Determine the end behavior, plot the [tex]$y$[/tex]-intercept, find and plot all real zeros, and plot at least one test value between each intercept. Then connect the points with a smooth curve. [tex]$f( x )=-10 x ^4-9 x ^3+59 x ^2+45 x -45$[/tex]
Choose the correct end behavior for [tex]$f ( x )$[/tex].
A. The ends of the graph will extend in the same direction, because the degree of the polynomial is odd.
B. The ends of the graph will extend in opposite directions, because the degree of the polynomial is odd.
C. The ends of the graph will extend in the same direction, because the degree of the polynomial is even.
D. The ends of the graph will extend in opposite directions, because the degree of the polynomial is even.
The [tex]$y$[/tex]-intercept is -45. (Type an integer or a simplified fraction.)
The zeros of the polynomial are [tex]$x =$[/tex] $\square$.
(Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)
Answer (2)
The ends of the graph extend in the same direction because the degree of the polynomial is even.
The y -intercept is -45.
The zeros of the polynomial are x = − 5 , − 2 3 , 5 3 , 5 .
Test values between the zeros help determine the shape of the curve. x = − 5 , − 2 3 , 5 3 , 5
Explanation
Understanding the Problem We are given the function f ( x ) = − 10 x 4 − 9 x 3 + 59 x 2 + 45 x − 45 . We need to determine the end behavior, the y -intercept, the real zeros, and plot the graph of the function.
Determining End Behavior The degree of the polynomial is 4, which is even. The leading coefficient is -10, which is negative. Therefore, the ends of the graph will extend in the same direction (downwards).
Finding the y-intercept The y -intercept is the value of the function when x = 0 . So, f ( 0 ) = − 10 ( 0 ) 4 − 9 ( 0 ) 3 + 59 ( 0 ) 2 + 45 ( 0 ) − 45 = − 45 . The y -intercept is -45.
Finding the Real Zeros To find the real zeros of the polynomial, we need to solve the equation f ( x ) = − 10 x 4 − 9 x 3 + 59 x 2 + 45 x − 45 = 0 . By using a root-finding tool, we find the real roots to be approximately x = − 2.236 , − 1.5 , 0.6 , 2.236 . We can express these roots exactly as x = − 5 , − 2 3 , 5 3 , 5 .
Finding Test Values Now we need to find test values between each intercept. The zeros are approximately -2.236, -1.5, 0.6, and 2.236. We choose test values x = − 2 , − 0.5 , 1.25 , 2 . We calculate the function values at these test points:
f ( − 2 ) = − 10 ( − 2 ) 4 − 9 ( − 2 ) 3 + 59 ( − 2 ) 2 + 45 ( − 2 ) − 45 = − 160 + 72 + 236 − 90 − 45 = 13 f ( − 0.5 ) = − 10 ( − 0.5 ) 4 − 9 ( − 0.5 ) 3 + 59 ( − 0.5 ) 2 + 45 ( − 0.5 ) − 45 = − 0.625 + 1.125 + 14.75 − 22.5 − 45 = − 52.25 f ( 1.25 ) = − 10 ( 1.25 ) 4 − 9 ( 1.25 ) 3 + 59 ( 1.25 ) 2 + 45 ( 1.25 ) − 45 = − 24.414 + − 17.578 + 92.188 + 56.25 − 45 = 61.446 f ( 2 ) = − 10 ( 2 ) 4 − 9 ( 2 ) 3 + 59 ( 2 ) 2 + 45 ( 2 ) − 45 = − 160 − 72 + 236 + 90 − 45 = 49
Final Answer The ends of the graph will extend in the same direction, because the degree of the polynomial is even. The y -intercept is -45. The zeros of the polynomial are x = − 5 , − 2 3 , 5 3 , 5 .
Examples
Understanding the behavior of polynomial functions is crucial in various fields, such as engineering and physics. For example, when designing a bridge, engineers use polynomial functions to model the load distribution and ensure the structure's stability. By analyzing the end behavior, intercepts, and zeros of these functions, they can predict how the bridge will respond to different loads and make necessary adjustments to prevent failure. Similarly, in physics, polynomial functions are used to model the trajectory of projectiles, allowing scientists to predict their range and impact point.
The ends of the graph of the polynomial will extend in the same direction because the degree is even, and the leading coefficient is negative. The y -intercept is − 45 , and the real zeros are approximately x = − 2.236 , − 1.5 , 0.6 , 2.236 .
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