The given equation defines a function whose graph has one of...

The given equation defines a function whose graph has one of the following general shapes. In each case, identify the appropriate shape or shapes. (In some cases, there may be two possibilities.) GRAPH CAN'T COPY.
$G(x)=-\frac{2}{3} x^5+x^4-3 x^3+2$

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Key Concepts

Polynomial Functions

A polynomial function is an expression consisting of variables and coefficients combined using only addition, subtraction, multiplication, and non-negative integer exponents. The general behavior and shape of its graph depend on its degree and the coefficients of its terms.

Degree of the Polynomial

The degree of a polynomial is the highest power of the variable present in the expression. It determines the overall shape of the graph and the maximum number of turning points that the function can have. In this context, a fifth degree (quintic) function may have up to four turning points.

Leading Coefficient Test

The leading coefficient test is used to determine the end behavior of a polynomial function. For a fifth degree polynomial, if the leading coefficient is negative, the graph will rise to positive infinity as x approaches negative infinity and fall to negative infinity as x approaches positive infinity.

End Behavior

End behavior describes how the values of a polynomial function behave as x approaches positive or negative infinity. This is primarily determined by the leading term. In the case of a quintic function with a negative leading coefficient, the end behavior will be asymmetrical, with opposite directions at the two ends of the graph.

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