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Basic Classes of Functions: Fresh Take

  • Recognize the degree of a polynomial, find the roots of quadratic polynomials, and describe the graphs of basic odd and even polynomial functions
  • Graph a piecewise-defined function
  • Explain the difference between algebraic and transcendental functions
  • Sketch the graph of a function that has been shifted, stretched, or reflected from its initial graph position

Polynomial Functions

The Main Idea 

  • A polynomial function is of the form f(x)=ax+ax¹+...+ax+a, where n is a non-negative integer.
  • Key Terms:
    • Degree: The highest power of x in the polynomial
    • Leading Term: The term with the highest degree
    • Leading Coefficient: The coefficient of the leading term
  • Types of Polynomials:
    • Linear: Degree 1 (e.g., f(x)=mx+b)
    • Quadratic: Degree 2 (e.g., f(x)=ax²+bx+c)
    • Cubic: Degree 3
    • Higher degrees: Quartic (4), Quintic (5), etc.
  • The zero function f(x)=0 is considered a polynomial of degree 0.

Identify the degree, leading term, and leading coefficient for each of the following functions:

  1. h(y)=2y3y²+5y7
  2. k(s)=0.5s³+2s1
  3. m(w)=w+3w2w²+w

Zeros of Polynomial Functions

The Main Idea 

  • For quadratic functions f(x)=ax²+bx+c, zeros can be found using:
    • Factoring (if possible)
    • The quadratic formula: x=b±(b²4ac)(2a)
  • The discriminant (b²4ac) determines the nature of quadratic solutions:
    • Positive: Two distinct real roots
    • Zero: One real root (double root)
    • Negative: No real roots
  • Higher-degree polynomials may have more complex methods for finding zeros.

Find the zeros of the quadratic function g(x)=2x²+5x3.

Graphs of Polynomial Functions Basics

The Main Idea 

  • The degree of a polynomial greatly influences its graph’s shape.
  • Even-degree polynomials:
    • Both ends of the graph point in the same direction (up or down)
    • Similar to a parabola, but may be flatter near the origin
  • Odd-degree polynomials:
    • Ends of the graph point in opposite directions
    • One end goes up, the other goes down
  • The sign of the leading coefficient determines the direction of the graph’s ends:
    • Positive leading coefficient: right end goes up
    • Negative leading coefficient: right end goes down
  • As the degree increases, graphs tend to flatten near the origin and become steeper away from it.

Describe the end behavior and basic shape of the graph for the polynomial function:

f(x)=2x+3xx²+7

Piecewise-Defined Functions

The Main Idea 

  • A piecewise-defined function uses different formulas for different parts of its domain.
  • Each piece of the function has its own subdomain, which together form the function’s complete domain.
  • Piecewise functions may be continuous or discontinuous, depending on how the pieces connect.
  • Each piece is graphed separately within its specified domain, paying special attention to the transition points.

Sketch a graph of the function

f(x)={2x,x2x+2,x>2

The cost of mailing a letter is a function of the weight of the letter. Suppose the cost of mailing a letter is 49¢ for the first ounce and 21¢ for each additional ounce. Write a piecewise-defined function describing the cost C as a function of the weight x for 0<x3, where C is measured in cents and x is measured in ounces.

Algebraic Functions and Transcendental Functions

The Main Idea 

  • Algebraic Functions:
    • Involve addition, subtraction, multiplication, division, rational powers, and roots
    • Include polynomial, rational, and root functions
    • Examples: f(x)=x³+2x1,g(x)=(x²+1)(x2),h(x)=(x+3)
  • Transcendental Functions:
    • Cannot be expressed using a finite number of algebraic operations
    • Main types: trigonometric, exponential, and logarithmic functions
    • Examples: sin(x),ex,log(x)

Is f(x)=x2 an algebraic or a transcendental function?

Classify the following functions as algebraic or transcendental, and identify their specific type if possible:

  1. f(x)=(x+2x²5)(x1)
  2. g(x)=sin(x²)+cos(x)
  3. h(x)=3(x²+1)+log(x)
  4. k(x)=(2x+3x)x

Transformations of Functions 

The Main Idea 

  • Vertical Shift: f(x)±c
    • +c shifts up, c shifts down
    • Does not change shape of graph
  • Horizontal Shift: f(x±c)
    • c shifts right, +c shifts left
    • Does not change shape of graph
  • Vertical Scaling: cf(x)
    • |c|>1 stretches vertically, 0<|c|<1 compresses vertically
    • Changes height of graph
  • Horizontal Scaling: f(cx)
    • 0<|c|<1stretches horizontally, |c|>1 compresses horizontally
    • Changes width of graph
  • Reflections:
    • f(x) reflects over x-axis
    • f(x) reflects over y-axis
  • Multiple Transformations: y=cf(a(x+b))+d
    • Apply in order: horizontal shift, horizontal scaling, vertical scaling, vertical shift

Transform the following function which has the base tool-kit function f(x)=x:

g(x)=2(x1)+3