Question

A company's weekly profit \(P\) (in hundreds of dollars) from a product is given by the model \(P(x)=80+20 x-0.5 x^{2}, \quad 0 \leq x \leq 20\) where \(x\) is the amount (in hundreds of dollars) spent on advertising. (a) Use a graphing utility to graph the profit function. (b) The company estimates that taxes and operating costs will increase by an average of $$\$ 2500$$ per week during the next year. Rewrite the profit equation to reflect this expected decrease in profits. Identify the type of transformation applied to the graph of the equation. (c) Rewrite the profit equation so that \(x\) measures advertising expenditures in dollars. [Find \(P(x / 100) .]\) Identify the type of transformation applied to the graph of the profit function.

Short answer

The graph of profit function is a downward-opening parabola that decreases by 25 units to reflect the increased operating costs, and stretched horizontally by a factor of 100 to reflect the measure of advertising expenditures in dollars.

Step-by-step solution

Graphing the Profit Function

Use a graphing utility to graph the function \(P(x)=80+20 x-0.5 x^{2}\). This function represents the company's weekly profit in hundreds of dollars.

Adjusting for Increased Operating Expenses

Increase in operating costs and taxes by $2500 will impact the overall profit. Since our profit function is given in hundreds of dollars, we convert $2500 to $25 in our equation. The new function reflecting this decrease in profits is \(P(x)=80+20x-0.5x^{2}-25\). The graph of the function decreases vertically by a value of 25. This is called a vertical translation.

Rewriting Equation for Advertising Expenditures in Dollars

To rewrite the profit equation with \(x\) as advertising expenditures in actual dollars instead of hundreds of dollars, you replace \(x\) with \(x/100\) in our original equation to get \(P(x/100)=80+20(x/100)-0.5(x/100)^{2}\). This transformation stretches the graph horizontally by a factor of 100, also known as horizontal scaling.

Key concepts

Graphing Utility

When it comes to visualizing complex functions, graphing utilities are an invaluable tool for students. These utilities, which can be software or online applications, allow one to input a mathematical function and immediately see the graph that represents it. This visual representation makes understanding the behavior of the function much more intuitive.

For the task at hand, a graphing utility would enable students to quickly graph the company's profit function, making it easier to identify features such as intercepts, maxima, or minima. By inputting the profit function equation, the utility plots the revenue outcomes against different levels of advertising expenditures. For students who are visual learners, this approach can significantly aid in comprehending how the profit varies with the advertisement spending.

Vertical Translation

A vertical translation is a shift of a graph up or down along the y-axis. In our exercise, the company is facing an increase in taxes and operating costs, which translates to a decrease in profit by \(2500 per week, or \)25 in hundreds of dollars. When we subtract 25 from our original profit function, this action moves the entire graph down by 25 units, representing this new expense. The result is a parallel graph that is lower than the original, showing how the increased costs affect the company's profits. Understanding how the graph shifts can help students visualize the real-world impact of financial changes on a company's bottom line.

Advertising Expenditures

The variable advertising expenditures directly relates to how much a company is spending to advertise its product. In our context, 'x' in the profit function represents this variable in hundreds of dollars. By rewriting the function with 'x' in actual dollars, not only do we gain a more precise mathematical representation, but we also align the function more closely with realistic financial data.

Rewriting the function to reflect advertising expenditures in actual dollars changes the scale of the x-axis, which can affect how students interpret the data. More granular data could reveal more insights into the relationship between the profits and advertising spending, clarifying the importance of budgeting for marketing purposes.

Operating Costs Adjustment

The concept of operating costs adjustment is crucial in the context of managing a business's profitability. When we increase the operating costs in the profit function, we reflect additional expenses the company will have to endure. From an educational standpoint, showing this adjustment in the profit function allows students to understand how fixed costs, like taxes and operational expenses, can eat into the profit margins.

Being able to quantify this change and express it in a function's parameters is a critical skill. Students are better equipped to grasp the financial implications for a business when they can see the effects of cost adjustments within the mathematical structure of a profit function.

Horizontal Scaling

Altering the scale of the x-axis in a function graph is known as horizontal scaling. This transformation stretches or compresses the graph along the horizontal axis. In the exercise, converting the advertising expenditure to dollars required dividing the 'x' variable by 100. Doing so resulted in the profit function graph stretching horizontally by a factor of 100. Each unit on the x-axis now represents a single dollar rather than a hundred dollars.

Understanding horizontal scaling helps students recognize how data presentation impacts our interpretation of information. When they see the stretched graph, they can better understand how different levels of detail in the scale might affect decisions based on the visual analysis of the graph.