Chapter 3

Properties of power function, f(x) = xⁿ, n=positive even integer

1. f is an even function, so its graph is symmetric with respect to the y-axis. 2. The domain is the set of all real numbers. The range is the set of non negative real numbers. 3. The graph always contains the points (-1,1), (0,0), and (1,1). 4. As the exponent n increases in magnitude, the graph becomes more vertical when x<-1 or x>1; but for x near the origin, the graph tends to flatten out and lie closer to the x-axis.

Factor Theorem

1. if f(c) = 0, then x - c is a factor of f(x). 2. if x - c is a factor of f(c) , then f(c) = 0

Finding Horizontal or Oblique Asymptote of a Rational function.

1. if its proper fraction between exponents, then the horizontal asymptote is y = 0. 2. if exponents are equal, the y = a/b → the quotient of the leading coefficients. 3. If it's improper fraction between exponents, then use long division to find the equation of the asymptote.

Properties of power functions, f(x) = xⁿ, n=positive ODD integer

1.f is an odd function, so its graph is symmetric with respect to the origin. 2. the domain and the range are the set of all real numbers. 3. The graph always contains the points (-1,-1), (0,0),and (1,1) 4. As the exponent n is increase\es in magnitude, the graph becomes more vertical when x<-1 or x>1; bur for x near the origin, the graph tends to flatten out and lie closer to the x-axis.

Number of real zeros

A polynomial function cannot have more real zeros than its degree.

Locating vertical asymptotes

A rational function in its LOWEST TERMS (completely factored) has a vertical asymptote of x=r if and only if (x -r ) is a real zero of the denominator. In other words, the real zeros of the denominator will be the vertical asymptotes of a rational function.

Turning Points (theorem)

If f is a polynomial function of degree n, then the graph of f has at most n-1 turning points. if the graph of a polynomial function f has n-1 turning points, the degree of f is at least n.

Definition

Let R denote a function. if, as x → -∞ or as x→∞∞, the values of R(x) approach some fixed number L, then the line y = L is a HORIZONTAL ASYMPTOTE of the graph of R. if, as x approaches some number c, the values |R(x)|→ ∞ [R(x)→∞ or R(x) → -∞, then the line x = c is a VERTICAL ASYMPTOTE of the graph of R. The graph of R never intersects a vertical asymptote.

If r is a zero of odd multiplicity

The sing of f(x) changes from one side to the other side of r. The graph of f CROSSES the x-axis at r.

If r is a zero of even multiplicity

The sing of f(x) does not change from one side to the other side of r. the graph of f TOUCHES the x-axis at r.

Zero of multiplicity N of F

if (x - r)ⁿ is a factor of a polynomial f and (x - r )ⁿ⁺¹ is not a factor of f, then r is called a ZERO OF MULTIPLICITY N OF F

The remainder theorem

let f be a polynomial function. If F(x) is dived by x - c, the the remainder is f(x)