AP PreCalculus Practice Quiz: Polynomial Functions and Rates of Change

Correct Answer: A

A nonconstant polynomial function, p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{0}, requires n to be a positive integer. Option A fits this definition with n=2. Option B is not a polynomial because the exponent 1/2 is not an integer. Option C is a rational function. Option D is a constant polynomial, not a nonconstant one.

Correct Answer: B

The provided content states that a local maximum or minimum occurs where a polynomial function switches between increasing and decreasing. A local maximum specifically occurs where the function's output values stop increasing and start decreasing.

Correct Answer: D

The content states that 'Between every two distinct real zeros of a nonconstant polynomial function, there must be at least one input value corresponding to a local maximum or local minimum.' Since there are distinct real zeros at -4 and 6, there must be at least one local extremum between them.

Correct Answer: A

The content states that 'Polynomial functions of an even degree will have either a global maximum or a global minimum.' This function has a degree of 4, which is even. Because the leading coefficient (-5) is negative, both ends of the graph will tend towards negative infinity, which guarantees the existence of a global maximum.

Correct Answer: D

This is a direct application of the definition provided in the content: 'Points of inflection of a polynomial function occur at input values where the rate of change of the function changes from increasing to decreasing or from decreasing to increasing.'

Correct Answer: B

According to the content, between every two distinct real zeros, there must be at least one local extremum. There is an interval between -1 and 3, and another interval between 3 and 7. Therefore, there must be at least one local extremum in (-1, 3) and at least one in (3, 7), for a minimum of two local extrema.

Correct Answer: C

The definition of a polynomial function states that n must be a positive integer. In the function r(x) = 4x^-2 + 3x, the term 4x^-2 has an exponent of -2, which is not a positive integer. Therefore, r(x) is not a polynomial function.

Correct Answer: B

The content states that 'Between every two distinct real zeros... there must be at least one... local maximum or local minimum.' If a function has no local extrema, it cannot have two distinct real zeros, because that would violate this rule. Therefore, the function can have one real zero (like p(x)=x^3) or no real zeros, meaning it can have at most one.

Correct Answer: B

The function has an even degree (6), so it must have a global maximum or a global minimum. Since the leading coefficient is positive, the function's values will tend to positive infinity as x approaches both positive and negative infinity. This 'U' shape guarantees that the function must have a global minimum value.

Correct Answer: C

The content specifies two conditions for local extrema: 'Where a polynomial function switches between increasing and decreasing, or at the included endpoint of a polynomial with a restricted domain.' Therefore, a local minimum can occur either where the behavior changes from decreasing to increasing or at one of the endpoints.

Correct Answer: C

A point of inflection is defined as an input value 'where the rate of change of the function changes from increasing to decreasing or from decreasing to increasing.' A point where a quantity (in this case, the rate of change) switches from increasing to decreasing or vice versa is, by definition, a local maximum or local minimum of that quantity.

Correct Answer: C

The provided content states that 'Polynomial functions of an even degree will have either a global maximum or a global minimum.' By logical extension, polynomial functions of an odd degree do not have this property. Their end behavior goes in opposite directions (one end to +∞, the other to -∞), meaning the function's range is all real numbers. Therefore, it cannot have a global maximum or a global minimum.