2.5: GRAPHS OF EXPENSE AND REVENUE FUNCTIONS OBJECTIVES

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1 Section 2.5: GRAPHS OF EXPENSE AND REVENUE FUNCTIONS OBJECTIVES Write, graph and interpret the expense function. Write, graph and interpret the revenue function. Identify the points of intersection of the expense and revenue functions. Identify breakeven points, and explain them in the context of the problem. Key Terms nonlinear function second degree equation quadratic equation parabola leading coefficient maximum value vertex of a parabola axis of symmetry Chapter 2: Modeling a Business 1

2 Vocabulary Review: Expenses & Revenue 1. Expenses represent the cost to manufacture an item or provide a service. A contributor to consumer demand is the price at which the item is sold. Recall from Section 2.4: Expense relies on the quantity produced and demand relies on price, so the expense function can be written in terms of price. 2.Revenue is the total amount a business collects from the sale of a product or service. Recall from Section 2.4: Revenue depends on the demand for a product, which is a function of the price of the product. Vocabulary Review: Expenses & Revenue The relationship between price, demand, expense and revenue is better understood when the functions are graphed on a coordinate plane. Both the demand and expense functions are linear. However, when revenue is expressed in terms of price, the function is a nonlinear function, more specifically, a second degree equation or a quadratic equation. where 0 The graph of a quadratic equation is called a parabola. Chapter 2: Modeling a Business 2

3 Review: Characteristics of a Parabola is the axis of symmetry of the parabola., is the vertex of the parabola. value is the axis of symmetry. y value is found by plugging in in for. The vertex of the parabola can be classified as a maximum or minimum. Parabola with a positive leading coefficient General form of a Quadratic Equation where 0 is called the leading coefficient If 0, then the parabola opens upward. When a parabola opens upward, the vertex is a minimum point. Chapter 2: Modeling a Business 3

4 Parabola with a negative leading coefficient where 0 If 0, then the parabola opens downward. When a parabola opens downward, the vertex is a maximum point. The downward parabola models the revenue function. Example 1 A particular item in the Picasso Paints product line costs $7.00 each to manufacture. The fixed costs are $28,000. The demand function is q = 500p + 30,000 where q is the quantity the public will buy given the price, p. Graph the expense function in terms of price on the coordinate plane and provide the viewing window. Note: Use the intercepts of the function to determine the size of the window on the calculator. Choose an appropriate scale for each axis. Chapter 2: Modeling a Business 4

5 Example 2 What is the revenue equation for the Picasso Paints product? Write the revenue equation in terms of the price. Example 3 Determine the revenue if the price per item is set at $ Example 4 Graph the revenue equation on a coordinate plane and provide the viewing window. Example 5 Use the graph in Example 4, which price would yield the higher revenue, $28 or $40? Chapter 2: Modeling a Business 5

6 Example 6 The revenue and expense functions are graphed on the same set of axes. The points of intersection are labeled A and B. Explain what is happening at those two points. Example 7 Why is using the prices of $7.50 and $61.00 not in the best interest of the company? Explain your answer. Chapter 2: Modeling a Business 6

7 Example 8 An electronics company manufactures earphones for portable music devices. Each earphone costs $5 to manufacture. Fixed costs are $20,000. The demand function is q = 200p + 40,000. Write the expense function in terms of q and determine a suitable viewing window for that function. Graph the expense function. Chapter 2: Modeling a Business 7