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 = a for x < 0 and y = b for x > 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'.
   


2. 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.
   
   


3. 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

where a is any real number.

Slide the value of a in the graph below to observe the changes of graph for

1. a > 1
   
   1. a is odd number
   2. a is even number
   3. a is other number
2. 0 < a < 1
3. a < 0
   
   1. a is odd number
   2. a is even number
   3. a is other number

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Quadratic-Linear

A rational function with quadratic as numerator and linear as denominator might be able to be simplified to
.
Slide the values of a, b, c, d and e to view the changes of the graph and the asymptotes. Take note that the asymptotes are x = e (red dotted line) and (blue dotted line).


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Quadratic-Quadratic

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

.
Slides the values of a, b, c, d, e and f  to view the changes of the graph and the position of the asymptotes. Take note that there are 2 vertical asymptotes at x = e and x = f, and 1 horizontal asymptote, .


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Linear-Quadratic

A rational function with linear function as numerator and quadratic function as denominator might be able to be written as:
.
You may slide the values of a, b, c and d to see the changes of the graphs and asymptotes. You might discover that there are two vertical asymptotes, x = c and x = d, and one horizontal asymptote y = 0.


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Linear-Linear

A rational function with linear function for both denominator and numerator is in the form of
.
Slide the values of a, b, c and d to observe the changes of the graph and the position of the asymptotes. Observe that the asymptotes are and .


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Constant-Linear

A rational function with constant function as numerator and linear function as denominator is in the form of
    or    .
The first graph below shows the first function mentioned above. Slide the values of a and b to look at the changes of the graphs and also the position of the asymptotes. Observe that the vertical asymptote is x = b and the horizontal asymptote is always y = 0.


The following graph shows the second function. Slide the values of a, b and c to look at the changes of the graphs and also the position of the asymptotes. Observe that the vertical asymptote is now  ; whereas the horizontal asymptote remains as y = 0 .



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Rational Function

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


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 available on top.

Area Between Two Curves

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



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)
;
b) g(x) - f(x) from x = a to x = b if g(x) > f(x)
.
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 Below The Curve

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 . Let's take a look at the graph below:


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?

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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. Here, we are going to look at the estimation of area using this method by changing the number of rectangles (n) built between x = a to x = b.

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 . 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 ways to estimate the area below a curve y = f (x) for a < x < b is using the method of Upper Sum. Here, we are going to look at the area estimation using this method by changing the number of rectangles (n) built between x = a to x = b.

Look at the graph below, let's consider when n = 5 (5 rectangles are built). 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 . 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. The blue solid curve is the graph of f (x), and the green dotted curve is f '(x).



Recall the concept f '(a) is the slope of f(x) at x = a, while comparing with the graph above:
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 any function (remember, a function is a one-to-one or many-to-one relationship) and you may try to sketch its derivative graph.

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Logarithmic Function

A basic logarithmic function has a form of f (x) = log_base (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 log_b (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 (the value of b) of the red curve to e, which is approximately 2.7 and compare with the blue one.



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

1. a > 0, the graph is increasing;
2. a = 0, the graph is constant (which is not considered as an exponential function)
3. 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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Exponential Function

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


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

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

What other features that you can find for exponential function?

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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.

1. 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.
   
   


2. 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.
   
   


3. 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.
   
   


4. 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.
   
   

There are many types of functions. I listed out some of them with graphs provided. You may click on any topic under Function through clicking "Content" on top to have further understanding on function and also the shifting/scaling/reflecting of the function.

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Cotangent Function

Cotangent function is a reciprocal for tangent function. If , then . Generally, cotangent function can be written as .

Now, consider only ,
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.


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

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Secant Function

Secant function is a reciprocal for cosine function. If , then . Generally, secant function can be written as .

Now, consider only ,

1. as cos(x) approaches to 0, sec(x) approaches to ∞ or -∞,
2. when cos(x) = 1, sec(x) = 1,
3. when 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.



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


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Cosecant Function

Cosecant function is a reciprocal of sine function. If , then . Generally, cosecant function can be written as .

