Which of the following graphs could be the graph of the function [tex]f(x)=-0.08 x(x^2-11 x+1)[/tex]?
Answer (2)
The function is a cubic polynomial with a negative leading coefficient, indicating it starts from the top left and goes to the bottom right.
Find the roots of the function by setting f ( x ) = 0 , which gives x = 0 and x 2 − 11 x + 1 = 0 .
Use the quadratic formula to find the roots of x 2 − 11 x + 1 = 0 , resulting in x = 2 11 ± 117 .
Approximate the roots to be x = 0 , x ≈ 0.092 , and x ≈ 10.908 , which are the points where the graph crosses the x-axis. Therefore, the graph should have these characteristics.
Explanation
Analyzing the Function We are given the function f ( x ) = − 0.08 x ( x 2 − 11 x + 1 ) and we need to determine which graph could represent this function. First, let's analyze the function.
Determining the End Behavior The function is a cubic polynomial. The leading coefficient is − 0.08 , which means that as x approaches infinity ( x → ∞ ), f ( x ) approaches negative infinity ( f ( x ) → − ∞ ), and as x approaches negative infinity ( x → − ∞ ), f ( x ) approaches infinity ( f ( x ) → ∞ ). This tells us that the graph will start from the top left and go to the bottom right.
Finding the Roots To find the roots of the function, we set f ( x ) = 0 . This gives us − 0.08 x ( x 2 − 11 x + 1 ) = 0 . The roots are x = 0 and the roots of x 2 − 11 x + 1 = 0 .
Using the Quadratic Formula We use the quadratic formula to find the roots of x 2 − 11 x + 1 = 0 . The quadratic formula is x = 2 a − b ± b 2 − 4 a c . In this case, a = 1 , b = − 11 , and c = 1 . So, x = 2 ( 1 ) 11 ± ( − 11 ) 2 − 4 ( 1 ) ( 1 ) = 2 11 ± 121 − 4 = 2 11 ± 117 .
Approximating the Roots The roots are x = 0 , x = 2 11 + 117 and x = 2 11 − 117 . Approximating these values, we have x = 0 , x = 2 11 + 10.81665 ≈ 10.908 and x = 2 11 − 10.81665 ≈ 0.092 .
Sketching the Graph Therefore, the graph should cross the x-axis at approximately x = 0 , x = 0.092 , and x = 10.908 . Since the leading coefficient is negative, the graph will start from the top left and go to the bottom right. The graph should cross the x-axis at approximately x = 0 , x = 0.092 , and x = 10.908 .
Examples
Understanding the behavior of polynomial functions like this one is crucial in many real-world applications. For instance, engineers use polynomial functions to model the trajectory of projectiles, such as rockets or baseballs. By analyzing the roots and the leading coefficient, they can predict the range and maximum height of the projectile. Similarly, economists use polynomial functions to model cost and revenue curves, helping businesses optimize their production and pricing strategies. This problem demonstrates how analyzing roots and end behavior can provide valuable insights into the behavior of a function, which is essential for making informed decisions in various fields.
The function f ( x ) = − 0.08 x ( x 2 − 11 x + 1 ) is a cubic polynomial with roots at approximately x = 0 , x ≈ 0.092 , and x ≈ 10.908 . The graph starts from the top left and goes to the bottom right, intersecting the x-axis at these points. This leads us to identify a corresponding graph that reflects this behavior and intercepts.
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