4.1 More Nonlinear Functions and Their Graphs Learn terminology about polynomial functions Find extrema of a function Identify symmetry in a graph of a function Determine if a function is odd, even, or neither

Polynomial Functions (1 of 2) A polynomial function f of degree n in the variable x can be represented by where each coefficient ak is a real number, an ≠ 0, and n is a nonnegative integer. The leading coefficient is an and the degree is n.

Polynomial Functions (2 of 2) The domain of a polynomial function is all real numbers, and its graph is continuous and smooth without breaks or sharp edges.

Absolute and Local Extrema (1 of 2)

Finding Extrema (1 of 2) Maximum and minimum values that are either absolute or local are called extrema. A function may have several local extrema, but at most one absolute maximum and one absolute minimum.

Finding Extrema (2 of 2) It is possible for a function to assume an absolute extremum at two values of x. The absolute maximum is 11. It is a local maximum as well, because near x = −2 and x = 2 it is the largest y- value.

Absolute and Local Extrema (2 of 2) Let c be in the domain of f. f(c) is an absolute (global) maximum if f(c) ≥ f(x) for all x in the domain of f. f(c) is an absolute (global) minimum if f(c) ≤ f(x) for all x in the domain of f. f(c) is an local (relative) maximum if f(c) ≥ f(x) when x is near c. f(c) is an local (relative) minimum if f(c) ≤ f(x) when x is near c.

Example: Modeling ocean temperatures (1 of 4) The monthly average ocean temperature in degrees Fahrenheit at Bermuda can be modeled by a. Graph f in [1, 12, 1] by [50, 90, 10]. b. Estimate the absolute extrema. Interpret the results.

Example: Modeling ocean temperatures (2 of 4) Solution a. Here is the graph.

Example: Modeling ocean temperatures (3 of 4) b. Many graphing calculators have the capability to find maximum and minimum y- values. An absolute minimum of about 61.5 corresponds to the point (2.01, 61.5). This means the monthly average ocean temperature is coldest during February, when it reaches a minimum of 61.5 ºF.

Example: Modeling ocean temperatures (4 of 4) An absolute maximum of about 82 corresponds to the point (7.61, 82.0), meaning that the warmest average ocean temperature occurs in August when it reaches a maximum of 82 ºF. (You could say late July since x ≈ 7.61.)

Even Symmetry If a graph was folded along the y-axis, and the right and left sides match, then the graph would be symmetric with respect to the y-axis. A function whose graph satisfies this characteristic is called an even function.

Even Function A function f is an even function if f(−x) = f(x) for every x in its domain. The graph of an even function is symmetric with respect to the y-axis.

Odd Symmetry Another type of symmetry occurs with respect to the origin. If the graph could rotate, the original graph would reappear after half a turn. This represents an odd function.

Odd Function A function f is an odd function if f(−x) = −f(x) for every x in its domain. The graph of an odd function is symmetric with respect to the origin.

Identifying Odd and Even Functions The terms odd and even have special meaning when they are applied to a polynomial function f. If f(x) contains terms that have only odd powers of x, then f is an odd function. Similarly, if f(x) contains terms that have only even powers of x (and possibly a constant term), then f is an even function.

Example: Identifying odd and even functions (1 of 2) Identify whether the function is odd, even, or neither. x –3 –2 –1 1 2 3 f(x) 10.5 –0.5 Solution The function defined by the table has domain D = {−3, −2, −1, 0, 1, 2, 3}. Notice that f(−3) = 10.5 = f(3), f(−2) = 2 = f(2), f(−1) = −0.5 = f(1) and f(−0) = –2 = f(0). The function f satisfies the statement f(−x) = f(x) for every x in D. Thus f is an even function.

Example: Identifying odd and even functions (2 of 2) Identify whether the function f(x) = x³ − 5x is odd, even, or neither. Solution Since f is a polynomial containing only odd powers of x, it is an odd function. This also can be shown symbolically as follows.