Problem 68 - College Algebra

Question

A department store has two locations in a city. From 1998 through \(2002,\) the profits for each of the store's two branches are modeled by the functions \(f(x)=-0.44 x+13.62\) and \(g(x)=0.51 x+11.14 .\) In each model, \(x\) represents the number of years after 1998 and \(f\) and \(g\) represent the profit, in millions of dollars. a. What is the slope of \(f ?\) Describe what this means. b. What is the slope of \(g\) ? Describe what this means. c. Find \(f+g .\) What is the slope of this function? What does this mean?

Step-by-Step Solution

Answer

a. The slope of function f is -0.44, indicating a decrease in profit over time. b. The slope of function g is 0.51, indicating an increase in profit over time. c. The function of f+g is 0.07x + 24.76, with a slope of 0.07 representing a slow increase in combined profit over time.

Step 1: Find the Slope of f

The slope of function f is -0.44. In this context, slope refers to the rate of change in a function, specifically, how much the profit changes for each year after 1998. A negative slope means the profit is decreasing over the given time period.

Step 2: Find the Slope of g

The slope of function g is 0.51. In similar manner to step 1, this positive slope indicates that the profit is increasing over the given time period.

Step 3: Find the Sum of f and g

To find the sum of f and g (notated as \(f+g\)), we add together the similar terms from each function. This results in a function \(f+g = (-0.44+0.51)x + (13.62+11.14) = 0.07x + 24.76.\n The slope of \(f+g\) is therefore 0.07, indicating that the combined profit of both stores is increasing over time, although more slowly than store g on its own.

Key Concepts

Understanding Slope

In mathematics, the slope is the measure of the steepness or incline of a line, often found in a linear equation. It is represented by the letter 'm' in the slope-intercept form of a line, which is \(y = mx + c\). In the context of the department store's profits:

• The slope of the function \(f(x) = -0.44x + 13.62\) is \(-0.44\).
• The slope indicates that the profits of the first store decrease by \(-0.44\) million dollars per year.

A negative slope tells us that as time progresses, the profit is falling, illustrating a decline over the specified years.
In contrast, the slope of \(g(x) = 0.51x + 11.14\) is \(0.51\). This signifies that the profits for the second store are actually increasing by \(0.51\) million dollars annually.
Understanding the slope in these functions allows us to analyze the trend and rate of profit increase or decrease over time.

Rate of Change

Rate of change quantifies how one quantity changes in relation to another. In the context of profit functions, it represents how quickly profits are changing over the years.
For function \(f(x)\), the rate of change, or the slope, is \(-0.44\). This means profits decrease by \(-0.44\) million each year.
For function \(g(x)\), the rate of change is \(0.51\), indicating an annual profit increase of \(0.51\) million.

• Positive rate of change: Profit increases.
• Negative rate of change: Profit decreases.

This rate of change gives us insights into which store is performing better over time, showing a direct relationship between the number of years and profit changes.

Profit Function

A profit function is a mathematical way to depict a company's profit over time based on certain variables. Here, \(f(x)\) and \(g(x)\) represent the profits from two store locations, where \(x\) is the number of years after 1998.

• Function \(f(x) = -0.44x + 13.62\): Indicates a decrease in profit over time for the first store.
• Function \(g(x) = 0.51x + 11.14\): Shows an increase in profit for the second store.

These functions allow businesses to predict future profits and make informed decisions. By analyzing them, they can understand whether their profits are cultivating growth or facing downturns.
This modeling helps in strategic planning and better allocation of resources.

Function Addition

Function addition involves combining two function expressions to create a new function. In this exercise:

• Given \(f(x) = -0.44x + 13.62\) and \(g(x) = 0.51x + 11.14\).
• Adding \(f(x)\) and \(g(x)\) results in \(f(x) + g(x) = 0.07x + 24.76\).

This new function represents the total combined profit of both stores. The slope of this combined function is \(0.07\), suggesting a modest increase in combined profits each year.
Function addition helps to analyze the overall trend across multiple entities or datasets, allowing a clearer picture of combined performances.