Sketch the graph of the function defined by the given equation. G(x)=(1)/(2)(x^3-x)

Name: Problem 16, Chapter 6: Modern Analytic Geometry
Uploaded: 2020-10-24T13:19:57-08:00
Duration: 1 min 40 s
Channel: Dushyant Barot
Description: Sketch the graph of the function defined by the given equation. G(x)=(1)/(2)(x^3-x)

step 1 We're given the function G of x is x cubed by 2. This is calculated g of minus x, it's minus x w

Sketch the graph of the function defined by the given...

Key Concepts

Polynomial Functions

A polynomial function is a function that can be expressed in the form a_n*x^n + a_(n-1)*x^(n-1) + … + a_0, where the coefficients and exponents are constants. Understanding polynomial functions involves analyzing their degree, coefficients, and behavior, which is essential for graphing and identifying key features such as intercepts, symmetry, and end behavior.

Cubic Functions

A cubic function, which is a type of polynomial function with a degree of three, typically exhibits an S-shaped curve. Its graph can have up to two turning points and usually has a point of inflection where the curvature changes sign. Recognizing the structure of a cubic function is crucial for understanding its overall shape and behavior on the graph.

Intercepts

Intercepts are the points where the graph of a function crosses the axes. The x-intercepts (or roots) are found by setting the function equal to zero and solving for x, while the y-intercept is found by evaluating the function at x = 0. These points are fundamental in sketching the overall shape and location of the graph.

Symmetry

Symmetry in functions, often identified as even or odd symmetry, helps simplify graphing. An odd function, which satisfies f(-x) = -f(x), is symmetric with respect to the origin. Recognizing this property can reduce the work required in graphing by allowing you to reflect points accordingly.

End Behavior

The end behavior of a function describes how the values of the function behave as x approaches positive or negative infinity. For polynomials, the leading term dominates the function's behavior at the extremes. In a cubic function, the sign of the leading coefficient indicates whether the graph rises or falls as x moves towards infinity or negative infinity.

Critical Points

Critical points are values of x where the function's derivative is zero or undefined. These points correspond to local maximums, local minimums, or saddle points and are key in understanding and sketching the function's overall structure, particularly in identifying the turning points of the graph.

Inflection Points

An inflection point is where the function changes concavity (from concave up to concave down or vice versa). It can be found by analyzing the second derivative of the function. Inflection points provide insight into the curvature of the graph and contribute to a more accurate sketch of the function's behavior.

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