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Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. In our exercise, we are working with a cubic polynomial function, because cubic functions can have one local maximum and one local minimum. For a function to be a polynomial, its powers need to be whole numbers.

For a cubic function, the general form is: \[ f(x) = ax^3 + bx^2 + cx + d \]Here, a, b, c, and d are coefficients. Understanding how these coefficients change the shape of the graph is key.

For instance, the term \[ ax^3 \] determines the end behavior of the function - whether it goes to infinity or negative infinity as x becomes very large. The coefficient 'a' does more than just stretch or compress the graph; it determines whether the graph opens upwards (positive 'a') or downwards (negative 'a').

By analyzing polynomial functions, we can also understand the symmetry of the graph and potential points of inflection.

When graphing polynomial functions like the cubic function \[ f(x) = -x^3 + x - 3 \], there are several essential techniques:

1. **Intercepts** - Determine where the function crosses the x-axis (x-intercepts) and y-axis (y-intercepts). Our given function has intercepts at (0, -3) and (3, 0). These points help in plotting the graph correctly.

2. **Critical Points** - Find the critical points by taking the first derivative and setting it equal to zero. This helps locate the local maxima and minima. For our function, the local maximum is at x = -1 and the local minimum is at x = 2.

3. **Behavior Analysis** - Examine the end behavior of the graph. Since our function is cubic, it has one tail going to positive or negative infinity and the other tail going to the opposite infinity.

4. **First Derivative Test** - Checking where the first derivative transitions from positive to negative or negative to positive helps us identify whether we have a local maximum or a minimum at the critical points.

By using these techniques, you can graph any polynomial function accurately.

Local maximums and minimums are specific points where the function reaches a highest or lowest value within a certain interval.

For our cubic function, the local maximum is at x = -1, with a value of -2. This means as you move left to right through x = -1, the function increases to -2 and then starts to decrease.

The local minimum is at x = 2, with a value of -6. This means as you move left to right through x = 2, the function decreases to -6 and then starts to increase.

To find these points analytically:

• Take the derivative of the function.
• Set it to zero to find potential maxima and minima.
• Use the second derivative to confirm whether each critical point is indeed a maximum or minimum.

Here's the derivative of our function: \[ f'(x) = 3ax^2 + 2bx + c \]
We use the critical points found earlier (\[ f'(-1) = 0 \rightarrow 3a - 2b + c = 0 \] and \[ f'(2) = 0 \rightarrow 12a + 4b + c = 0 \]) to identify and confirm the local extrema.

Understanding local maxima and minima not only helps in graphing but also has applications in various optimization problems.