Which of the following graphs could be the graph of the function [tex]$f(x)=-0.08 x(x^2-11 x+18)$[/tex]?
Answer (2)
Find the roots of the function f ( x ) = − 0.08 x ( x 2 − 11 x + 18 ) by setting f ( x ) = 0 .
Factor the quadratic to find the roots: x = 0 , 2 , 9 .
Determine the end behavior based on the negative leading coefficient: as x \tto \tinfty , f ( x ) \tto − \tinfty and as x \tto − \tinfty , f ( x ) \tto \tinfty .
Identify the graph that matches these roots and end behavior.
Explanation
Understanding the Problem We are given the function f ( x ) = − 0.08 x \t ( x 2 − 11 x + 18 ) and we need to determine which graph represents this function.
Finding the Roots First, let's find the roots of the function by setting f ( x ) = 0 . This means we need to solve the equation − 0.08 x \t ( x 2 − 11 x + 18 ) = 0 .
Factoring the Quadratic We can factor the quadratic term x 2 − 11 x + 18 . We are looking for two numbers that multiply to 18 and add up to 11. These numbers are 2 and 9. So, x 2 − 11 x + 18 = ( x − 2 ) ( x − 9 ) .
Identifying the Roots Now we have f ( x ) = − 0.08 x ( x − 2 ) ( x − 9 ) . Setting each factor to zero gives us the roots: x = 0 , x = 2 , and x = 9 . These are the points where the graph intersects the x-axis.
Determining End Behavior The leading term of the polynomial is − 0.08 x 3 . The coefficient is negative, which means that as x approaches positive infinity, f ( x ) approaches negative infinity, and as x approaches negative infinity, f ( x ) approaches positive infinity. This tells us about the end behavior of the graph.
Sketching the Graph Now we know the roots are 0, 2, and 9. The function is a cubic with a negative leading coefficient. This means the graph starts in the second quadrant (top left), crosses the x-axis at 0, goes up to a local maximum, comes down crossing the x-axis at 2, goes down to a local minimum, and then goes up crossing the x-axis at 9, and then goes down towards negative infinity.
Identifying the Graph Based on the roots and end behavior, we can identify the correct graph.
Examples
Understanding polynomial functions and their graphs is crucial in many fields. For example, engineers use polynomial functions to model curves and surfaces in computer-aided design. Economists use them to model cost and revenue functions. In physics, projectile motion can be modeled using quadratic functions, which are a type of polynomial. By analyzing the roots and end behavior of these functions, we can predict the behavior of the system being modeled. For instance, knowing the roots of a revenue function can tell a business when they will break even.
The function f ( x ) = − 0.08 x ( x 2 − 11 x + 18 ) has roots at x = 0 , x = 2 , and x = 9 . Its end behavior indicates that as x approaches positive infinity, f ( x ) goes to negative infinity, and as x approaches negative infinity, f ( x ) goes to positive infinity. Therefore, the correct graph will cross the x-axis at these roots and has the specified end behavior.
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