Graphing Polynomial Functions

4.1 Graphing Polynomial Functions A. VOCABULARY & CONCEPTS Polynomial: a monomial or sum of monomials. The exponents are whole numbers, coefficients are real numbers (non imaginary) (0, 1, 2, 3, ...) (no negatives) (f(x) = 3x^4 + 2x^2 - x^2 + \frac{1}{2}x - 7) ← polynomial function (g(x) = 2x^3 - x^2 + x^{-1} + 5) ← NOT a polynomial function Standard form of a polynomial function: Terms are written in descending order of exponents from left to right. (f(x) = x - 2x^3 + 5x^4 + 4 - \frac{1}{3}x^2) (f(x) = 5x^4 - 2x^3 - \frac{1}{3}x^2 + x + 4) ← Standard form Degree of a polynomial: The largest exponent is the degree of a polynomial. (f(x) = 5x^4 - x^2 + 2) degree: 4 (g(x) = x + 1) degree: 1 Leading coefficient: The first coefficient (number in front of the variable) when written in standard form. (f(x) = 7 - 1.6x^2 - 5x) (f(x) = -1.6x^2 - 5x + 7) Leading coefficient: -1.6 Common Polynomial Functions | Degree | Type | Example | |---|---|---| | 0 | Constant | (f(x) = 14) | | 1 | Linear | (f(x) = 5x - 7) | | 2 | Quadratic | (f(x) = 2x^2 + x - 9) | | 3 | Cubic | (f(x) = x^3 - x^2 + 3x) | | 4 | Quartic | (f(x) = x^4 + 2x - 1) | 6. End behavior: Symbols Meaning (f(x) \rightarrow +\infty) as (x \rightarrow +\infty) The function approaches positive infinity as x approaches positive infinity. (As x gets bigger, the function goes up) (f(x) \rightarrow +\infty) as (x \rightarrow -\infty) The function approaches positive infinity as x approaches negative infinity (As x gets smaller, the function goes up) (f(x) \rightarrow -\infty) as (x \rightarrow +\infty) The function approaches negative infinity as x approaches positive infinity (As x gets bigger, the function goes down) (f(x) \rightarrow -\infty) as (x \rightarrow -\infty) The function approaches negative infinity as x approaches negative infinity (As x gets smaller, the function goes down)   End Behavior of Polynomial Functions Degree: odd Degree: odd Leading coefficient: positive Leading coefficient: negative \(f(x) \rightarrow +\infty\) \(f(x) \rightarrow +\infty\) as \(x \rightarrow +\infty\) as \(x \rightarrow -\infty\) \(f(x) \rightarrow -\infty\) \(f(x) \rightarrow -\infty\) as \(x \rightarrow -\infty\) as \(x \rightarrow +\infty\) Degree: even Degree: even Leading coefficient: positive Leading coefficient: negative \(f(x) \rightarrow +\infty\) \(f(x) \rightarrow +\infty\) as \(x \rightarrow -\infty\) as \(x \rightarrow +\infty\) \(f(x) \rightarrow -\infty\) \(f(x) \rightarrow -\infty\) as \(x \rightarrow -\infty\) as \(x \rightarrow +\infty\) B. EXAMPLES Decide whether each function is a polynomial function. If so, write in standard form and state its degree, type, and leading coefficient.   a. (f(x) = -2x^{3} + 5x + 8) already in standard form degree: 3 type: cubic LC: -2 b. (g(x) = -0.8x^{3} + \sqrt{2}x^{4} - 12) yes Standard form: (\sqrt{2}x^{4} - 0.8x^{3} - 12) degree: 4 type: quartic LC: √2 c. (h(x) = -x^{2} + 7x^{-1} + 4x) Not a polynomial function because exponent -1 is not a whole number.