a polynomial, a trigonometric, and an exponential function.

Question

a) Keeping in mind the rest of the question, write out algebraically and sketch an example of a polynomial, a trigonometric, and an exponential function.

b) How can you tell from looking at your function if it is polynomial, trigonometric or exponential?

c) Generate a table of values for each of your function. [1 mark each, 3 total] Explain how you can tell from looking at your table of values that a function is polynomial, trigonometric or exponential?

d) State the domain and range of each function in a).

e) Give an example of a real life application of each function, and explain how it can be used. [2 marks for each function, 6 total] Provide a detailed solution and an interpretation for each of your functions under that real life application.

Answer 1

poly is of the form ax^2 + bx + c, a trinomial, binomial just 2 terms, variable is the base, constant is the power, a and b are coefficients, c is a constant term

exponential is of the form a2^b + c where b is the power, a is a coefficent, 2 and c are constant terms

trig is like asinx where x is a variable, a is a coefficient, sin = sine = a trig function which is cyclical

poly may appear cyclical inside, but end points approach infinity or negative infinity

exponential approaches infinity at one end, but is asymptotic at the other

you can easily tell the trig function, as it has either sin, cos, tan, csc, sec or cot

you can distinguish poly from exp by the base and power. if a variable base it's poly if variable power it's exp

but sinx = (e^ix - e^ix)/2x

and Euler's identity: e^ix = cosx + isinx

so, trig functions are interchangeable with exponential functions

as one can convert to the other

a table of values will be cyclical if trig

end behavior will distinguish exp from poly. exp will have one end asymptotic, while poly won't

domain for poly is all real numbers, same for exp. but trig is all real numbered angles, usually in degrees or radians, while domain for poly and exp are unitless numbers

range for exp is from just above the asymptote to infinity. range for the poly is either all real numbers or from the y value of the vertex or global extremum to neg/pos infinity. range for trig can be all reals, or between -1 and 1, or all reals except between -1 and 1,

in real life, exp are useful in finance for amortizing loans, calculating interest rates, monthly payments, future values

poly is useful as a trinomial for calculating height of a falling object

trig is useful at sea where a boat has lost radio contact, radar and any communications, left only with a telescope and angle measurements to see how far they are from land and a lighthouse