How to Use the Quadratic Function

Oct 13, 2025 by miodeya.com 39 views

**A Company's Profit Model: Understanding the Quadratic Function** ===========================================================

Introduction

In the world of business, understanding the factors that influence a company's profits is crucial for making informed decisions. One way to model a company's profits is by using a quadratic function. In this article, we will explore a quadratic function used by a company to model its profits, and we will answer some frequently asked questions related to this topic.

The Quadratic Function

The quadratic function used by the company is given by:

f(x) = -18x^2 + 202.61x + 1,282.46

where $x$ is the number of years since 2006.

What is the Meaning of the Quadratic Function?

The quadratic function represents the company's profits in thousands of dollars, where $x$ is the number of years since 2006. This means that if we plug in a value of $x$ , we can find the company's profits for that particular year.

How to Use the Quadratic Function

To use the quadratic function, we simply need to plug in a value of $x$ and solve for $f(x)$ . For example, if we want to find the company's profits in 2020, we would plug in $x = 14$ (since 2020 is 14 years after 2006).

Q&A

Q: What is the value of $f(14)$ ?

A: To find the value of $f(14)$ , we simply need to plug in $x = 14$ into the quadratic function:

f(14) = -18(14)^2 + 202.61(14) + 1,282.46

Simplifying the expression, we get:

f(14) = -18(196) + 202.61(14) + 1,282.46

f(14) = -3,528 + 2,835.54 + 1,282.46

f(14) = 589.00

Therefore, the company's profits in 2020 were approximately $589,000.

Q: What is the vertex of the quadratic function?

A: The vertex of the quadratic function can be found using the formula:

x = -\frac{b}{2a}

where $a$ and $b$ are the coefficients of the quadratic function.

In this case, $a = -18$ and $b = 202.61$ . Plugging these values into the formula, we get:

x = -\frac{202.61}{2(-18)}

x = -\frac{202.61}{-36}

x = 5.64

Therefore, the vertex of the quadratic function is at $x = 5.64$ .

Q: What is the x-intercept of the quadratic function?

A: The x-intercept of the quadratic function can be found by setting $f(x) = 0$ and solving for $x$ .

In this case, we set $f(x) = 0$ and solve for $x$ :

-18x^2 + 202.61x + 1,282.46 = 0

Solving this quadratic equation, we get:

x = -\frac{202.61 \pm \sqrt{202.61^2 - 4(-18)(1,282.46)}}{2(-18)}

x = -\frac{202.61 \pm \sqrt{41,111.32 + 92,311.52}}{-36}

x = -\frac{202.61 \pm \sqrt{133,422.84}}{-36}

x = -\frac{202.61 \pm 364.51}{-36}

Therefore, the x-intercepts of the quadratic function are at $x = -\frac{202.61 + 364.51}{-36}$ and $x = -\frac{202.61 - 364.51}{-36}$ .

Q: What is the y-intercept of the quadratic function?

A: The y-intercept of the quadratic function can be found by setting $x = 0$ and solving for $f(x)$ .

In this case, we set $x = 0$ and solve for $f(x)$ :

f(0) = -18(0)^2 + 202.61(0) + 1,282.46

f(0) = 1,282.46

Therefore, the y-intercept of the quadratic function is at $f(x) = 1,282.46$ .

Q: How to graph the quadratic function?

A: To graph the quadratic function, we can use a graphing calculator or a computer program. We can also use the vertex and x-intercepts to graph the function.

To graph the function, we can start by plotting the vertex at $x = 5.64$ and $f(x) = 1,282.46$ . We can then plot the x-intercepts at $x = -\frac{202.61 + 364.51}{-36}$ and $x = -\frac{202.61 - 364.51}{-36}$ .

We can also use the quadratic function to graph the function by plugging in different values of $x$ and plotting the corresponding values of $f(x)$ .

Conclusion

In this article, we explored a quadratic function used by a company to model its profits. We also answered some frequently asked questions related to this topic, including the value of $f(14)$ , the vertex of the quadratic function, the x-intercept of the quadratic function, the y-intercept of the quadratic function, and how to graph the quadratic function.