Sketch a graph of the function showing all extreme, intercepts and asymptotes. f(x)=x^4+2 x-1

Name: Problem 11, Chapter 0: Calculus: Early Transcendental Functions
Uploaded: 2020-07-08T17:37:26-08:00
Duration: 54 s
Channel: John Irizar
Description: Sketch a graph of the function showing all extreme, intercepts and asymptotes. f(x)=x^4+2 x-1

step 1 I've gone ahead and used our graphing utility to graph the function x to the 4th plus 2x minus 1

Sketch a graph of the function showing all extreme,...

Key Concepts

Graph Sketching Techniques

Graph sketching involves combining all available information about the function, including intercepts, extremes, and end behavior, to create an accurate visual representation. By understanding the nature of the function and analyzing its crucial features, one can effectively sketch a graph that illustrates the key characteristics of the function, such as turning points and the overall shape of the graph.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as the independent variable approaches large positive or negative values. For polynomial functions, this is determined by the leading term. Typically, polynomials do not have vertical, horizontal, or slant asymptotes since they extend to infinity in a predictable manner as dictated by their highest power term.

Extreme Values

Extreme values refer to the local or global minima and maxima of a function. They can be identified by finding the derivative of the function, setting it equal to zero to determine critical points, and then analyzing these points using the first or second derivative tests. Extreme values play a crucial role in understanding the peaks and valleys of the graph.

Intercepts

Intercepts are the points where the graph of the function crosses the axes. The x-intercepts (or roots) are found by solving the equation f(x)=0, whereas the y-intercept is determined by evaluating f(0). These points provide valuable information about the location and shape of the graph.

Polynomial Functions

Polynomial functions are algebraic expressions consisting of terms with non-negative integer exponents. They are smooth and continuous, which means they have no breaks, holes, or asymptotes. The degree of the polynomial determines the overall shape and end behavior of the graph, particularly how the function behaves as x approaches positive or negative infinity.

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