Polynomial Functions and Modeling Section 4.1 Polynomial Functions and Modeling

Objectives Determine the behavior of the graph of a polynomial function using the leading-term test. Factor polynomial functions and find the zeros and their multiplicities. Use a graphing calculator to graph a polynomial function and find its real-number zeros, its relative maximum and minimum values, and its domain and range. Solve applied problems using polynomial models; fit linear, quadratic, power, cubic, and quartic polynomial functions to data.

Polynomial Function A polynomial function P is given by where the coefficients an, an - 1, …, a1, a0 are real numbers and the exponents are whole numbers.

Quadratic Function Function: f(x) = x2 – 2x – 3 = (x + 1)(x – 3) Zeros: −1, 3 x-intercepts: (−1, 0), (3, 0) y-intercept: (0, −3) Minimum: −4 at x = 1 Maximum: None Domain: All real numbers, Range:

Cubic Function Function: g(x) = x3 + 2x2 – 11x – 12 = (x + 4)(x + 1)(x – 3) Zeros: −4, −1, 3 x-intercepts: (−4, 0), (−1, 0), (3, 0) y-intercept: (0, −12) Relative minimum: −20.7 at x = 1.4 Relative maximum: 12.6 at x = −2.7 Domain: All real numbers, Range: All real numbers,

Examples of Polynomial Functions

Examples of Nonpolynomial Functions

Polynomial Functions The graph of a polynomial function is continuous; that is, it has not holes or breaks. It is also smooth; there are no sharp corners. Furthermore, the domain of a polynomial function is the set of all real numbers.

The Leading-Term Test If is the leading term of a polynomial function, then the behavior of the graph as or as can be described in one of the four following ways.

Example Using the leading term-test, match each of the following functions with one of the graphs AD that follow. a) b) c) d)

Graphs a. b. c. d.

Solution C Negative Even d) x6 A Positive Odd c) x5 B b) 5x3 D a) 3x4 Graph Sign of Leading Coeff. Degree of Leading Term Leading Term

Finding Zeros of Factored Polynomial Functions If c is a real zero of a function (that is, f (c) = 0), then (c, 0) is an x-intercept of the graph of the function.

Example Find the zeros of To solve the equation f(x) = 0, we use the principle of zero products, solving x  2 = 0 and x + 1 = 0. The zeros of f(x) are 2 and 1. See graph on right.

Even and Odd Multiplicity If (x  c)k, k  1, is a factor of a polynomial function P(x) and (x  c)k + 1 is not a factor and:  k is odd, then the graph crosses the x-axis at (c, 0);  k is even, then the graph is tangent to the x-axis at (c, 0).

Example Find the zeros of f (x) = x3 – 2x2 – 9x + 18. Solution We factor by grouping. f (x) = x3 – 2x2 – 9x + 18 = x2(x – 2) – 9(x – 2). By the principle of zero products, the solutions of the equation f(x) = 0, are 2, –3, and 3.

Example Find the zeros of f (x) = x4 + 4x2 – 45. We factor as follows: f (x) = x4 + 4x2 – 45 = (x2 − 5)(x2 + 9). Solve the equation f(x) = 0 to determine the zeros. We use the principle of zero products.

Example Find the zeros of f (x) = 0.1x3 – 0.6x2 – 0.1x + 2. Approximate the zeros to three decimal places. Solution Use a graphing calculator to create a graph. Look for points where the graph crosses the x-axis. We use the ZERO feature to find them. The zeros are approximately –1.680, 2,154, and 5.526.

Example The polynomial function can be used to estimate the number of milligrams of the pain relief medication ibuprofen in the bloodstream t hours after 400 mg of the medication has been taken. a) Find the number of milligrams in the bloodstream at t = 0, 0.5, 1, 1.5, and so on, up to 6 hr. Round the function values to the nearest tenth. Solution Using a calculator, we compute the function values.

Example continued Using a calculator, we compute the function values.

Example continued b) Find the domain, the relative maximum and where it occurs, and the range. Solution The implications of this application restrict the domain of the function. After the medication has been taken, M(t) will be positive for a period of time and eventually decrease back to 0 when t = 6 and not increase again (unless another dose is taken). Thus the restricted domain is [0, 6].

Example continued To determine the range, we find the relative maximum value of the function using the MAXIMUM feature. The maximum is about 345.8 mg. It occurs approximately 2.15 hr, or 2 hr 9 min, after the initial dose has been taken. The range is about [0, 345.8].