How To Graph Polynomial Functions Using End Behavior, Multiplicity & Zeros

This paragraph introduces the topic of graphing polynomial functions, emphasizing the importance of understanding end behavior, multiplicity, and zeros. It begins with a discussion on the graph of y equals positive x squared, highlighting its characteristics as an even function with a degree of two. The end behavior is described as 'up, up,' meaning as x approaches both positive and negative infinity, y also approaches positive infinity. The paragraph then contrasts this with the graph of y equals negative x squared, which is also even but has a 'down, down' end behavior. The concept of even and odd exponents and their impact on the graph's end behavior is introduced, along with a brief mention of graphing y equals positive x cubed and y equals negative x cubed.

This paragraph focuses on summarizing the rules for determining the end behavior of polynomial functions based on their degree and leading coefficient. For even-degree polynomials with a positive leading coefficient, the end behavior is 'up, up,' while for those with a negative leading coefficient, it is 'down, down.' The paragraph then addresses odd-degree polynomials, explaining that the end behavior alternates, with 'down, up' for a positive leading coefficient and 'up, down' for a negative one. A table is mentioned as a helpful tool for remembering these rules, and the discussion shifts to the behavior of graphs near x-intercepts, considering different multiplicities of the intercepts.

The focus of this paragraph is on identifying the x-intercepts (zeros) of a polynomial function and determining their multiplicity. Using the polynomial function y equals x plus 2 times x minus 1 squared times x minus 4 as an example, the process of setting each factor equal to zero to find the x-intercepts is explained. The concept of multiplicity is introduced, with the behavior of the graph at each x-intercept depending on the multiplicity of the factor. The overall degree of the polynomial is determined by adding the exponents, which in this case results in a degree four polynomial.

This paragraph delves into the process of sketching the graph of a polynomial function with an even degree and a positive leading coefficient. The end behavior is identified as 'up, up,' and the method for sketching the graph from the x-intercepts is described. The behavior at each x-intercept is determined by the multiplicity of the corresponding factor, with a straight line crossing the x-axis for a multiplicity of 1, and a bounce off the x-axis for a multiplicity of 2. The graph is then sketched by connecting the x-intercepts, taking into account the behavior at each intercept. The relative heights of the peaks are determined by plugging in values, and in this example, the peaks are found to be of the same height due to the symmetry of the graph about x equals 1.

The final paragraph addresses the determination of peak heights in the graph of a polynomial function and the importance of symmetry in the graph. It is noted that in the specific example given, the two peaks have the same height due to the symmetry of the graph about the line x equals 1. This symmetry is attributed to the equal multiplicities of the two x-intercepts and their equal distance apart. The paragraph emphasizes that while this is the case for the given example, it may not hold true for all polynomial functions, and plugging in numbers is the best way to determine the relative heights of the peaks.