. You may try to change the base of the red curve to e, which is approximately 2.7 and compare with the blue one.

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Slide the values of a, b, c, d and e to see the changes of the graph. You may find out that as c > 0 AND

a > 0, the graph is increasing;
a = 0, the graph is constant (which is not considered as an exponential function)
a < 0, the graph is decreasing.

What other features that you can find for logarithmic function?

Logarithmic function is actually the inverse of exponential function. Graphically, inverse reflects at each other through the line y = x. The graph given below is the form of f(x) = a log (bx+c) + d (base 10). Can you find the general form of exponential function given in blue dotted curve? You may check your answer by clicking the checkbox at the bottom right corner of the graph below.

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Betty, May 21, 2011, Created with GeoGebra

Exponential Function

An basic exponential function has a form of f(x) = ax, where a>0 and a≠1. The graph given below is extended to the form of f(x) = abx+c + d in order to shift or scale the function.

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Slide the values of a, b, c and d to see the change of the graph. You may find out that for

0 < a < 1, the graph is decreasing;
a = 1, the graph is constant (which is not considered as an exponential function)
a > 1, the graph is increasing.

What other features that you can find for exponential function?

Betty, Created with GeoGebra

Function

Function is a kind of relationship. There are 4 types of relationship:

1) One-to-one
2) Many-to-one
3) One-to-many
4) Many-to-many

Function is either Type 1) or 2) relationship. e.g.

a) y = 5x + 2 is a function because one value of x yields one value of y. Thus, this function is a one-to-one relationship. Point A is a point on the line y = 5x + 2. No matter how you move the position of point A, you will see that one value of x gives to one value of y.

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b) y = x2 - 1 is a function because two values of x yields to one value of y for all values of x except at y = -1, which is produced by one value of x only, i.e. x = 0. Therefore, this function is a many-to-one relationship. Drag point A and you will find that point A1 has the same y-coordinate with point A but different x-coordinate.

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c) y2 = x is NOT a function because one value of x yields two values of y. For example, when x = 1, we get y = 1, -1. Try to move point A. You can see that there is a corresponding point B (which has the same x-coordinate) has different value of y-coordinate from point A. Thus, this equation is a one-to-many relationship.

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d) x2 + y2 = 25 is NOT a function because this is a many-to-many relationship. For example, when x = 3, we get y = 4, -4. On the other hand, when y = 4, we get x = 3, -3. Try to move point A and look at the changes of position of point B, C and D.

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There are many types of functions. I listed out some of them with graphs provided. You may click on any topic under Function at the left side bar to have further understanding on Function and also the shifting/scaling/reflecting of the function.

Betty, Created with GeoGebra

Friday, May 20, 2011

Cotangent Function

Cotangent function is a reciprocal for tangent function. If $f(x)=\tan x$, then $g(x)=\frac{1}{\tan x}=\cot x$. Generally, cotangent function can be written as $f(x) = a \cot (bx+c) +d$.

Now, consider only $f(x) = \cot x=\frac{1}{\tan x}$,
as tan(x) approaches to 0, cot(x) approaches ∞ or -∞,
as tan(x) approaches ∞ or -∞, cot(x) approaches 0.

You may change the values of a = 1, b = 1, c = 0 and d = 0 and think about the construction of cot(x) from tan(x). After that, you may alter the values to look at the shifting, scaling or reflecting of the graphs.

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Slide the values of a, b, c and d to observe the changes of the graph $f(x) = a \cot (bx+c) +d$. What can you say about the effects of these values towards the function cot (x)?

Betty, Created with GeoGebra

Secant Function

Secant function is a reciprocal for cosine function. If $f(x) = \cos x$, then $g(x)=\frac{1}{\cos x}=\sec x$. Generally, secant function can be written as $f(x) = a \sec (bx+c) +d$.

Now, consider only $f(x) = \sec x=\frac{1}{\cos x}$,
as cos(x) approaches to 0, sec(x) approaches ∞ or -∞,
as cos(x) = 1, sec(x) = 1,
as cos(x) = -1, sec(x) = -1.

You may change the values of a = 1, b = 1, c = 0 and d = 0 and think about the construction of sec(x) from cos(x). After that , you may alter the values to look at the shifting, scaling or reflecting of the graphs.

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Slide the values of a, b, c and d to observe the changes of the graph $f(x)=a\sec(bx+c)+d$. What can you say about the effects of these values towards the function sec (x)?

Betty, Created with GeoGebra