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Exponential Functions

October 24, 2009 Leave a comment

Exponential functions are any function of the form
y = a^{bx} latex for some constants a and b.

If a and b are both positive then the graph will be an upward curve which tends to infinity as x tends to infinity and tends to 0 as x tends to negative infinity and looks something like the below. Note that all exponential graphs cut the y axis at 1.

The graph of y = 2^x (y equals 2 to the power x)

The graph of y = 2^x (y equals 2 to the power x)

If a is positive and b is negative the graph is simply a reflection of this about the y axis to give the following graph:

The graph of y=2^-x (y equals 2 to the power of minus x)

The graph of y=2^-x (y equals 2 to the power of minus x)

The most import exponential graph is y=e^x because the gradient of this graph is always equal to the value of e^x at that point.

Categories: algebra, calculus Tags: , ,

Tan Graph – y=tan(x)

September 1, 2009 Leave a comment

The graph of y=tanx is different from the other cos and sin graphs as it has a range from -∞ to ∞ and a period of 180° or π radians. The graph of y=tan(x) in radians is shown below

Graph of y=tan(x) in radians

Graph of y=tan(x) in radians

As can be seen from the graph the curve passes through the origin. It has  vertical asymtopes (lines it tends toward but never touches — in this case where the graph goes to infinity) at x =π/2,3π/2,5π/2 and x=-π/2,-3π/2 etc radians or at x=90,270,450 and x=-90,-270 etc degrees.

The graph has a stationary (flat) point whenver it crosses the x-axis.

Sine Graph

The sine function is a periodic function meaning that it repeats itself every so many (in the case of sine 2pi radians or 360o). It has a range of -1 to 1 and has a domain for -∞ to ∞. Starting at the origin it increase to 1 at 90<sup>o</sup> or pi/2 radians and then decrease to -1 at 270<sup>0</sup> or 3pi/2 radians and then returns to 0 and 360<sup>o</sup> or 2pi radians.

On the graph below the angle, in radians, is along the x axis and the value of the sine function for that angle is on the y axis.

Graph of y=sin(x)

Graph of y=sin(x)

Categories: maths, trigonometry Tags: , ,