Answer :
The behavior of the graph with the equation y=x66x5+50x3+45x2108x108 at each of its zeros will be as follows: two of them will resemble a quadratic function, while one of them will resemble a linear function.
This is further explained below.
What is a graph?
Generally, a diagram that illustrates the relationship between variable quantities, usually consisting of two variables, with each variable being measured along one of a pair of axes that are intersected at right angles.
In conclusion, At each of its zeroes, the graph y=x66x5+50x3+45x2108x108 will exhibit one linear behavior and two behaviors that resemble quadratic functions.
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Final answer:
The graph y=-x⁶-6x⁵+50x³+45x²-108x-108 intersects or touches the x-axis at its zeroes, showing changes or constancy in sign. With its even degree and negative leading coefficient, it starts from the top left and ends at the bottom right, suggesting that it will cross the x-axis at most 6 times.
Explanation:
The behavior of the graph y=-x6-6x5+50x3+45x2-108x-108 at each of its zeros can be described by examining how the function changes as it crosses these points. The zeros of a polynomial are the points where the function intersects the x-axis, i.e., where y=0.
The degree of the polynomial, which is 6 in this case, suggests it will have at most 6 real zeros. The leading coefficient, which is negative, implies that the graph starts from the top left and ends at the bottom right of the coordinate plane, assuming large positive and negative values of x, respectively.
To determine the exact behavior at each zero, one might need to perform further analysis, such as factorizing the polynomial if possible, or using numerical methods or graphing technology to approximate the zeros. The end behavior of the polynomial, based on the negative leading coefficient and even degree, will be that as x → ∞, y → -∞, and as x → -∞, y → -∞ as well.
