Question

For each polynomial function, identify its graph from choices A–F.ƒ(x)=−(x−2)2(x−5) ƒ(x)=$$-(x-2)^2(x-5)$$

Accepted Answer

Start by identifying the degree of the polynomial function ƒ(x) = $$-(x-2)^2(x-5). $$Since (x-2) is squared and (x-5) is to the first power, the degree is 2 + 1 = 3, which means the polynomial is cubic. Determine the leading term by multiplying the highest degree terms from each factor: from $$(x-2)^2$$, the highest term is $$x^2$$, and from (x-5), it is x. Multiplying these gives $$x^3$$, and considering the negative sign, the leading term is $$-x^3. $$Analyze the end behavior of the graph using the leading term $$-x^3$$: as x approaches positive infinity, ƒ(x) approaches negative infinity; as x approaches negative infinity, ƒ(x) approaches positive infinity. Identify the zeros of the function from the factors: x = 2 (with multiplicity 2) and x = 5 (with multiplicity 1). The multiplicity affects how the graph behaves at these points—at x=2, the graph touches and turns around (because of even multiplicity), and at x=5, it crosses the x-axis (because of odd multiplicity). Use the information about degree, leading coefficient, end behavior, and zeros to match the function to the correct graph among choices A–F, focusing on the shape near the zeros and the overall direction of the graph.

For each polynomial function, identify its graph from choices A–F.
$ƒ (x) = - (x - 2)^{2} (x - 5)$

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Key Concepts

Polynomial Function and Degree

Zeros and Their Multiplicities

End Behavior of Polynomial Graphs

For each polynomial function, identify its graph from choices A–F.ƒ(x)=−(x−2)2(x−5) ƒ(x)=-(x-2)^2(x-5)