Lec - 25 B - Examples of Quadratic Functions

Summary

The script explains how to determine the minimum or maximum values of a quadratic function by analyzing its vertex and coefficients. Using a mathematical example of a quadratic function, it illustrates the process of finding the domain, range, and the minimum value. Then, the script applies this understanding to a real-life problem involving a tour bus company in Chennai aiming to maximize its income by adjusting fares. Finally, it introduces the concept of slope in both linear and quadratic functions and sets the stage for further exploration of the slope in quadratic functions.

Takeaways

• 🔢 The given function is f(x) = x² - 6x + 9, and we are tasked with finding whether it has a minimum or maximum value.
• 📏 The domain of the function is the entire real line, meaning there are no restrictions on x.
• 📉 Since the coefficient of x² (a = 1) is positive, the parabola opens upwards, indicating the function has a minimum value.
• 📍 The vertex of the parabola is the key to finding the minimum value, using the formula -b/2a.
• ➕ After correcting the math, the vertex is at x = 3, giving f(3) = 0, meaning the minimum value is 0.
• 📊 The range of the function is all values greater than or equal to 0, written as R ∩ {f(x) ≥ 0}.
• 💼 In a real-world example involving a tour bus service, the company loses 10 passengers per Rs. 4 hike in fare, starting at Rs. 40 per person.
• 💰 The company's income can be modeled by the quadratic expression: Income = (500 - 10x)(40 + 4x) = -40x² + 1600x + 20,000.
• 📈 Since the quadratic function has a negative coefficient for x², it will have a maximum value, representing the highest possible income.
• 🏷️ The optimal fare is Rs. 120, achieved by increasing the fare by Rs. 80 (or 20 units of Rs. 4 hikes), resulting in maximum profit for the company.

Q & A

What is the function provided in the first part of the script?

The function given is f(x) = x² - 6x + 9.

What is the domain of the function f(x) = x² - 6x + 9?

The domain of the function is the entire real line, or all real numbers (−∞, ∞).

How do you determine whether the function f(x) = x² - 6x + 9 has a minimum or maximum value?

Since the coefficient of x² (a = 1) is positive, the function opens upwards and therefore has a minimum value.

How is the vertex of the parabola f(x) = x² - 6x + 9 calculated?

The x-coordinate of the vertex is calculated using the formula -b/2a. Here, b = -6 and a = 1, so the vertex is at x = 3.

What is the minimum value of the function f(x) = x² - 6x + 9?

The minimum value is 0, which occurs when x = 3. Substituting x = 3 into the function gives f(3) = 0.

What is the range of the function f(x) = x² - 6x + 9?

Since the function has a minimum value of 0, the range is [0, ∞), meaning the function takes values greater than or equal to 0.

What is the business problem discussed in the second part of the script?

A tour bus in Chennai serves 500 customers daily, charging ₹40 per person. The company estimates it will lose 10 passengers for every ₹4 fare hike, and it wants to maximize its profit by adjusting the fare.

How is the income of the bus company modeled as a function of fare hikes?

Let x represent the number of ₹4 fare hikes. The income function is modeled as (500 - 10x)(40 + 4x), which simplifies to -40x² + 1600x + 20,000.

Why does the income function of the bus company have a maximum value?

Since the coefficient of x² is negative (a = -40), the income function is a downward-opening parabola, meaning it has a maximum value.

At what fare does the bus company maximize its income, and what is the new fare?

The income is maximized when x = 20, which means the company should increase the fare by 20 units of ₹4 each. The new fare will be 40 + 4(20) = ₹120.