Question

Sketch the graph of each polynomial function. $$f(x)=2 x^{4}$$

   Sketch the graph of each polynomial function.
$$f(x)=2 x^{4}$$

Precalculus

Margaret L. Lial,… 4th Edition

Chapter 3, Problem 9

Instant Answer

Solved by Expert James Kiss

07/27/2021

Step 1

The function $f(x)=2x^{4}$ is a polynomial function of degree 4 with a positive leading coefficient. This tells us that the end behavior of the function is going to be up in both directions, i.e., as $x$ approaches negative infinity, $f(x)$ approaches positive Show more…

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Sketch the graph of each polynomial function. $$f(x)=2 x^{4}$$

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Key Concepts

Polynomial Functions

A polynomial function is an expression involving a sum of powers in one or more variables multiplied by coefficients. Its graph is continuous and smooth, and the behavior of the function is largely dictated by the degree (the highest power) and the leading coefficient. These factors determine the overall shape and end behavior of the graph.

Degree and Leading Coefficient

The degree of a polynomial indicates the maximum number of turning points and the end behavior of the graph. For even-degree polynomials with a positive leading coefficient, both ends of the graph tend to positive infinity, while a negative leading coefficient results in both ends going to negative infinity. The degree and leading coefficient together help in predicting how the graph behaves as x becomes very large or very small.

Even Functions and Symmetry

An even function is one that satisfies the condition f(x) = f(-x) for all x in its domain, which means its graph is symmetric about the y-axis. This symmetry simplifies the graphing process because you can reflect the part of the graph for x ? 0 across the y-axis to obtain the complete graph.

Graph Sketching Techniques

When sketching polynomial functions, it is important to identify key features such as intercepts, turning points, and end behavior. For even-degree polynomials, determine the behavior at the extremes based on the leading coefficient, use symmetry to plot corresponding points on both sides of the y-axis, and then connect the points with a smooth curve to produce an accurate sketch of the graph.

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