University:
High School
Course:
High School Algebra
Academic year:

2020
Views:

292

Pages:

4
Author:

Destinee Fuller

4.1 Graphing Polynomial Functions A. VOCABULARY & CONCEPTS Polynomial Exponents are whole numbers (0, 1, 2, 3, ...) Coefficients are real numbers (non-imaginary) Example of a polynomial: f(x) = 3x⁴ + 2x³ + ½x - 7 Example of a non-polynomial: f(x) = 2x³ - x² - 2x⁻¹ + 5 Standard Form Polynomial is written from greatest to lowest exponent Example: f(x) = x - 2x³ + 5x⁴ + 4 - 1/3x² → 5x⁴ - 2x³ - 1/3x² + x + 4 Degree Greatest exponent Example: f(x) = 5x⁴ - x² + 2 Degree = 4 Leading Coefficient Coefficient of the term with the greatest exponent Example: f(x) = 7 - 1.6x² - 5x -1.6x² - 5x + 7 ↑ leading coefficient = -1.6 Evaluating Substituting (plugging in) numbers Example: g(x) = 2x⁵ - 3x⁴ - 7x + 8 g(-2) = 2(-2)⁵ - 3(-2)⁴ - 7(-2) + 8 = 2(-32) - 3(16) + 14 + 8 = -64 - 48 + 14 + 8 = -90 Common Polynomial Functions Degree Type Example 0 Constant y = # 1 Linear f(x) = 5x + 7 2 Quadratic f(x) = 2x² - x + 9 3 Cubic f(x) = x³ - x² + 3x 4 Quartic f(x) = x⁴ + 2x - 1 B. EXAMPLES Decide whether each function is a polynomial function. If so, write in standard form and state its degree, type, and leading coefficient. 1 1. a. g(x) = -0.8x⁴ + √2x - 12 g(x) = √2x⁴ - 0.8x³ - 12 **YES** degree: quartic type: quartic LC = √2 b. h(x) = -x² + 7x⁻¹ + 4x **Not a polynomial function** **because the exponent -1 is not a whole number.** Describe the end behavior of the graph of the function. a. f(x) = -0.5x⁴ + 2.5x² + x - 1 **even** **negative LC** As x → ∞, f(x) → -∞ As x → -∞, f(x) → -∞ b. f(x) = 0.25x³ - x² - 1 **odd** **positive LC** As x → ∞, f(x) → ∞ As x → -∞, f(x) → -∞ 3. Describe the degree (odd or even?) and the leading coefficient (positive or negative?) a. Ends matching = even degree Facing up = positive LC b. Opposite ends = odd degree positive LC c. Even degree Negative LC 4. Graph the function a. f(x) = x⁴ - 2x³ - 4x² + 4 LC: 1 ↑↑ A table of values is shown: | x | y | |---|---| | -2 | 12 | | -1 | 2 | | 0 | 4 | | 1 | 0 | | 2 | -4 | The graph of the function is plotted on a coordinate plane. The graph shows a curve that increases from left to right, passing through the points (-2, 12), (-1, 2), (0, 4), (1, 0), and (2, -4).