Problem 3
Question
For Activities 1 through \(6, \quad\) for each linear model a. give the slope of the line defined by the equation. b. write the rate of change of the function in a sentence of interpretation. c. evaluate and give a sentence of interpretation for \(f(0)\). The profit is \(f(x)=2 x-4.5\) thousand dollars when \(x\) hundred units are sold. The profit is \(f(x)=2 x-4.5\) thousand dollars when \(x\) hundred units are sold.
Step-by-Step Solution
Verified Answer
Slope is 2; profit increases by $2000 per 100 units sold; profit is -$4500 when no units are sold.
1Step 1: Identify the Slope of the Line
The linear function given is \( f(x) = 2x - 4.5 \). In a linear equation of the form \( f(x) = mx + b \), the coefficient \( m \) is the slope of the line. Here, \( m = 2 \). So, the slope of the line is 2.
2Step 2: Interpret the Rate of Change
The rate of change in a linear function is represented by its slope. The slope of 2 indicates that for every increase of 100 units sold, the profit increases by $2000. Therefore, the interpretation is: for each additional hundred units sold, the profit increases by $2000.
3Step 3: Evaluate and Interpret f(0)
To evaluate \( f(0) \), substitute \( x = 0 \) into the equation \( f(x) = 2x - 4.5 \). This gives \( f(0) = 2(0) - 4.5 = -4.5 \). This means when no units are sold (0 hundred), the profit is -$4500, indicating a loss.
Key Concepts
Understanding the SlopeExploring the Rate of ChangeMastering Function EvaluationAnalyzing Profit in Business
Understanding the Slope
The term **slope** is a fundamental concept in linear equations. In the context of a linear equation like \( f(x) = mx + b \), the slope is represented by the coefficient \( m \). This value tells us how steep or flat a line is, which in practical terms indicates how the function's output changes in response to its input.
For the linear function \( f(x) = 2x - 4.5 \), the coefficient of \( x \) is 2. This means that the slope is 2, signifying a consistent increase. In the context of profit, each additional 100 units sold increases profit by $2000. This constant rate changes no matter where you start on the x-axis. Understanding the slope helps in predicting outcomes and making informed business decisions.
For the linear function \( f(x) = 2x - 4.5 \), the coefficient of \( x \) is 2. This means that the slope is 2, signifying a consistent increase. In the context of profit, each additional 100 units sold increases profit by $2000. This constant rate changes no matter where you start on the x-axis. Understanding the slope helps in predicting outcomes and making informed business decisions.
Exploring the Rate of Change
The **rate of change** in a linear function exhibits how a variation in one quantity corresponds to a change in another. It's closely linked to the concept of slope. In our linear equation, the rate of change is identical to the slope, which is 2.
This indicates that for every 100 additional units sold, the profit rises by $2000. It's crucial for businesses as it offers them a glimpse into how changes in sales volume directly influence profitability, allowing them to plan and strategize efficiently. Being a constant, this makes rate of change easier to predict and manage.
This indicates that for every 100 additional units sold, the profit rises by $2000. It's crucial for businesses as it offers them a glimpse into how changes in sales volume directly influence profitability, allowing them to plan and strategize efficiently. Being a constant, this makes rate of change easier to predict and manage.
Mastering Function Evaluation
**Function evaluation** involves substituting a specific value for the variable in the equation to calculate the output. For instance, finding \( f(0) \) means you plug 0 in place of \( x \), resulting in \( f(0) = 2(0) - 4.5 = -4.5 \). This process simplifies a function into a single, solvable calculation.
The output, \( -4.5 \), is usually interpreted with respect to the context. Here it signifies the profit (or loss) when no units are sold, resulting in -$4500, indicating losses that could reflect fixed costs or starting baseline costs. Function evaluation is vital in predicting specific outcomes within linear models.
The output, \( -4.5 \), is usually interpreted with respect to the context. Here it signifies the profit (or loss) when no units are sold, resulting in -$4500, indicating losses that could reflect fixed costs or starting baseline costs. Function evaluation is vital in predicting specific outcomes within linear models.
Analyzing Profit in Business
In any business, **profit analysis** using linear functions helps in understanding the relationship between the volume of sales and the profit generated. For the function \( f(x) = 2x - 4.5 \), \( x \) represents sales in hundreds, and \( f(x) \), the profit in thousands.
Analyzing the profit function, we see that profit increases by \(2000 for every additional 100 units sold, which is critical information for making sales strategies. Also, knowing that no sales lead to a loss of \)4500 gives businesses an idea of the baseline costs incurred even without selling anything. Profit analysis through linear equations aids in visualizing financial outcomes and planning accordingly.
Analyzing the profit function, we see that profit increases by \(2000 for every additional 100 units sold, which is critical information for making sales strategies. Also, knowing that no sales lead to a loss of \)4500 gives businesses an idea of the baseline costs incurred even without selling anything. Profit analysis through linear equations aids in visualizing financial outcomes and planning accordingly.
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