Determine the end behavior, plot the $y$-intercept, find and plot all real zeros, and plot at least one test value between each intercept.
$f(x)=x^3-x^2-16 x-20$
Choose the correct end behavior for $f ( x )$.
A. The ends of the graph will extend in opposite directions, because the degree of the polynomial is odd.
B. The ends of the graph will extend in the same direction, because the degree of the polynomial is odd.
C. The ends of the graph will extend in opposite directions, because the degree of the polynomial is even.
D. The ends of the graph will extend in the same direction, because the degree of the polynomial is even.
Answer (2)
Determine the end behavior: Since the degree is odd and the leading coefficient is positive, the graph extends in opposite directions.
Find the y-intercept: Evaluate f ( 0 ) = − 20 , so the y-intercept is ( 0 , − 20 ) .
Find the real zeros: Solve f ( x ) = 0 to find the zeros x = − 2 and x = 5 .
Plot test values: Choose test values between the zeros, such as x = − 3 , x = 1 , and x = 6 , and evaluate the function to find the corresponding points. The end behavior is: The ends of the graph will extend in opposite directions, because the degree of the polynomial is odd. T h e e n d s o f t h e g r a p h w i ll e x t e n d in o pp os i t e d i rec t i o n s , b ec a u se t h e d e g ree o f t h e p o l y n o mia l i s o dd .
Explanation
Understanding the Problem The problem asks us to analyze the polynomial f ( x ) = x 3 − x 2 − 16 x − 20 . We need to determine its end behavior, find and plot the y-intercept, find and plot all real zeros, and plot test values between the intercepts to sketch the curve.
Determining End Behavior Since the degree of the polynomial is 3 (odd) and the leading coefficient is 1 (positive), the end behavior is as follows: as x approaches negative infinity, f ( x ) approaches negative infinity, and as x approaches positive infinity, f ( x ) approaches positive infinity. Therefore, the ends of the graph will extend in opposite directions, because the degree of the polynomial is odd.
Finding the y-intercept To find the y-intercept, we set x = 0 and evaluate f ( 0 ) : f ( 0 ) = ( 0 ) 3 − ( 0 ) 2 − 16 ( 0 ) − 20 = − 20 So, the y-intercept is ( 0 , − 20 ) .
Finding the Real Zeros To find the real zeros, we need to solve the equation f ( x ) = 0 : x 3 − x 2 − 16 x − 20 = 0 Using a root-finding tool, we find that the real roots are x = − 2 and x = 5 .
Plotting Intercepts and Zeros Now we plot the y-intercept ( 0 , − 20 ) and the real zeros ( − 2 , 0 ) and ( 5 , 0 ) on the coordinate plane.
Finding Test Values We need to choose test values between the zeros to determine the behavior of the function in these intervals. We have the intervals ( − ∞ , − 2 ) , ( − 2 , 5 ) , and ( 5 , ∞ ) .
Let's choose the test values x = − 3 , x = 1 , and x = 6 .
For x = − 3 : f ( − 3 ) = ( − 3 ) 3 − ( − 3 ) 2 − 16 ( − 3 ) − 20 = − 27 − 9 + 48 − 20 = − 8 So, the point is ( − 3 , − 8 ) .
For x = 1 : f ( 1 ) = ( 1 ) 3 − ( 1 ) 2 − 16 ( 1 ) − 20 = 1 − 1 − 16 − 20 = − 36 So, the point is ( 1 , − 36 ) .
For x = 6 : f ( 6 ) = ( 6 ) 3 − ( 6 ) 2 − 16 ( 6 ) − 20 = 216 − 36 − 96 − 20 = 64 So, the point is ( 6 , 64 ) .
Connecting the Points Now we plot the points ( − 3 , − 8 ) , ( 1 , − 36 ) , and ( 6 , 64 ) on the coordinate plane. Finally, we connect the plotted points with a smooth curve, considering the end behavior of the polynomial.
Examples
Polynomial functions are used to model various real-world phenomena, such as the trajectory of a projectile, the growth of a population, or the behavior of financial markets. By analyzing the end behavior, intercepts, and zeros of a polynomial function, we can gain insights into the underlying process and make predictions about its future behavior. For example, engineers use polynomial functions to design bridges and buildings, while economists use them to forecast economic trends. Understanding polynomial functions is essential for solving many practical problems in science, engineering, and finance. The function f ( x ) = x 3 − x 2 − 16 x − 20 can describe the volume of a container depending on its dimensions, and finding its roots helps determine the dimensions for a specific volume.
The end behavior of the polynomial f ( x ) = x 3 − x 2 − 16 x − 20 is that the ends of the graph will extend in opposite directions, as the degree is odd. The y-intercept is at the point ( 0 , − 20 ) , and the real zeros are at x = − 2 and x = 5 . Test values in between the intercepts show how the function behaves in those regions.
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