Use the input-output table to answer the questions.

Question

𝑥 𝑦
−4 10
−1 5
2 −20
5 −35
A. Write the linear equation that represents the input-output pairs in the table.
B. Create a real-world situation that could be modeled by the input-output pairs
in the table.
C. Explain the meaning of the slope and y-intercept in the context of the
situation described.
D. Create a graph for the table.
E. Use the linear equation and graph to answer the following:
• What is the change in the output value per unit change in the input�value?
• What is the output value when the input value is 0?
• What is the output value when the input value is 10?
• Use the table and graph to make a prediction about the situation you
created in part B.

Answer 1

To answer your questions based on the input-output table provided, we need to analyze the relationship between the input variable \( x \) and the output variable \( y \):

\[ \begin{array}{|c|c|} \hline x & y \ \hline -4 & 10 \ -1 & 5 \ 2 & -20 \ 5 & -35 \ \hline \end{array} \]

A. Write the linear equation that represents the input-output pairs in the table.

First, we can determine the slope \( m \) using two points from the table. Let's take \( (-1, 5) \) and \( (2, -20) \):

\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-20 - 5}{2 - (-1)} = \frac{-25}{3} = -\frac{25}{3} \]

Now let's find the y-intercept \( b \) using one of the points. Using \( (x, y) = (-1, 5) \):

\[ y = mx + b \ 5 = -\frac{25}{3}(-1) + b \ 5 = \frac{25}{3} + b \ b = 5 - \frac{25}{3} \ b = \frac{15}{3} - \frac{25}{3} = -\frac{10}{3} \]

Thus, the linear equation is:

\[ y = -\frac{25}{3}x - \frac{10}{3} \]

B. Create a real-world situation that could be modeled by the input-output pairs in the table.

Let's consider a situation involving the cost of producing a certain item where \( x \) is the number of items produced (or hours worked) and \( y \) is the amount of profit (or loss). For instance, if a factory incurs a loss when producing certain items, the table could represent the financial outcome in terms of profit/loss at different production levels.

C. Explain the meaning of the slope and y-intercept in the context of the situation described.

• Slope: The slope of \( -\frac{25}{3} \) indicates that for every additional unit produced (or hour worked), the factory will lose approximately $8.33. This suggests that as production increases, the losses grow at a proportional rate.

• Y-intercept: The y-intercept of \( -\frac{10}{3} \) (approximately -3.33) implies that if no items are produced (or no hours are worked), the factory still incurs a fixed cost of around $3.33 (for example, overhead costs), which results in a loss even without production.

D. Create a graph for the table.

To graph the equation \( y = -\frac{25}{3}x - \frac{10}{3} \) based on the points:

1. Plot the points from the table: \((-4, 10)\), \((-1, 5)\), \( (2, -20)\), and \( (5, -35)\).
2. Draw a straight line through these points to represent the linear equation.

E. Use the linear equation and graph to answer the following:

1. What is the change in the output value per unit change in the input value? The change in output value per unit change in input, represented by the slope, is approximately $-8.33 per additional item produced (i.e., \( -\frac{25}{3} \)).

2. What is the output value when the input value is 0? Substitute \( x = 0 \) into the linear equation:

   \[ y = -\frac{25}{3}(0) - \frac{10}{3} = -\frac{10}{3} \approx -3.33 \]

   So when no items are produced, the output value (loss) will be about -3.33.

3. What is the output value when the input value is 10? Substitute \( x = 10 \) into the linear equation:

   \[ y = -\frac{25}{3}(10) - \frac{10}{3} = -\frac{250}{3} - \frac{10}{3} = -\frac{260}{3} \approx -86.67 \]

   The output value (loss) when 10 items are produced will be approximately -86.67.

4. Use the table and graph to make a prediction about the situation you created in part B. Based on the trend shown in the table and the graph, as production increases, the losses will continue to grow. If the factory maintains or increases production without addressing the cost, it can expect to incur even larger financial losses. This indicates a need for revising production strategies or reducing costs to improve profitability.

