Problem 4 - Algebra 2

Question

For Exercises \(3-5,\) use the following information. The projected sales of e-books in millions of dollars can be modeled by the function \(S(x)=-17 x^{3}+200 x^{2}-113 x+44,\) where \(x\) is the number of years since 2000 . Use direct substitution to evaluate \(S(8)\)

Step-by-Step Solution

Answer

The value of \(S(8)\) is 3236.

Step 1: Understand the function

The function given is a polynomial function that models the projected sales of e-books. The number of years since 2000 is represented by \(x\), and \(S(x)\) gives the sales in millions of dollars.

Step 2: Substitute the given value into the function

We are asked to evaluate \(S(8)\). This means we need to substitute \(x = 8\) into the function \(S(x) = -17x^3 + 200x^2 - 113x + 44\).

Step 3: Calculate the value of the cubic term

First, calculate \(-17 \times 8^3\). Compute \(8^3 = 512\), then multiply by \(-17\) to get \(-17 \times 512 = -8704\).

Step 4: Calculate the value of the quadratic term

Next, calculate \(200 \times 8^2\). Compute \(8^2 = 64\), then multiply by \(200\) to get \(200 \times 64 = 12800\).

Step 5: Calculate the value of the linear term

Now calculate \(-113 \times 8\), which is \(-113 \times 8 = -904\).

Step 6: Calculate the constant term

The constant term is \(44\), which remains unchanged regardless of \(x\).

Step 7: Sum all the calculated terms

Add all the terms together: \(-8704 + 12800 - 904 + 44\). Calculate this step-by-step: \(-8704 + 12800 = 4096\), \(4096 - 904 = 3192\), and \(3192 + 44 = 3236\).

Step 8: State the result

The value of \(S(8)\) is 3236.

Key Concepts

Substitution Method

The substitution method is a valuable tool when working with functions. It allows you to find the value of a function for a specific input by replacing the variable in an equation with a given number. In our exercise, we're using direct substitution to find out what the function's output is when the input is a particular number of years.

• Identify the function: In your problem, the function given is a polynomial, specifically a cubic one.
• Substitute: Use the value given for the variable (in this case, 8) and replace every instance of that variable in the equation.
• Calculate: Carry out the operations according to the substituted numbers to evaluate the function's output.

The substitution method is straightforward but powerful, enabling you to evaluate complex functions one step at a time.

Cubic Polynomial

A cubic polynomial is a polynomial of degree three. This means that the highest power of the variable (often denoted as "x") is three. Cubic polynomials typically take the form \(ax^3 + bx^2 + cx + d\), where \(a\), \(b\), \(c\), and \(d\) are constants.

• The "cubic" refers to the fact that the variable has an exponent of three, meaning it will produce a curve that is usually not a straight line.
• This type of polynomial can have up to three real roots, and its curve can have various shapes, depending on the coefficients.
• In your function \(S(x) = -17x^3 + 200x^2 - 113x + 44\), the coefficient of the cubic term is \(-17\), shaping the behavior of the graph and affecting its growth rate over time.

Cubic polynomials are versatile for modeling and can depict various real-world scenarios, such as the sales projections in this problem.

Evaluating Functions

Evaluating functions involves computing the function's value based on a specific input value. This is often done for polynomial functions, like the one described in the exercise.

• Plug in the value: Substitute the given value into the polynomial to replace every instance of the variable "x".
• Follow the order of operations: Start by computing powers, then carry out multiplications, and finally, add or subtract the results.
• Summarize your result: Once all terms have been computed and combined, the final number represents the function's value for that input.

While the individual steps might seem tedious, they form the basis for solving polynomial equations accurately. Evaluating functions is a skill that enhances understanding mathematical relationships and patterns.