Which of the following is the graph of the function [tex]f(x)=(x-2)^2(x+3)(x+1)[/tex]?
Answer (1)
Find the roots of the function: x = − 3 , − 1 , 2 .
Determine the multiplicity of each root: x = 2 has multiplicity 2, x = − 3 and x = − 1 have multiplicity 1.
Determine the end behavior: the graph opens upwards on both ends.
Calculate the y-intercept: f ( 0 ) = 12 .
The graph has roots at x = − 3 , − 1 , 2 , touches the x-axis at x = 2 , crosses the x-axis at x = − 3 and x = − 1 , opens upwards on both ends, and has a y-intercept of 12.
Explanation
Understanding the Function We are given the function f ( x ) = ( x − 2 ) 2 ( x + 3 ) ( x + 1 ) and asked to identify its graph. To do this, we need to find the roots, their multiplicities, the end behavior, and the y-intercept of the function.
Finding the Roots The roots of the function are the values of x for which f ( x ) = 0 . From the factored form, we can see that the roots are x = 2 , x = − 3 , and x = − 1 .
Determining Multiplicities The multiplicity of a root is the number of times the corresponding factor appears in the factored form of the polynomial. The root x = 2 comes from the factor ( x − 2 ) 2 , so its multiplicity is 2. The roots x = − 3 and x = − 1 come from the factors ( x + 3 ) and ( x + 1 ) respectively, so their multiplicities are both 1.
Analyzing End Behavior The end behavior of a polynomial is determined by its leading term. Expanding the function, we have f ( x ) = ( x 2 − 4 x + 4 ) ( x 2 + 4 x + 3 ) = x 4 + … . Since the leading term is x 4 , the function has degree 4, which is even, and a positive leading coefficient, which means that as x approaches positive or negative infinity, f ( x ) approaches positive infinity. In other words, the graph opens upwards on both ends.
Calculating the Y-Intercept The y-intercept is the value of f ( x ) when x = 0 . Plugging in x = 0 , we get f ( 0 ) = ( 0 − 2 ) 2 ( 0 + 3 ) ( 0 + 1 ) = ( 4 ) ( 3 ) ( 1 ) = 12 . So the y-intercept is 12.
Sketching the Graph Now we can sketch the graph. The roots are x = − 3 , − 1 , 2 . The graph touches the x-axis at x = 2 (since the multiplicity is 2) and crosses the x-axis at x = − 3 and x = − 1 (since the multiplicities are 1). The graph opens upwards on both ends, and the y-intercept is 12.
Conclusion Based on the above analysis, we can conclude that the graph of the function f ( x ) = ( x − 2 ) 2 ( x + 3 ) ( x + 1 ) has roots at x = − 3 , − 1 , 2 , touches the x-axis at x = 2 , crosses the x-axis at x = − 3 and x = − 1 , opens upwards on both ends, and has a y-intercept of 12.
Examples
Understanding the behavior of polynomial functions like this is crucial in many fields. For instance, engineers use polynomial functions to model curves and surfaces in design. Economists might use them to model cost functions, where understanding the roots and intercepts can help determine break-even points and optimal production levels. In computer graphics, polynomials are used to create smooth curves and realistic shapes. By analyzing roots, end behavior, and intercepts, professionals can make informed decisions and predictions in their respective domains.
