Question

A medical researcher establishes that a patient's reaction time, \(r\), in minutes, to a dose of a particular drug is \(r(d)=-3 d^{3}+3 d^{2},\) where \(d\) is the amount of the drug, in millilitres, that is absorbed into the patient's blood. a) What type of polynomial function is \(r(d) ?\) b) What are the leading coefficient and constant of this function? c) Make a sketch of what you think the function will look like. Then, graph the function using technology. How does it compare to your sketch? d) What are the restrictions on the domain of this function? Explain why you selected those restrictions.

Short answer

a) Cubic polynomial. b) Leading coefficient: \(-3\), constant: \(0\). c) Matches cubic trends. d) Domain: \([0, \infty)\).

Step-by-step solution

Identify the type of polynomial

The given function is \(r(d) = -3d^3 + 3d^2\). This is a polynomial function. Since the highest exponent of the variable \(d\) is 3, this function is specifically a cubic polynomial.

Determine the leading coefficient and the constant term

In the polynomial \(r(d)=-3d^3+3d^2\), the leading term is \(-3d^3\). Hence, the leading coefficient is \(-3\). There is no constant term in this polynomial, or equivalently, the constant term is \(0\).

Sketch the function

To sketch the function, note that it is a cubic polynomial with a negative leading coefficient. This typically implies that as \(d\) approaches positive infinity, \(r(d)\) will tend towards negative infinity, and as \(d\) approaches negative infinity, \(r(d)\) will tend towards positive infinity. Draw a rough sketch showing these trends with a point of inflection due to the cubic term.

Graph the function using technology

Use graphing software or a graphing calculator to plot \(r(d)=-3d^3+3d^2\). Check if the plot matches your sketch. The graph should show a steep downfall as \(d\) moves towards positive infinity and a rising curvature past an inflection point.

Determine restrictions on the domain

Consider the real-world context: \(d\) represents milliliters of a drug absorbed in the blood, so \(d\) must be non-negative. Therefore, the domain is restricted to \([0, \infty)\). This aligns with the fact that negative amounts of the drug do not make physical sense.

Key concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers of a variable multiplied by coefficients. The general form of a polynomial function in terms of a variable \(x\) is given by \[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \]Here, \(a_n, a_{n-1}, ..., a_1, \) and \(a_0\) are constants, with \(a_n eq 0\). Each term consists of a coefficient and a variable raised to a non-negative integer exponent.
The degree of the polynomial is the highest exponent of the variable in the function. For example, in the function \(r(d) = -3d^3 + 3d^2\), the highest exponent is 3, so it is a cubic polynomial.

• Polynomials can have different degrees: constant (0), linear (1), quadratic (2), cubic (3), etc.
• The degree of the polynomial determines its graph's general shape or form.
• Polynomials are continuous and smooth functions, meaning they have no breaks, holes, or sharp corners.

Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. For the function \(r(d) = -3d^3 + 3d^2\), the leading term is \(-3d^3\), so the leading coefficient is \(-3\).
The leading coefficient plays a crucial role in determining the end behavior of a polynomial function.

• If the leading coefficient is positive, the function will rise to positive infinity as the variable increases.
• If the leading coefficient is negative, the function will fall to negative infinity as the variable increases.

In our specific example, because the leading coefficient is \(-3\), the function \(r(d)\) will tend towards negative infinity as \(d\) goes to positive infinity, which is a key point in sketching the function.

Domain Restrictions

The domain of a function is the set of all possible input values (in this case, values of \(d\)) for which the function is defined.
For polynomial functions, the domain is generally all real numbers. However, in real-world applications, we may need to consider domain restrictions based on the context. Here, the function \(r(d) = -3d^3 + 3d^2\) represents a patient's reaction time to a drug dose.

• Since \(d\) represents the amount of drug in milliliters absorbed in the blood, it must be non-negative.
• This means the domain is restricted to \([0, \infty)\).
• Negative values wouldn't make sense in this scenario, as you can't have a negative amount of drug absorbed.

Taking into account such restrictions ensures the function makes sense in the context it's applied.

Function Sketching

Sketching a function involves drawing a rough graph based on some key properties of the function. For \(r(d) = -3d^3 + 3d^2\), here are the steps to sketch it:

• Identify key features: Intercepts, end behavior, and turning points.
• Intercepts: Set \(r(d)\) to zero and solve for \(d\) to find where the graph crosses the axes.
• End behavior: Since the leading coefficient is negative and the highest degree term is cubic, as \(d \rightarrow \infty\), \(r(d) \rightarrow -\infty\).

Begin by plotting the intercepts and then connecting them showing the general trend as \(d\) increases.
Once your sketch is ready, use graphing tools like calculators or software to plot the exact function. This will help verify the accuracy of your initial sketch and give a more precise picture of the function's behavior.