Problem 100 - College Algebra Essentials

Question

A department store has two locations in a city. From 2008 through 2012 , 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 2008 , 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 \(f(x)\) is -0.44, which means the profit decreases by $0.44 million per year. b. The slope of \(g(x)\) is 0.51, meaning the profits increase by $0.51 million per year. c. The function \(f+g\) is \(0.07x + 24.76\), and its slope is 0.07, meaning the combined yearly profit of the two branches increases by $0.07 million.

Step 1: Understanding slope

The slope of a function describes the rate of change of the function. In these models, it represents the change in profits per year.

Step 2: Finding slope of function f

The given function \(f(x)=-0.44 x+13.62\) is in the form of \(y=mx+b\) where m is the slope. The slope of function \(f\) is -0.44. This means that the profit of the store's first branch decreases by $0.44 million per year.

Step 3: Finding slope of function g

The second function \(g(x) = 0.51x+11.14\) also has a slope, which is 0.51. This means that the profit of the store's second branch increases by $0.51 million per year.

Step 4: Finding sum of functions f and g

To find the sum \(f+g\), we add the two functions: \((f+g)(x) = f(x) + g(x) = (-0.44x+13.62) + (0.51x+11.14) \). This simplifies to \((f+g)(x) = 0.07x+24.76\).

Step 5: Evaluating slope of sum function

In the sum function \((f+g)(x) = 0.07x+24.76\), the slope is 0.07. This means that the combined yearly profit of the two branches increases by $0.07 million per year.

Key Concepts

Slope

The concept of slope is fundamental in understanding linear functions as it essentially describes the function's "steepness" or tilt. In mathematical terms, slope is the change in the dependent variable ( y ) per unit change in the independent variable ( x ). For the functions representing our department store's profits, the slope informs us how these profits progress year by year.

• The slope is found by identifying the coefficient of x in the function y = mx + b .
• For our first function, f(x) = -0.44x + 13.62 , the slope is -0.44 .
• A negative slope (-0.44) indicates a decrease in profit by $0.44 million every subsequent year.
• For the second store, represented by g(x) = 0.51x + 11.14 , the slope is 0.51 , which suggests that the profit increases by $0.51 million per year.

The interpretative value of slope helps not only in profit analysis but also in appreciating the linear trends visible within data. Recognizing whether a slope is positive or negative aids in predicting future outcomes and strategizing accordingly.

Rate of Change

The rate of change is closely related to the slope and signifies how a quantity progresses over time. In the context of functions describing profits, it offers a view of how quickly or slowly profits are increasing or decreasing.

• For any linear function described by y = mx + b , the rate of change is given by the slope m .
• In function f(x) = -0.44x + 13.62 , the negative rate of change (-0.44) illustrates a reduction in revenue, highlighting declining financial performance over time.
• Conversely, g(x) = 0.51x + 11.14 , showcases a positive rate of change (0.51), indicative of improving business prospects each year.

This insight into rates of change is crucial, especially in economics and business, as it aids in adjusting business strategies and identifying potential problems or opportunities early on.

Profit Modeling

Profit modeling uses mathematical functions to quantify how a company's profits evolve over a certain period. Linear functions like f(x) and g(x) serve as simple yet powerful tools in predicting financial growth or contraction, driven by various factors.

• Each function characterizes the profit trend of a store location over time.
• With function f(x) = -0.44x + 13.62 , the negative slope indicates a modeled decrease in profit over the given years.
• In contrast, g(x) = 0.51x + 11.14 suggests a prospective increase, providing a forecast useful for decision-making.

These linear models are essential in financial strategies, aiding businesses in focusing efforts on more profitable sectors and planning necessary interventions to manage losses. Thus, profit modeling is a valuable approach to evaluating business performance efficiently over time.

Sum of Functions

Adding two functions, known as the sum of functions, combines their respective growth trends into a single expression. This aggregated function offers insights into the collective performance of multiple entities.

• To find the sum of f(x) and g(x) , simply add their expressions: (f+g)(x) = f(x) + g(x) .
• For our exercise, this sum simplifies to (f+g)(x) = 0.07x + 24.76 .
• The slope of this new function, 0.07 , indicates an overall increase in joint profits by $0.07 million per year.

This combined outcome suggests a collaboration impact, reflecting how different branches together can create a stable or improved financial landscape, which individual analysis might not reveal. Utilizing function sums allows businesses to evaluate holistic performance, crucial for long-term planning and investment.