Answer 2

can make the real world situation about a bakery

Answer 3

Sure! Let's create a real-world situation for a bakery that could be modeled by the input-output pairs in the table you provided.

B. Real-World Situation for a Bakery

Situation: The bakery produces gourmet cupcakes. For every batch produced, there are certain fixed and variable costs. If the bakery generates a loss at specific production levels, it reflects the costs not being covered by sales, indicating inefficiencies or pricing issues.

• Input (\( x \)): The number of batches of cupcakes baked.
• Output (\( y \)): The profit or loss generated from baking and selling those batches.

Context Based on Table Values:

• The bakery incurs losses when it produces specific amounts of cupcakes because, despite selling some, the costs (like ingredients, labor, and overhead) are too high relative to revenue.
• For example, when they bake:
  • 0 batches, the bakery incurs a fixed overhead loss (perhaps $3.33).
  • 2 batches, they incur a deeper loss of $20, suggesting revenue does not cover the high initial costs yet.
  • By the time they produce 5 batches, their losses deepen to about $35, reflecting inefficient pricing or overproduction relative to demand.

C. Explanation of Slope and Y-Intercept in the Bakery Context

1. Slope: The slope of \( -\frac{25}{3} \) means that for every additional batch of cupcakes produced, the bakery loses approximately $8.33. This suggests that the costs of producing an extra batch (combined with fixed costs) significantly outweighs the revenue generated from selling that batch.

2. Y-Intercept: The y-intercept of \( -\frac{10}{3} \) (approximately -3.33) indicates that at no production (0 batches), the bakery still incurs a loss of about $3.33. This reflects fixed costs, such as rent, utilities, and salaries that the bakery must pay regardless of production, meaning they start in a deficit before even making any sales.

Summary

Through this scenario, we see that the bakery is struggling to balance production costs against the revenue generated from cupcake sales. The data points illustrate financial losses at various production levels, suggesting a need for strategic adjustments in pricing, production amounts, or cost management to optimize profitability.

Answer 4

can use the bakery idea for part E point 4

Answer 5

Absolutely! Let's use the bakery scenario to provide a prediction based on the data and the graph related to input-output pairs.

E. Use the table and graph to answer the following:

4. Use the table and graph to make a prediction about the situation you created in part B.

Given the context of the bakery's operations, we can make the following predictions:

1. Financial Forecast: Based on the trend illustrated in the graph, the bakery is currently operating at a loss as production increases. If this trend continues and no changes are applied to their pricing strategy or production efficiency, we can predict that increasing production to more than 5 batches will likely result in even greater financial losses. For example, the output value when producing 10 batches was calculated to be approximately -$86.67. This significant loss indicates that unless the bakery revises its approach (e.g., increasing prices, reducing production costs, or improving sales strategies), it is on a path to experiencing unsustainable financial performance.

2. Recommendation for Change: The bakery may need to investigate their costs more closely. For instance, if producing more than 5 batches continues to result in increased losses, the bakery could consider reducing the batch size to a level where costs are manageable or optimizing their ingredient sourcing to lower the overall production cost per batch.

3. Market Research: Additionally, the bakery could look into conducting market research to determine if there is sufficient demand to justify producing larger quantities. If market analysis suggests there's a demand for more cupcakes, then marketing efforts could be revamped to increase sales per batch produced.

4. Sales Promotions: The bakery could also implement strategies such as promotions or loyalty programs to encourage more sales and improve cash flow, thus potentially covering fixed costs and transitioning from losses to profitability.

Conclusion

In summary, based on the current trends and losses illustrated in the table and graph, the bakery should consider implementing cost-cutting measures, reviewing pricing strategies, and focusing on marketing initiatives to foster a more financially successful operation moving forward. If they do not make these changes, increasing production will likely lead to deeper losses, threatening the business's sustainability.