Section 1.2 Flashcards
Mathematical Model
A mathematical model is a mathematical description (often by means of a function or an equation) of a real-world phenomenon such as the size of a population the demand for a product, the speed of a falling object, the concentration of a product in a chemical reaction, the life expectancy of a person at birth, or the cost of emission reductions.
Linear Function
If y is a linear function of x, it means that the graph of the function is a line, so we can use the slop-intercept form of the equation of a line to write a formula for the function as y=f(x)=mx+b where m is the slop of the line and b is the y-intercept.
Empirical Model
Empirical models are used when there is no physical law or principle to help formula a model, and are based entirely on collected to find a curve that “fits” the data in a sense that it captures the basic trend of the data points.
Polynomial
A function P is called a polynomial if P(x)=a(sub n)xn + a(sub n-x)x(n-1) + … + a(sub 2)x*2 + a(sub 1)x + a(sub 0) where n is a nonnegative integer and the numbers a(sub 0), a(sub 1), a(sub 2), … , a(sub n) are constants called the coefficients of the polynomial.
Domain of Polynomial
The domain of all polynomials is all real numbers.
Degree of Polynomial
If the leading coefficient a(sub n) doesn’t equal 0, than the degree of the polynomial is n.
Polynomial of 1st degree
P(x) = mx+b or linear function
Polynomial of 2nd degree
P(x) = ax*2 + bx + c or quadratic function
Polynomial of 3rd degree
P(x) = ax3 + bx2 + cx + d (where a doesn’t equal 0) or cubic function
Power Function
A function of the form f(x) = x*a, where a is a constant, is called a power function.
Power Function a=n, where n is positive integer
The general shape of the graph of f(x) = xn depends on whether n is even or odd. If n is even, then f(x) = xn is an even function and its graph is similar to a parabola y=x2. If n is odd, then f(x) = xn is an odd function and its graph is similar to that of y = x3. As n increase, the graph of y = xn becomes flatter near 0 and steeper when |x| is greater than or equal to 1. (If x is small, then x2 is smaller, x3 is even smaller, etc.)
Power Function a=1/n, where n is a positive integer
Root Function - For even values of n, the graph of y = nrt(x) is similar to that of y = sqrtx (half sideways parabola)
- For n=3 we have the cube root function f(x) = 3rt(x) whose domain is all real numbers (recall that every real number has a cube root) and whose graph is a sideways x3 graph, which is similar to other xodd numbers/
Power Function a=-1
Reciprocal Function - f(x) = x*(-1) = 1/x. Its graph has the equation y=1/x or xy=1=, and is a hyperbola with the coordinate axes as its asymptotes.
Rational Functions
Rational function f is a ration of two polynomials: f(x) = (P(x))/ (Q(x)) where P and Q are polynomials. The domain is all values of x such that Q(x) doesn’t equal 0.
Algebraic Function
A function f is called an algebraic function if it can by constructed using algebraic operations (addition, subtraction, multiplication, division, and taking roots) starting with polynomials. Any rational function is a algebraic function.
