M3U4D10 Warm Up Write the equation in vertex form if the directrix is y = 10 and the focus is (3,4). y = -1/12 (x – 3) Collect Warmups Quiz

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Graphing Polynomial Functions OBJ: Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. (F-IF.8a)

Polynomial Functions The largest exponent within the polynomial determines the degree of the polynomial. Polynomial Function in General Form Degree Name of Function 1Linear 2Quadratic 3Cubic 4Quartic

Polynomial Functions f(x) = 3 ConstantFunction Degree = 0 Maximum Number of Zeros: 0

f(x) = x + 2 LinearFunction Degree = 1 Maximum Number of Zeros: 1 Polynomial Functions

f(x) = x 2 + 3x + 2 QuadraticFunction Degree = 2 Maximum Number of Zeros: 2 Polynomial Functions

f(x) = x 3 + 4x Cubic Function Degree = 3 Maximum Number of Zeros: 3 Polynomial Functions

Quartic Function Degree = 4 Maximum Number of Zeros: 4 Polynomial Functions

Leading Coefficient The leading coefficient is the coefficient of the first term in a polynomial when the terms are written in descending order by degrees. For example, the quartic function f(x) = -2x 4 + x 3 – 5x 2 – 10 has a leading coefficient of -2.

The Leading Coefficient Test As x increases or decreases without bound, the graph of the polynomial function f (x)  a n x n  a n-1 x n-1  a n-2 x n-2  …  a 1 x  a 0 (a n  0) eventually rises or falls. In particular, For n odd:a n  0 a n  0 As x increases or decreases without bound, the graph of the polynomial function f (x)  a n x n  a n-1 x n-1  a n-2 x n-2  …  a 1 x  a 0 (a n  0) eventually rises or falls. In particular, For n odd:a n  0 a n  0 If the leading coefficient is positive, the graph falls to the left and rises to the right. If the leading coefficient is negative, the graph rises to the left and falls to the right. Rises right Falls left Falls right Rises left

As x increases or decreases without bound, the graph of the polynomial function f (x)  a n x n  a n-1 x n-1  a n-2 x n-2  …  a 1 x  a 0 (a n  0) eventually rises or falls. In particular, For n even:a n  0 a n  0 As x increases or decreases without bound, the graph of the polynomial function f (x)  a n x n  a n-1 x n-1  a n-2 x n-2  …  a 1 x  a 0 (a n  0) eventually rises or falls. In particular, For n even:a n  0 a n  0 If the leading coefficient is positive, the graph rises to the left and to the right. If the leading coefficient is negative, the graph falls to the left and to the right. Rises right Rises left Falls left Falls right The Leading Coefficient Test

Example Use the Leading Coefficient Test to determine the end behavior of the graph of f (x)  x 3  3x 2  x  3. Falls left y Rises right x

Determining End Behavior Match each function with its graph. A. B. C. D. C B A D

x-Intercepts (Real Zeros) Number Of x-Intercepts of a Polynomial Function A polynomial function of degree n will have a maximum of n x- intercepts (real zeros). Find all zeros of f (x) = -x 4 + 4x 3 - 4x 2.  x 4  4x 3  4x 2  We now have a polynomial equation. x 4  4x 3  4x 2  0 Multiply both sides by  1. (optional step) x 2 (x 2  4x  4)  0 Factor out x 2. x 2 (x  2) 2  0 Factor completely. x 2  or (x  2) 2  0 Set each factor equal to zero. x  0 x  2 Solve for x. The zeroes are (0,0) and (2,0)

Extrema Turning points – where the graph of a function changes from increasing to decreasing or vice versa. The number of turning points of the graph of a polynomial function of degree n  1 is at most n – 1. Local maximum point – highest point or “peak” in an interval o function values at these points are called local maxima Local minimum point – lowest point or “valley” in an interval o function values at these points are called local minima Extrema – plural of extremum, includes all local maxima and local minima

Extrema

Number of Local Extrema A linear function has degree 1 and no local extrema. A quadratic function has degree 2 with one extreme point. A cubic function has degree 3 with at most two local extrema. A quartic function has degree 4 with at most three local extrema. How does this relate to the number of turning points?

Comprehensive Graphs The most important features of the graph of a polynomial function are: 1.intercepts, 2.extrema, 3.end behavior. A comprehensive graph of a polynomial function will exhibit the following features: 1.all x-intercepts (if any), 2.the y-intercept, 3.all extreme points (if any), 4.enough of the graph to exhibit end behavior.

Classwork M4U3D9-10 Graphing Polynomial Functions p. 30 COMPLETE TOGETHER

Homework U4D9 Graphing Polynomial Functions WS p. 31