- The total expense function is a linear equation \(E(q) = 10q + 20\).
- The graph begins at the y-intercept \((0, 20)\), which represents the fixed costs.
- The function has a constant positive slope of 10, meaning each additional repair adds \(\$10\) to the total expense.
- The resulting graph is a straight line starting at \((0, 20)\) and rising linearly as the number of repairs \(q\) increases.
Explanation
- Analyzing the Expense Function
The expense function is given by \(E(q) = 20 + 10q\), where \(q\) represents the number of bicycles repaired and \(E(q)\) represents the total expense in dollars. This is a linear function of the form \(E(q) = mq + c\), where \(m\) is the slope and \(c\) is the y-intercept. Here, the slope \(m = 10\) and the y-intercept \(c = 20\). The variable \(q\) must be non-negative (\(q \ge 0\)) because it represents the quantity of bicycles repaired.
- Determining the Graph Characteristics
To understand the shape of the graph, we identify its key features. The y-intercept occurs when \(q = 0\). Substituting this into the function gives \(E(0) = 20 + 10(0) = 20\). This means the graph starts at the point \((0, 20)\) on the vertical axis. The slope of \(m = 10\) indicates that for every additional bicycle repaired (\(\Delta q = 1\)), the total expense increases by \(\$10\) (\(\Delta E = 10\)). For example, if 1 bicycle is repaired, \(E(1) = 20 + 10(1) = 30\). If 2 bicycles are repaired, \(E(2) = 20 + 10(2) = 40\). The graph is a straight line starting at \((0, 20)\) and extending to the right with a constant positive slope.
- Conclusion on the Graph Representation
The relationship between the number of bicycles repaired and the total expenses is linear. The graph is a ray starting at the point \((0, 20)\) and passing through points such as \((1, 30)\), \((2, 40)\), and \((3, 50)\). As the number of bicycles repaired increases, the total expenses increase linearly, reflecting both the fixed costs of \(\$20\) and the variable costs of \(\$10\) per bicycle. The final representation of the expense function is a straight line segment or ray originating from \((0, 20)\) with a positive gradient of 10.
Examples
In a real-life scenario, businesses often use linear expense functions like this to predict costs. For example, a small catering business might have a fixed cost for renting kitchen equipment and a variable cost for the ingredients used in each meal. By modeling their total expenses, they can easily determine the budget required for different numbers of events. This helps in pricing their services correctly to ensure profitability while remaining competitive. Understanding these graphs is a fundamental tool for financial planning and resource management in any service-based industry.