Unformatted text preview: Increasing A function is increasing on an interval I if F (x1 ) < f (x2 ) whenever x1 < x2 in I . Decreasing A function is decreasing on an interval I if F (x1 ) > f (x2 ) whenever x1 < x2 in I . Section 1.2 Mathematical Models
Polynomial Functions A function is called a polynomial if it is of the form: P (x) = an xn + an−1 xn−1 + · · · + a2 x2 + a1 x + a0 where n is a nonnegative integer and the a’s are constants called the coeﬃcients of the polynomial. The domain of the polynomial is the reals, and the degree of the polynomial is n. A polynomial of degree one is a linear function: P (x) = mx + b. A polynomial of degree 2 is called quadratic: P (x) = ax2 + bx + c. A polynomial of degree 3 is a cubic function: P (x) = ax3 + bx2 + cx + d. Power Functions A power function is of the form: f (x) = xa . And we have several cases: a = n, n is a positive integer This reduces in essence to the polynomials. a = 1/n, n is a positive integer This is a root function. f (x) = x1/n =n For n = 1 it is the square root. √ x. a = −1 This is the reciprocal function. It is a hyperbola with the coordinate axes as its asymptotes. yx = 1. Rational Functions are just the ratio of two polynomials. f (x) = and has the domain all values x such that Q(x) = 0. P (x) Q(x) Chapter 1 Review Algebraic Functions are functions that are constructed using algebraic operations on polyno functions can have extremely bizarre looking graphs! Trigonometric Functions Know the graphs! sin x, cos x, tan x (2π radians = 360 degrees) Exponential Functions Exponential functions are of the form f (x) = ax where a is called is a positive constant. Much more later. Logarithmic Functions The log functions are the inverse of the exponential functions. Muc Transcendental Functions Any function which is not algebraic is transcendental. The trig, and their inverses are examples of transcendental functions. A transcendental function is usua ized by an inﬁnite series expansion, and there are a huge number of transcendental functions. Section 1.3: New Functions From Old Functions
Vertical and Horizontal Shifts: Suppose c > 0, to obtain the graph of y = f (x) + c, shift the graph of y = f (x) a distance c units upward y = f (x) − c, shift the graph of y = f (x) a distance c units downward y = f (x − c), shift the graph of y = f (x) a distance c units to the right y = f (x + c), shift the graph of y = f (x) a distance c units to the left Vertical and Horizontal Stretching and Reﬂecting: Stretching: If c > 1 then the graph is the graph of y = f (x) stretched by a factor of c in the vertical direction. Reﬂection: The graph of y = −f (x) is the graph of y = f (x) reﬂected about the x-axis, ordered pair has changed from (x, f (x)) to (x, −f (x)). ...
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