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Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. They can range from simple linear equations to more complex forms, such as quadratic or cubic functions. In our exercise, we're dealing with a **cubic polynomial function** given by the equation \[ h(t) = -t^3 + 54t^2 + 480t + 20. \]These functions are characterized by their degree, which, in this case, is 3, due to the presence of the term \(-t^3\).Cubic functions can have several interesting features:

• They may have up to three real roots where the polynomial equals zero.
• They can have one or two turning points, depending on the nature of their roots.
• They show different end behaviors at each infinity, as we'll discuss later.

Understanding these characteristics helps in sketching the curve accurately and predicting its behavior over different intervals of \(t\).

Critical points of a polynomial function occur where the slope of the tangent is zero. For the given rocket altitude function, these points signify moments during the flight where the rocket changes its altitude direction, either from rising to falling or vice versa.To find these critical points, we use the derivative of the function. The derivative, \( h'(t) = -3t^2 + 108t + 480 \),when set equal to zero, helps us find the specific \(t\) values where these changes occur.Solving the derivative equation:\[ -3t^2 + 108t + 480 = 0 \]requires using the quadratic formula. By identifying the coefficients \(a = -3\), \(b = 108\), and \(c = 480\), we solve for \(t\) to determine the critical points. These points help define where the rocket has its highest or lowest altitudes, critical for sketching the polynomial function.

End behavior in polynomial functions describes how the values of the function behave as the variable becomes extremely large or small. Observing the end behavior provides insight into what the graph looks like towards infinity in both directions.For the cubic polynomial \(h(t) = -t^3 + 54t^2 + 480t + 20\), the end behavior is determined primarily by the term with the highest degree, which is \(-t^3\).Because this coefficient is negative, the end behavior indicates:

• As \(t \to \infty\), the function \(h(t) \to -\infty\), meaning the graph will drop towards negative infinity.
• As \(t \to -\infty\), \(h(t) \to \infty\), meaning the graph will rise towards positive infinity.

This results in a characteristic "N" shape of a negative cubic graph, dropping from positive to negative as \(t\) increases.

Calculating the derivative is essential to understand the rate of change of a function. In this case, the derivative of the polynomial function reflects how the rocket's altitude changes with time, pinpointing where its ascent or descent slows to a stop.To derive \(h(t) = -t^3 + 54t^2 + 480t + 20\), use the power rule: multiply each term by the power of \(t\) and reduce the power by one. This gives us:\[ h'(t) = -3t^2 + 108t + 480. \]By setting this equation to zero, we can solve for \(t\), revealing the critical points where the slope of the graph is flat. The quadratic formula helps in finding these exact values. The solutions to \(h'(t) = 0\) guide us in marking the significant turning points in the graph where changes in direction occur.