Find a polynomial function f of least degree having the graph shown. (Hint: See the NOTE following Example 4.)

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Step 1: Identify the roots of the polynomial from the graph. The graph crosses the x-axis at x = -3 and x = 3, so these are roots of the polynomial. Since the graph touches the x-axis and turns around at these points, the roots have even multiplicity (likely multiplicity 2).

Step 2: Note the local maximum at (0, 16). This point gives us the y-intercept of the polynomial, so when x = 0, f(x) = 16.

Step 3: Write the general form of the polynomial using the roots and their multiplicities. Since the roots are at x = -3 and x = 3 with multiplicity 2, the polynomial can be expressed as \(f(x) = a(x + 3)^2 (x - 3)^2\) where \(a\) is a constant to be determined.

Step 4: Use the y-intercept to find the value of \(a\). Substitute x = 0 and f(0) = 16 into the polynomial: \(16 = a(0 + 3)^2 (0 - 3)^2\). Simplify and solve for \(a\).

Step 5: Write the final polynomial function with the determined value of \(a\). This polynomial will be of least degree and match the graph shown.

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주요 개념

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Polynomial Function and Degree

A polynomial function is an expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, with non-negative integer exponents. The degree of the polynomial is the highest exponent of the variable, which determines the general shape and number of turning points of the graph.

Zeros and Their Multiplicities

Zeros of a polynomial are the x-values where the function equals zero. The multiplicity of a zero affects the graph's behavior at that point: an odd multiplicity causes the graph to cross the x-axis, while an even multiplicity causes it to touch and turn around without crossing.

Local Maximum and Critical Points

A local maximum is a point where the function changes from increasing to decreasing, creating a peak on the graph. Identifying local maxima helps determine the polynomial's turning points, which relate to the degree and shape of the polynomial function.