Question

A snowboard manufacturer determines that its profit, \(P,\) in dollars, can be modelled by the function \(P(x)=1000 x+x^{4}-3000\) where \(x\) represents the number, in hundreds, of snowboards sold. a) What is the degree of the function \(P(x) ?\) b) What are the leading coefficient and constant of this function? What does the constant represent? c) Describe the end behaviour of the graph of this function. d) What are the restrictions on the domain of this function? Explain why you selected those restrictions. e) What do the \(x\) -intercepts of the graph represent for this situation? f) What is the profit from the sale of 1500 snowboards?

Short answer

a) Degree: 4 b) Leading coefficient: 1, Constant: -3000 c) As \(x \to \infty\), \(P(x) \to \infty\); As \(x \to -\infty\), \(P(x) \to \infty\) d) Domain: \(x \ge 0\) e) x-intercepts: No profit points f) Profit for 1500 snowboards: 59625

Step-by-step solution

Identify the Degree of the Function

The degree of a polynomial function is the highest power of the variable x. For the function \(P(x) = 1000x + x^4 - 3000\), the highest power of x is 4. Therefore, the degree of the function is 4.

Identify the Leading Coefficient and Constant

The leading coefficient is the coefficient of the term with the highest power. In \(P(x) = 1000x + x^4 - 3000\), the highest power is 4, and the coefficient of \(x^4\) is 1. Therefore, the leading coefficient is 1. The constant term is the term without any variables, which is -3000.

Determine the Representation of the Constant

The constant term in a profit function usually represents fixed costs or initial investment. In this case, the constant -3000 may represent the fixed costs that are subtracted from the total profit.

Describe the End Behaviour of the Function

The end behaviour of a polynomial function depends on the degree and the leading coefficient. Since the function \(P(x) = 1000x + x^4 - 3000\) has a degree of 4 and a positive leading coefficient, the end behaviour can be described as follows: - As \(x \to \infty\), \(P(x) \to \infty\). - As \(x \to -\infty\), \(P(x) \to \infty\).

Determine the Restrictions on the Domain

Since the variable x represents the number of snowboards sold, measured in hundreds, it doesn't make sense for x to be negative or a non-integer value. Therefore, the domain is restricted to non-negative integers, so \(x \geq 0\).

Explain the Significance of the x-Intercepts

The x-intercepts of the graph of the function \(P(x) = 1000x + x^4 - 3000\) represent the number of snowboards (in hundreds) that result in zero profit. These intercepts are the values of x that make \(P(x) = 0\).

Calculate the Profit for 1500 Snowboards

To find the profit from the sale of 1500 snowboards, first convert 1500 to hundreds by dividing by 100, which gives x = 15. Now, substitute x = 15 into the function: \[P(15) = 1000(15) + 15^4 - 3000\] Calculate: \[P(15) = 15000 + 50625 - 3000\] \[P(15) = 62625 - 3000\] \[P(15) = 59625\]

Key concepts

degree of a polynomial

In polynomial functions, the degree is a key characteristic. It's the highest power of the variable, like in the function given: \(P(x) = 1000x + x^4 - 3000\). Here, the highest power of \(x\) is 4, making the degree of this function 4. The degree tells us a lot about the function's shape and behavior. For instance, higher-degree polynomials have more twists and turns in their graphs than lower-degree polynomials. Always look for the term with the largest exponent when determining the degree of a polynomial function.

leading coefficient

The leading coefficient is another important aspect of polynomial functions. It's the coefficient of the term with the highest degree. For our function, \(P(x) = 1000x + x^4 - 3000\), the term with the highest degree is \(x^4\), and its coefficient is 1. So, the leading coefficient is 1. This coefficient affects the steepness and direction of the graph. If the leading coefficient is positive, like in our example, the graph will rise to infinity as \(x\) moves towards infinity.

end behavior

End behavior describes how the values of a polynomial function behave as \(x\) approaches very large positive or negative values. For the function \(P(x) = 1000x + x^4 - 3000\), which has a degree of 4 and a positive leading coefficient, the end behavior is:

• As \(x \to \infty\), \(P(x) \to \infty\).
• As \(x \to -\infty\), \(P(x) \to \infty\).

Regardless of the degree, if the leading coefficient is positive, both ends of the graph will infinitely rise or fall depending on whether the degree is even or odd.

domain restrictions

Domain restrictions help us understand the possible values of \(x\) in a function. For our polynomial \(P(x)\), \(x\) represents the number of snowboards sold, measured in hundreds. Therefore, \(x\) must be a non-negative integer. This makes sense because you can't sell a negative number of snowboards, and it would be impractical to consider fractions of snowboards. Hence, the domain of \(P(x)\) is all non-negative integers: \(x \geq 0\).

x-intercepts

The \(x\)-intercepts of a polynomial function are the points where the graph crosses the \(x\)-axis. These intercepts are vital because they show the values of \(x\) that make the function equal to zero. For \(P(x) = 1000x + x^4 - 3000\), solving for \(x\) when \(P(x) = 0\) will give you the intercepts. In terms of our problem, these \(x\)-intercepts represent the number of snowboards that result in zero profit. Interpreting these points can provide insights into break-even points for the manufacturer.

profit calculation

Profit calculation involves using the polynomial function to find the profit for a specific number of units sold. For the function provided, \(P(x) = 1000x + x^4 - 3000\), if you want to find the profit for 1500 snowboards, convert 1500 into hundreds, giving \(x = 15\). Substitute \(x = 15\) into the function:

• \(P(15) = 1000(15) + 15^4 - 3000\)
• \(P(15) = 15000 + 50625 - 3000\)
• \(P(15) = 62625 - 3000\)
• \(P(15) = 59625\)

So, the profit from selling 1500 snowboards is $59,625. This type of calculation is crucial in business to forecast earnings and make informed decisions.