SECTION 4.1 Polynomial Functions 191 Solution (a) p is a polynomial function of degree 3, and it is in standard form. The leading term is x5 ,3 and the constant term is 9. − (b) f is a polynomial function of degree 4. Its standard form is f x x x 3 2. 4 ( ) = − + + The leading term is x3 ,4 − and the constant term is 2. (c) g is not a polynomial function because g x x x , 1 2 ( ) = = so the variable x is raised to the 1 2 power, which is not a nonnegative integer. (d) h is not a polynomial function. It is the ratio of two distinct polynomials, and the polynomial in the denominator is of positive degree. (e) G is a nonzero constant polynomial function, so it is of degree 0.The polynomial is in standard form. The leading term and constant term are both 8. (f) H x x x x x x x x x 2 1 2 2 1 2 4 2 . 3 2 3 2 5 4 3 ( ) ( ) ( ) =− − =− −+=−+ − So, H is a polynomial function of degree 5.The leading term is x2 .5 − Since no constant term is shown, the constant term is 0. Degree Form Name Graph No degree ( ) = f x 0 Zero function The x -axis 0 ( ) = ≠ f x , 0 0 0 a a Constant function Horizontal line with y -intercept 0a 1 ( ) = + ≠ f x x , 0 1 0 1 a a a Linear function Nonvertical, nonhorizontal line with slope 1a and y -intercept 0a 2 ( ) = + + ≠ f x x x , 0 2 2 1 0 2 a a a a Quadratic function Parabola: graph is concave up if > 0; 2a graph is concave down if < 0 2a Table 1 Do you see a way to find the degree of H, in part (f), without multiplying it out? Now Work PROBLEMS 17 AND 21 We have discussed in detail polynomial functions of degrees 0, 1, and 2. See Table 1 for a summary of properties of the graphs of these polynomial functions. If you take a course in calculus, you will learn that the graph of every polynomial function is both smooth and continuous. By smooth , we mean that the graph contains no sharp corners or cusps; by continuous , we mean that the graph has no gaps or holes and can be drawn without lifting your pencil from the paper. See Figures 1(a) and (b). Figure 1 x y (a) Graph of a polynomial function: x y (b) Cannot be the graph of a polynomial function Corner Cusp Gap Hole smooth, continuous DEFINITION Power Function A power function of degree n is a monomial function of the form f x axn ( ) = (2) where a is a real number, a 0, ≠ and n 0 > is an integer. In Words A power function is defined by a single monomial. Power Functions Polynomial functions of degree n, n 0, > that have only one term are called power functions .
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