• Express a linear function using an equation, words, tables, and graphs
• Determine whether a linear function is increasing, decreasing, or constant
• Understand the slope of a line as the rate at which one quantity changes in relation to another
• Solve for the [latex]x[/latex]-intercept in a linear function equation

In today’s digital age, even traditional products like disco balls have found their place in modern business. When companies plan their inventory and sales strategies, they use mathematical models to make smart decisions. Let’s explore how a local manufacturing company uses linear functions to track their inventory and plan production.

Mirrored Marvels, an American manufacturing company, crafts high-quality disco balls that bring sparkle to events across the country. With each disco ball selling for [latex]$125[/latex], they carefully balance their fixed costs of [latex]$2,000[/latex] and production costs of [latex]$80[/latex] per ball. They currently have [latex]20[/latex] disco balls in stock.

Now that we understand how inventory changes as disco balls are sold, let’s look at the money coming in from these sales. Each time a disco ball leaves the warehouse, Mirrored Marvels receives payment. By tracking revenue (the money received from sales), the company can better understand their financial health.

Before we can understand the company’s total financial picture, we need to consider their costs. Some costs stay the same no matter how many disco balls are made – these are called fixed costs. For Mirrored Marvels, fixed costs include things like warehouse rent, basic utilities, and equipment maintenance.

In addition to fixed costs, Mirrored Marvels has to pay for materials and labor for each disco ball they make. These per-unit costs combine with the fixed costs to give us the total cost of production. Understanding this total cost helps the company make smart decisions about pricing and production.

Now that we understand both revenue and costs, we can figure out how much money Mirrored Marvels actually keeps from their disco ball sales. Profit tells us if the business is making or losing money by showing us what’s left after paying all the costs. Think of it like your own budget – if you earn [latex]$100[/latex] and spend [latex]$75[/latex], your “profit” would be [latex]$25[/latex].

Understanding profit helps businesses make crucial decisions, especially about their break-even point. For Mirrored Marvels, knowing their break-even point will tell them the minimum number of disco balls they need to sell just to cover their costs.

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Mirrored Marvels can use these functions to determine how many disco balls they must sell in order to break even. Breaking even occurs when revenue is the same as the total costs. Since a negative profit is where the company’s costs are higher than their revenue, and a positive profit is where the company’s revenue is higher than their costs, you can also think of the breaking-even point as where the profit is [latex]$0[/latex].