1. Financial Algebra © Cengage/South-Western Slide 1 2-5 GRAPHS OF EXPENSE AND REVENUE FUNCTIONS Find the vertex of the parabola with equation y = x 2 + 8x + 15. Vertex formula: (-b/2a, y) Warm Ups:

2. Financial Algebra © Cengage/South-Western Slide 2 2-5 GRAPHS OF EXPENSE AND REVENUE FUNCTIONS 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. OBJECTIVES

3. Financial Algebra © Cengage Learning/South-Western Slide 3 nonlinear function second-degree equation quadratic equation parabola leading coefficient maximum value vertex of a parabola axis of symmetry Key Terms

4. Financial Algebra © Cengage Learning/South-Western Nonlinear function - A function that has a graph that is not a straight line. Second-degree equation - A function with a variable raised to an exponent of 2. Quadratic equation - An equation written in the form y = ax2 + bx + c where a, b, and c are real numbers and a ≠ 0. Parabola - The shape of the graph of a quadratic function. Leading coefficient - The first coefficient in a quadratic equation when written in standard form, usually denoted by a. Maximum value - The peak or vertex of a parabola; the point where revenue can be maximized. Vertex of a parabola - The peak of the parabola and the point where revenue is maximized; the point of maximum value in a quadratic equation. Axis of symmetry - A vertical line that can be drawn through the vertex of a parabola so that the dissected parts of the parabola are mirror images of each other. Nonlinear function - A function that has a graph that is not a straight line. Second-degree equation - A function with a variable raised to an exponent of 2. Quadratic equation - An equation written in the form y = ax2 + bx + c where a, b, and c are real numbers and a ≠ 0. Parabola - The shape of the graph of a quadratic function. Leading coefficient - The first coefficient in a quadratic equation when written in standard form, usually denoted by a. Maximum value - The peak or vertex of a parabola; the point where revenue can be maximized. Vertex of a parabola - The peak of the parabola and the point where revenue is maximized; the point of maximum value in a quadratic equation. Axis of symmetry - A vertical line that can be drawn through the vertex of a parabola so that the dissected parts of the parabola are mirror images of each other. Slide 4

5. Financial Algebra © Cengage Learning/South-Western Slide 5 How can expense and revenue be graphed? How does price contribute to consumer demand? Name some other factors that might also play a role in the quantity of a product consumers purchase. Why does a non-vertical line have slope but a nonlinear function does not?

6. Financial Algebra © Cengage Learning/South-Western Slide 6 Parabola with a positive leading coefficient

7. Financial Algebra © Cengage Learning/South-Western Slide 7 Parabola with a negative leading coefficient

8. Financial Algebra © Cengage Learning/South-Western Slide 8 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 = –500 p + 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.

9. Financial Algebra © Cengage Learning/South-Western Slide 9 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 = –200 p + 40,000. Write the expense function in terms of q and determine a suitable viewing window for that function. Graph the expense function. CHECK YOUR UNDERSTANDING

10. Financial Algebra © Cengage Learning/South-Western Slide 10 Example 2 What is the revenue equation for the Picasso Paints product? Write the revenue equation in terms of the price.

11. Financial Algebra © Cengage Learning/South-Western Slide 11 Determine the revenue if the price per item is set at $25.00. CHECK YOUR UNDERSTANDING

12. Financial Algebra © Cengage Learning/South-Western Slide 12 EXAMPLE 3 Graph the revenue equation on a coordinate plane.

13. Financial Algebra © Cengage Learning/South-Western Slide 13 Use the graph in Example 3. Which price would yield the higher revenue, $28 or $40? CHECK YOUR UNDERSTANDING

14. Financial Algebra © Cengage Learning/South-Western Slide 14 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 4

15. Financial Algebra © Cengage Learning/South-Western Slide 15 Why is using the prices of $7.50 and $61.00 not in the best interest of the company? CHECK YOUR UNDERSTANDING