2 5 GRAPHS OF EXPENSE AND REVENUE FUNCTIONS

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

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. Slide 2 Financial Algebra © Cengage/South-Western

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

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

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

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

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

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. Slide 8 Financial Algebra © Cengage Learning/South-Western

CHECK YOUR UNDERSTANDING 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. Slide 9 Financial Algebra © Cengage Learning/South-Western

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

CHECK YOUR UNDERSTANDING Determine the revenue if the price per item is set at $25. 00. Slide 11 Financial Algebra © Cengage Learning/South-Western

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

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

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

CHECK YOUR UNDERSTANDING Why is using the prices of $7. 50 and $61. 00 not in the best interest of the company? Slide 15 Financial Algebra © Cengage Learning/South-Western