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 Precalculus: Nonlinear Functions

Nonlinear functions are functions whose graphs are not straight lines. While there are many types of nonlinear functions, this course will focus on three that are commonly used in business: parabolic functions, demand functions, and exponential functions. A basic graphic representation of each of these functions is shown below.

In this course, you may notice that graphs of functions often appear only in the upper right-hand quadrant on a set of axes, in the region where both the x and y values are positive. This is because, in business situations, a function's x and y values usually stand for non-negative amounts, such as time, units produced, dollar amounts, and so on. Therefore, when the functions above are used in a business context, they will usually appear entirely in the upper right-hand quadrant (called the first quadrant) on a set of axes.

Parabolic Functions

A parabolic function is a symmetric, U-shaped function that has x2 as its highest term. The basic form of this function is organized into an equation below on the right.

This form is useful because it tells you a number of things about the shape of the parabolic function (illustrated above on the left), allowing you to graph the function quickly and to understand several key features.

 1 The direction of the parabola is determined by the constant c. If the constant c is positive, as in the graph above, the parabola opens upward in the shape of a U. If the constant c is negative, the parabola opens downward in the shape of an inverted U. 2 The sharpness of the parabola is also determined by c. The further the constant is from zero, the steeper the curve will be. 3 The parabola's highest or lowest point, known as the vertex, has the coordinates (a, b).

Consider the following production function, in which a company's marginal output in units per employee, f(x), depends on how many workers are employed, x.

You can recognize that this function is a parabola because its highest term, (x – 5)2, is squared. Simply by looking at the function's formula, you can determine a number of things about the parabola's shape.

• Because the constant c, –.5, is negative, the parabola opens downward.
• Because the absolute value of constant c, –.5, is between 0 and 1, the parabola is relatively wide.
• The parabola's highest point, the vertex, has the coordinates (5, 10). The vertex is the point at which marginal production is maximized. When 5 people are employed, 10 units of output are produced per employee.
• This function is graphed below.

The graph below illustrates how the constant c influences the width of a parabolic function. This graph illustrates the parabolic functions –.5(x – 5)2 + 10 and –2(x – 5)2 + 10. The constant c is –.5 and –2, respectively. Because the constants are negative, both functions open downward. As you can see, the further the constant is from 0, the sharper the curve. When c has the value of –2, the curve is sharper than when c is –.5.

Demand Function

A demand function has a general form , which can also be written as x-1. The demand function used often in business to describe demand; it is represented in equation form below on the right.

This form of the function tells you some things about the shape and behavior of the demand function, which is illustrated above on the left.

• Graphically, the constant c determines how close the graph will be to the x and y axes. The smaller c is, the closer the graph is to the origin, the point (0, 0).
• In general, demand functions will approach but will never cross the x or y axes.
• Consider the following production function, in which the demand for a company's cookies,
f(x), depends on the price of the cookies, x.

When graphed on a coordinate plane, demand functions have two sections that are mirror images of each other. In business situations, only the section in the first quadrant is used because the values being examined are positive. For example, the demand for cookies and the price of cookies would not be negative. Therefore, the company's demand function for cookies would be illustrated in this way.

The constant, 2, determines how close the graph is to the x and y axes. If the demand for cookies increases, the constant in the demand function would increase. How would this change the shape of the demand function? Consider the following graph, which shows the original demand function, where c is 2, and the increased demand function, where c is 5.

Notice that the larger c is, the further away the graph is from the origin (0, 0).

Exponential Function

Any function where a constant (a) is raised to a power of x is an exponential function. Exponential functions in business take the form of exponential growth and decay functions. In business scenarios, exponential functions often appear in the equation form that appears below on the right.

This form tells you two important things about the function's shape, which is graphed above on the left.

• The point (0, c) is the function's y-intercept.
• The sign of the exponent x (positive or negative) indicates the direction of the function.
• If the exponent is positive, the function increases to the right. (This is known as "exponential growth.")
• If the exponent is negative, the function decreases to the right. (This is known as "exponential decay.")

The exponential function is commonly used for investments with compounded interest. For example, imagine that you put \$10 into an account that compounds annually at a rate of 7 percent. The function used to find the value of this investment at a future point in time is below.

FV = the future value
N = the number of years in the future

Simply by looking at this function, you can tell that it is an exponential function because it has a constant, 1.07, raised to a power N. Using this function, you can quickly determine two things about its shape.

• Because the constant c is 10, the function's y-intercept is (0, 10).
• The exponent N is positive, which means the function is increasing as N increases to the right.
• Think about the business situation and you can see that this information makes sense. Consider the graph above with the following scenario—the y-intercept tells you that at the time of the initial investment, \$10 was deposited. This investment will increase over the years, as indicated by the positive exponent. You can find the exact shape of the function by calculating the value of the investment for a number of years and then plotting the coordinates.

1. Name the following nonlinear functions.

a. f(x) = 4(x – 2)2 + 6

b. f(x) = 6,400(1.05)x

c. f(x) =

2. Given the following function, answer the questions below.

f(x) = 20,000(1.1)x

a. What type of nonlinear function is this?

b. What is the y-intercept?

c. When x = 3, what does f(x) equal?

3. Given the following function, answer the questions below.

f(x) = 3(x – 4)2 + 7

a. What type of nonlinear function is this?

b. What is the vertex?

c. Is the function thin or wide compared to a parabola where c = 1?

4. An individual puts \$1,500 into a bank account that has an interest rate of 7 percent. The future value of the investment is modeled by the function FV = 1,500(1.07)T, where FV is the future value and N is the number of years in the future. Using this information, answer the following questions about this function.

a. What type of nonlinear function defines this relationship?

b. What is the y-intercept?

c. How much money will the individual have in 10 years?

5. The demand for book bags is modeled by f(x) = , where x equals the number of book bags demanded and f(x) is the price.

a. What is the price when nine book bags are demanded?

b. After it is found that the book bags have faulty straps, the demand changed to f(x) = , where x = number of book bags demanded and f(x) is the price. Is this new demand closer or further from the origin (0, 0)?

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