Functions-Graphing Logarithmic and Radicals?

Question

What is the standard equation for a radical function? What does the graph look like (shape)? What does the graph for a logarithmic function look like? How can these be used in real life?

Answer 1

Radical functions are f(x) = x ^ (rational number) like x^(1/2) "square root" x^(1/3) "cubed root" but also x^(7/5) the seventh power of the fifth root.

If the exponent is between 0 and 1, they all look like the square root function in the first quadrant, through the origin (0,0), concave down, going off to infinity slowly.

If the exponent is larger than 1, they all look like a parabola in the first quadrant, going off to infinity fatser and faster (concave up).

If the denominator is even, there is no graph for x<0 (domain issue). If the denominator is odd there is a graph for x<0. If the denominator is odd and the numerator is even, it is the first quadrant part flipped over the y-axis into quadrant II. If both numerator and denominator are odd, it is the first quadrant graph flipped through the oringin into quadrant III.

Log(x) has a vertical asymptote at x=0, an x-intercept at (1,0) and goes to infinity (but VERY slowly!!!! --- don't be fooled by graphing calculators, the log graph does NOT have a horizontal asymtptote).

is that enough? :-)

Oh yeah, apps....one interesting radical function application Isaw recently was in biology and related to the size of animals. The idea was energy comes in through our surface area (4*pi*r^2 for a sphere) while we use energy to feul our volume (4/3*pi*r^3) so there should be a relationship between body dimensions which is a radical (power law). The article I was reading was arguing over what that exponent was, 3:2 (conceptual value) or 4:3 (empirical value).

lots of apps for the logairthm scale: pH (acidity in chemistry), dB (decibels in physics), etc.

Answer 2

The standard equation of a radical is half a parabola, turned on its side. this is y=sqrt(x). It increases a lot at first, and then slows down, but still coninues in the upward and right directions.
A log function is quite similar to a radical function. This however, increaees much more slowly upwards. The range of y=log (x), unlike a radical is neg infinity to infinity. (radical is 0 to infinity, inclusive.) A log function has a verticle asymtote at x=0. This means that it will approach the y axis, in the negative direction, but NEVER touch it.
Logs are used in calculating the strength of earth quakes, The Richter Scale. It has to do with powers of 10, meaning that a 5 is 10 times worse than a 4, on the Richter Scale.
Hope that helps!