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Polynomial and Power Functions N N 1 2 F (X) = Anx + an 1X − +

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Polynomial and Power Functions N N 1 2 F (X) = Anx + an 1X − + polynomial functions polynomial functions Polynomial Functions MHF4U: Advanced Functions You have seen polynomial functions in previous classes. Common polynomial functions include constant, linear, quadratic and cubic functions. A polynomial function has the form Polynomial and Power Functions n n 1 2 f (x) = anx + an 1x − + ... + a2x + a1x + a0 − J. Garvin where n is a whole number and ak is a coefficient. The degree of the polynomial is the exponent of the greatest power. The leading coefficient is an, and the constant term is a0. J. Garvin— Polynomial and Power Functions Slide 1/17 Slide 2/17 polynomial functions polynomial functions Polynomial Functions Classifying Polynomial Functions Polynomial functions of degree 2 or greater can be drawn Example using a series of smooth, connected curves, as in the example Is the function f (x) = 3x5 x2 + 4x 1 a polynomial − − below. function? f (x) is a polynomial function. It has degree 5, a leading coefficient of 3, and a constant term of 1. − Note that there are no x3 and x4 terms. This is because the coeffients on these terms, a3 and a4, are both zero. J. Garvin— Polynomial and Power Functions J. Garvin— Polynomial and Power Functions Slide 3/17 Slide 4/17 polynomial functions polynomial functions Classifying Polynomial Functions Classifying Polynomial Functions Example Example Is the function f (x) = 2x 5 a polynomial function? 1 − Is the function f (x) = 3x 2 a polynomial function? f (x) is not a polynomial function, but an exponential f (x) is not a polynomial function, but a radical function. function. The exponents in a polynomial function are always whole Don’t be confused when the variable is the exponent. numbers. J. Garvin— Polynomial and Power Functions J. Garvin— Polynomial and Power Functions Slide 5/17 Slide 6/17 polynomial functions polynomial functions Classifying Polynomial Functions End Behaviour Example Recall that a function’s end behaviour describes how extreme Is the function f (x) = 2x4 + 5x2 3x 1 a polynomial values of the independent variable (x) affect those of the − − − function? dependent variable (f (x)). We can specify the end behaviour by using the quadrant f (x) is not a polynomial function, since the final term has a names, Q1-Q4. negative exponent. When we want to specify specific values, we say that “as x A graph of f (x) shows that this is not a polynomial function. approaches some value, f (x) approaches some other value.” This is denoted mathematically by “as x A, f (x) B” → → where both A and B are either constants or infinity. J. Garvin— Polynomial and Power Functions J. Garvin— Polynomial and Power Functions Slide 7/17 Slide 8/17 polynomial functions polynomial functions End Behaviour Power Functions Example A polynomial function with the form f (x) = axn is called a State the end behaviour of the polynomial function below. power function. The graphs of f (x) = x2, g(x) = x3, h(x) = x4 and j(x) = x5 are below. The function extends from Q3-Q1. As x , → −∞ f (x) , and as x , f (x) . J. Garvin— Polynomial and Power Functions J. Garvin— Polynomial and Power Functions Slide 9/17→ −∞ → ∞ → ∞ Slide 10/17 polynomial functions polynomial functions Power Functions Power Functions A polynomial function with degree four is called a quartic All power functions with even degree have the following function. characteristics: 4 The quartic function f (x) = x is similar in shape to the The x- and f (x)-intercept is the origin. quadratic function g(x) = x2. • The domain is ( , ). • −∞ ∞ The range is [0, ) if a > 0, and ( , 0] if a < 0. • ∞ −∞ The end behaviour is from Q2-Q1 (as x , • → −∞ f (x) and as x , f (x) ) if a > 0. → ∞ → ∞ → ∞ The end behaviour is from Q3-Q4 (as x , • → −∞ f (x) and as x , f (x) ) if a < 0. → −∞ → ∞ → −∞ The graph is symmetric in the f (x)-axis. • J. Garvin— Polynomial and Power Functions J. Garvin— Polynomial and Power Functions Slide 11/17 Slide 12/17 polynomial functions polynomial functions Power Functions Power Functions A polynomial function with degree five is called a quintic All power functions with odd degree have the following function. characteristics: 5 The quintic function f (x) = x is similar in shape to the The x- and f (x)-intercept is the origin. cubic function g(x) = x3. • The domain is ( , ). • −∞ ∞ The range is ( , ). • −∞ ∞ The end behaviour is from Q3-Q1 (as x , • → −∞ f (x) and as x , f (x) ) if a > 0. → −∞ → ∞ → ∞ The end behaviour is from Q2-Q4 (as x , • → −∞ f (x) and as x , f (x) ) if a < 0. → ∞ → ∞ → −∞ The graph is symmetric about the origin. • J. Garvin— Polynomial and Power Functions J. Garvin— Polynomial and Power Functions Slide 13/17 Slide 14/17 polynomial functions polynomial functions Power Functions Power Functions Example The end behaviour is Q2-Q4, so the power function has odd Describe the characteristics of a possible power function that degree. could describe the graph below. Given the end behaviour, the leading coefficient of the function is negative. The graph is symmetric about the origin, further confirming the odd degree. The function is “flatter” near the origin than the graph of g(x) = x3, implying one of two possible scenarios: the degree is 5 or greater. • there is a vertical compression. • J. Garvin— Polynomial and Power Functions J. Garvin— Polynomial and Power Functions Slide 15/17 Slide 16/17 polynomial functions Questions? J. Garvin— Polynomial and Power Functions Slide 17/17.
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