Which of the following statements is true in regard to sketching the graph of a polynomial function?
Choose the correct statement below.
A. The y-intercept can be found by setting f(x) equal to zero and solving for x.
B. The multiplicity of the real zeros can be used to determine the behavior of the graph at the x-intercepts.
C. The x-intercept(s) can be found by finding f(0).
D. The degree of the polynomial function can be used to determine the real zeros.
Answer (2)
The y-intercept is found by setting x = 0 , not by solving f ( x ) = 0 .
The multiplicity of real zeros determines the graph's behavior at x-intercepts.
The x-intercepts are found by solving f ( x ) = 0 , not by finding f ( 0 ) .
The degree of a polynomial indicates the maximum number of zeros, not the actual zeros, so the answer is B .
Explanation
Evaluate Statements We need to evaluate each statement to determine which one is true regarding the graph of a polynomial function.
Analyze Statement A Statement A says the y-intercept is found by setting f ( x ) = 0 and solving for x . This is incorrect. Setting f ( x ) = 0 gives the x-intercepts (also known as roots or zeros) of the function. The y-intercept is found by setting x = 0 and evaluating f ( 0 ) .
Analyze Statement B Statement B says the multiplicity of the real zeros determines the behavior of the graph at the x-intercepts. This is correct. If a zero has an even multiplicity, the graph touches the x-axis at that point and turns around (it's tangent to the x-axis). If a zero has an odd multiplicity, the graph crosses the x-axis at that point.
Analyze Statement C Statement C says the x-intercepts are found by finding f ( 0 ) . This is incorrect. Finding f ( 0 ) gives the y-intercept, not the x-intercepts. The x-intercepts are found by setting f ( x ) = 0 and solving for x .
Analyze Statement D Statement D says the degree of the polynomial function can be used to determine the real zeros. This is incorrect. The degree of the polynomial tells us the maximum possible number of zeros (real and complex), but it doesn't tell us what the actual zeros are. For example, a cubic polynomial (degree 3) can have 1, 2, or 3 real roots.
Conclusion Therefore, the correct statement is B. The multiplicity of the real zeros can be used to determine the behavior of the graph at the x-intercepts.
Examples
Understanding the behavior of polynomial functions is crucial in many fields. For instance, engineers use polynomial functions to model curves and surfaces in computer-aided design (CAD). The roots of these polynomials can represent critical points in the design, such as points of maximum stress or minimum curvature. Knowing how the multiplicity of these roots affects the shape of the curve allows engineers to optimize their designs for performance and stability. For example, a double root might indicate a point where a support structure is needed to prevent deformation.
The correct statement is B, which asserts that the multiplicity of the real zeros determines the behavior of the graph at the x-intercepts. Other statements regarding the y-intercept and the x-intercepts are incorrect. Understanding this concept is important for sketching polynomial functions accurately.
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