Section 3.2 Polynomial Functions and Their Graphs

1. Section 3.2Polynomial Functions and Their Graphs JMerrill 2005 Revised 2008

2. What is a polynomial? • An expression in the form of f(x) = anxn + an-1xn-1 + … + a2x2 + a1x + ao where n is a non-negative integer and a2, a1, and a0 are real numbers. • The function is called a polynomial function of x with degree n. • A polynomial is a monomial or a sum of terms that are monomials • Polynomials can NEVER have a negative exponent or a variable in the denominator! • The term containing the highest power of x is called the leading coefficient, and the power of x contained in the leading terms is called the degree of the polynomial.

3. Significant features • The graphs of polynomial functions are continuous (no breaks—you draw the entire graph without lifting your pencil). This is opposed to discontinuous functions (remember piecewise functions?). • This data is continuous as opposed to discrete.

4. Examples of Polynomials

5. Significant features • The graph of a polynomial function has only smooth turns. A function of degree n has at most n – 1 turns. • A 2nd degree polynomial has 1 turn • A 3rd degree polynomial has 2 turns • A 5th degree polynomial has…

6. Cubic Parent Function Draw the parent functions on the graphs. f(x) = x3

7. Quartic Parent Function Draw the parent functions on the graphs. f(x) = x4

8. Graph and Translate Start with the graph of y = x3. Stretch it by a factor of 2 in the y direction. Translate it 3 units to the right.

9. Graph and Translate Start with the graph of y = x4. Reflect it across the x-axis. Translate it 2 units down.

10. A parabola has a maximum or a minimum Any other polynomial function has a local max or a local min. (extrema) Max/Min Local max min max Local min

11. Leading Coefficient Test • As x moves without bound to the left or right, the graph of a polynomial function eventually rises or falls like this: • In an odd degree polynomial: • If the leading coefficient is positive, the graph falls to the left and rises on the right • If the leading coefficient is negative, the graph rises to the left and falls on the right • In an even degree polynomial: • If the leading coefficient is positive, the graph rises on the left and right • If the leading coefficient is negative, the graph falls to the left and right

12. End Behavior • If the leading coefficient of a polynomial function is positive, the graph rises to the right. y = x3 + … y = x5 + … y = x2

13. Finding Zeros of a Function • If f is a polynomial function and a is a real number, the following statements are equivalent: • x = a is a zero of the function • x = a is a solution of the polynomial equation f(x)=0 • (x-a) is a factor of f(x) • (a,0) is an x-intercept of f

14. Example • Find all zeros of f(x)=x3 – x2 – 2x • Set function = 0 0 = x3 – x2 – 2x • Factor 0 = x(x2 – x – 2) • Factor completely 0 = x(x – 2)(x + 1) • Set each factor = 0, solve 0 = x 0 = x – 2; so x = 2 0 = x + 1; so x = -1

15. You Do • f(x)=-2x4 + 2x2 • Degree of polynomial? • Even • End behavior? • Falls to the left and falls to the right • Zeros? • X = 0, 1, -1

16. How many roots? How many roots? Multiplicity (repeated zeros) 3 is a double root 3 is a double root 4 roots; x = 1, 3, 3, 4. 3 roots; x = 1, 3, 3.

17. How many roots? How many roots? Roots of Polynomials Triple root – lies flat then crosses axis Double roots Double roots 5 roots: x = 0, 0, 1, 3, 3. 0 and 3 are double roots 3 roots; x = 2, 2, 2