- 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)=a₁xⁿ+a₂xⁿ⁻¹+...+aₙ₋₁x+aₙ, 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:
- h(y)=2y⁴−3y²+5y−7
- k(s)=−0.5s³+2s−1
- m(w)=w⁵+3w⁴−2w²+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±√(b²−4ac)(2a)
- The discriminant (b²−4ac) 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+5x−3.
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)=−2x⁵+3x⁴−x²+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)={2−x,x≤2x+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<x≤3, 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+2x−1,g(x)=(x²+1)(x−2),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,log₂(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:
- f(x)=(x⁴+2x²−5)(x−1)
- g(x)=sin(x²)+cos(x)
- h(x)=3√(x²+1)+log₃(x)
- 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√(x−1)+3