Graph the polynomial function f(x)=-2(x-3)^2(x^2-25) using parts (a) through (e).
The real zero(s) of f is/are -5,3,5
(Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as needed. Type each answer only once.)
The smallest real zero is a zero of multiplicity 1, so the graph of f crosses the x-axis at x=-5. The middle real zero is a zero of multiplicity 2, so the graph of f the x-axis at x=3: The largest real zero is a zero of multiplicity 1, so the graph of f crosses the x-axis at x= 5.
(d) Determine the maximum number of turning points on the graph of the function.
The graph has, at most, 3 turning points.
(Type a whole number.)
(e) Use the information to draw a complete graph of the function. Choose the correct graph.
A.
B.
C.
D.
Answer (2)
The analysis of the polynomial function f ( x ) = − 2 ( x − 3 ) 2 ( x 2 − 25 ) reveals three real zeros with multiplicities indicating the graph crosses or touches the x-axis accordingly. The maximum number of turning points is 3, which corresponds to the graph opening downwards with a y-intercept of 450. The correct graph representing these behaviors is option C.
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The polynomial function is f ( x ) = − 2 ( x − 3 ) 2 ( x 2 − 25 ) .
The graph crosses the x-axis at x = − 5 and x = 5 , and touches the x-axis at x = 3 .
The y-intercept is 450, and the graph opens downwards.
Therefore, the correct graph is C.
Explanation
Analyze the polynomial function We are given the polynomial function f ( x ) = − 2 ( x − 3 ) 2 ( x 2 − 25 ) . We know the real zeros are -5, 3, and 5. The zero at x = − 5 has multiplicity 1, so the graph crosses the x-axis there. The zero at x = 3 has multiplicity 2, so the graph touches the x-axis there. The zero at x = 5 has multiplicity 1, so the graph crosses the x-axis there. The maximum number of turning points is 3. We need to choose the correct graph.
Expand the polynomial First, let's expand the polynomial to determine its degree and leading coefficient. We have f ( x ) = − 2 ( x − 3 ) 2 ( x 2 − 25 ) = − 2 ( x 2 − 6 x + 9 ) ( x 2 − 25 ) = − 2 ( x 4 − 6 x 3 + 9 x 2 − 25 x 2 + 150 x − 225 ) = − 2 ( x 4 − 6 x 3 − 16 x 2 + 150 x − 225 ) = − 2 x 4 + 12 x 3 + 32 x 2 − 300 x + 450 . The degree of the polynomial is 4, and the leading coefficient is -2.
Determine the end behavior Since the degree is 4 (even) and the leading coefficient is -2 (negative), the end behavior of the graph is that as x → − ∞ , f ( x ) → − ∞ , and as x → ∞ , f ( x ) → − ∞ . This means the graph opens downwards.
Analyze x-intercepts We know the x-intercepts are -5, 3, and 5. At x = − 5 and x = 5 , the graph crosses the x-axis. At x = 3 , the graph touches the x-axis (it's a turning point at the x-axis).
Find the y-intercept Let's find the y-intercept by setting x = 0 : f ( 0 ) = − 2 ( 0 − 3 ) 2 ( 0 2 − 25 ) = − 2 ( 9 ) ( − 25 ) = 450 . So the y-intercept is 450.
Choose the correct graph Now, let's analyze the given options. We are looking for a graph that opens downwards, crosses the x-axis at -5 and 5, touches the x-axis at 3, and has a y-intercept of 450. Option C satisfies these conditions.
Examples
Understanding polynomial functions helps in modeling various real-world scenarios, such as the trajectory of a projectile, the growth of a population, or the behavior of electrical circuits. By analyzing the zeros, multiplicity, and end behavior of a polynomial, we can predict and control the system's behavior. For example, engineers use polynomial functions to design stable bridges and architects use them to create aesthetically pleasing and structurally sound buildings. The function's y-intercept can represent an initial condition, such as the starting population or the initial voltage in a circuit. The roots of the polynomial can represent equilibrium points or critical values in the system.