Now, consider only ,

1. as sin(x) approaches to 0, csc(x) approaches to ∞ or -∞,
2. when sin(x) = 1, csc(x) = 1,
3. when sin(x) = -1, csc(x) = -1.

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



Slide the values of a, b, c and d to observe the changes of the graph . What can you say about the effects of these values towards the function csc (x)?


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Tangent Function

A tangent function generally can be written as f (x) = a tan (bx + c) + d.


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

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Cosine Function

A cosine function generally can be written as f (x) = a cos (bx + c) + d. The cycle repeated after every units. The amplitude of the curve is a units.


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


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Sine Function

A sine function generally can be written as f (x) = a sin (bx + c) + d. Try to move point A in the graph below, point B is the recurring point after one cycle. The period of the cycle is units and the amplitude is a units.


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

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Polynomials

Generally, polynomials are functions of

f (x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀,

where
n is the degree of the polynomials and it must be positive integer,
a_n is the leading coefficient of the polynomials and
a_i are coefficients.

Table below shows some examples of polynomials:

Function	Degree	Leading Coefficient	Type of Polynomial
4	0	4	Constant
5x + 1	1	5	Linear
25 - x2	2	-1	Quadratic
(x-1)2(1-2x)	3	-2	Cubic
(x-1)2(3x-1)2	4	9	Quartic

The domain for polynomial is always (-∞,∞); while the range is also (-∞,∞) for odd degree but for even degree we have to find it according to the function given. The graph shown below are the basic form of polynomials: f (x) = xⁿ and g(x) = -xⁿ. Try to slide the value of n to look at different types of the polynomials.

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Quartic Function

A quartic function is given as f (x) = ax⁴ + bx³ + cx² + dx + e where a, b, c, d and e are coefficients. a controls whether the graph ends upwards or downwards, while e tells us the y-intercept.


Slide the values of a, b, c, d and e to observe the changes of the graph f (x) = ax⁴ + bx³ + cx² + dx + e.

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Cubic Function

A cubic function is given as f (x) = ax³ + bx² + cx + d. The value of a decides the graph is mainly increasing or decreasing, while the value of d tells the y-intercept of the graph.


Slide the values of a, b, c and d to observe the changes of the graph f (x) = ax³ + bx² + cx + d.

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Quadratic Function

A quadratic function is given as f (x) = ax² + bx + c where a, b and c are coefficients. Sometimes, it is also written in the form of f (x) = a(x - h)² + k where (h, k) is the coordinates of the vertex. The graph below shows the quadratic curves in the first form (blue) and the second form (purple). The red dotted curve is the basic form of quadratic equation y = x².


Slide the values of a, b, c, h and k to observe the change of the graph f(x) = ax² + bx + c and f(x) = a(x - h)² + k. Can you get the formulae relating the values of a, b and c with the values of h and k?

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Linear Function

A linear function is given as f (x) = mx + c where m is the slope of the graph and c is the y-intercept.

Slide the values of m and c to observe the changes of graph f (x) = mx + c. You may realize that for

1. m > 0, the graph is increasing;
2. m = 0, the graph is constant;
3. m < 0, the graph is decreasing;

while for

1. c > 0, the graph has positive y-intercept;
2. c = 0, the graph passes through the origin;
3. c < 0, the graph has negative y-intercept.

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Constant Function

Constant function is given as f (x) = c where c is any real number.


Slide the value of c to observe the change of the graph f (x) = c. You may find out that for

1. c > 0, the line falls above the x-axis;
2. c = 0, the line falls on the x-axis;
3. c < 0, the line falls below the x-axis.

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Secant & Tangent

The graph below shows the process of getting instantaneous rate of change (slope of tangent) from average rate of change (slope of secant).

The red curve is given as y = g(x). Take note that the average rate of change of y from point A to point B is
.
Drag point B to observe the change of the slope of secant line, m_s. As point B is getting nearer to point A, the value of the slope of secant line is approaching to the value of the slope of tangent line at point A, m_t.



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