Question

Show that the graph of $y=a x^5+c x^3+e x$ has the property that if it contains the point $\left(x_1, y_1\right)$, then it also contains $\left(-x_1,-y_1\right)$. Any function whose graph has this property is called an odd function. An odd function is sometimes said to be symmetric with respect to the origin.

Name: Problem 28, Chapter 6: Modern Analytic Geometry
Uploaded: 2020-07-24T22:12:20-08:00
Duration: 1 min 37 s
Channel: Carson Merrill
Description: Show that the graph of y=a x^5+c x^3+e x has the property that if it contains the point (x1, y1), then it also contains (-x1,-y1). Any function whose graph has this property is called an odd function. An odd function is sometimes said to be symmetric with respect to the origin.

   Show that the graph of $y=a x^5+c x^3+e x$ has the property that if it contains the point $\left(x_1, y_1\right)$, then it also contains $\left(-x_1,-y_1\right)$. Any function whose graph has this property is called an odd function. An odd function is sometimes said to be symmetric with respect to the origin.

Modern Analytic Geometry

William Wooton,… 1st Edition

Chapter 6, Problem 28

Instant Answer

Solved by Expert Carson Merrill

07/24/2020

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We need to show that if the graph contains the point $ (x_1, y_1) $, then it also contains the point $ (-x_1, -y_1) $. Show more…

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Show that the graph of $y=a x^5+c x^3+e x$ has the property that if it contains the point $\left(x_1, y_1\right)$, then it also contains $\left(-x_1,-y_1\right)$. Any function whose graph has this property is called an odd function. An odd function is sometimes said to be symmetric with respect to the origin.

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Key Concepts

Odd Functions

An odd function is defined by the property that f(-x) = -f(x) for all values of x in its domain. This definition ensures that the function exhibits a specific type of symmetry, meaning that the output is negated when the input is negated.

Graph Symmetry with Respect to the Origin

Graph symmetry with respect to the origin means that if a point (x, y) is on the graph of the function, then the point (-x, -y) will also be on the graph. This is a direct consequence of the function being odd, as it ensures that the graph remains unchanged under a 180-degree rotation about the origin.

Properties of Polynomial Functions

Polynomial functions composed solely of odd-degree terms naturally exhibit the odd function property. When each term in the polynomial has an odd exponent, substituting -x for x results in the overall function value changing sign, thereby confirming that the polynomial is odd and its graph is symmetric with respect to the origin.

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