For each polynomial function: (a) List each real zero and...

For each polynomial function: (a) List each real zero and its multiplicity. (b) Determine whether the graph crosses or touches the $x$ -axis at each $x$ -intercept. (c) Determine the maximum number of turning points on the graph. (d) Determine the end behavior; that is, find the power function that the graph of fresembles for large values of $|x|$. $$ f(x)=\left(x-\frac{1}{3}\right)^{2}(x-1)^{3} $$

For each polynomial function:
(a) List each real zero and its multiplicity.
(b) Determine whether the graph crosses or touches the $x$ -axis at each $x$ -intercept.
(c) Determine the maximum number of turning points on the graph.
(d) Determine the end behavior; that is, find the power function that the graph of fresembles for large values of $|x|$.
$$
f(x)=\left(x-\frac{1}{3}\right)^{2}(x-1)^{3}
$$

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Key Concepts

End Behavior and Leading Term Analysis

End behavior describes how the graph of a polynomial function behaves as x approaches positive or negative infinity. This behavior is determined by the leading term of the polynomial, which dominates the function's value for large absolute values of x, effectively approximating the polynomial as a power function.

Turning Points of a Polynomial Graph

The maximum number of turning points (relative extrema) a polynomial graph can have is one less than the degree of the polynomial. This concept is crucial in understanding the overall shape and complexity of the graph as it indicates how many times the function can change direction.

Real Zeros and Their Multiplicity

A real zero of a polynomial is a value where the polynomial evaluates to zero. The multiplicity of a zero represents the number of times that zero is repeated as a factor in the polynomial. Understanding multiplicity is important because it affects how the graph of the function behaves at the corresponding x-intercept.

Graph Behavior at X-intercepts

The behavior of a polynomial's graph at its x-intercepts is determined by the multiplicity of the corresponding zero. If the multiplicity is odd, the graph crosses the x-axis at that intercept. If the multiplicity is even, the graph touches the x-axis and turns around at that point.

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