Given $p(x) = -4(x - 15)^2 + 2$, What Is The Value Of $p(7)$?

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Introduction

Quadratic functions are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. In this article, we will focus on evaluating a quadratic function, specifically the function p(x) = -4(x - 15)^2 + 2, at a given value of x, namely x = 7.

Understanding the Quadratic Function

The given quadratic function is p(x) = -4(x - 15)^2 + 2. This function has a few key features that are worth noting:

  • The function is in the form of a quadratic function, which is a polynomial function of degree two.
  • The coefficient of the squared term is -4, which means the parabola opens downward.
  • The vertex of the parabola is at (15, 2), which is the point where the function reaches its maximum or minimum value.
  • The function has a constant term of 2, which means that the function will always be 2 units above the x-axis.

Evaluating the Function at x = 7

To evaluate the function p(x) = -4(x - 15)^2 + 2 at x = 7, we need to substitute x = 7 into the function and simplify the expression.

p(7) = -4(7 - 15)^2 + 2

Step 1: Evaluate the expression inside the parentheses

First, we need to evaluate the expression inside the parentheses, which is (7 - 15).

(7 - 15) = -8

Step 2: Square the result

Next, we need to square the result, which is (-8)^2.

(-8)^2 = 64

Step 3: Multiply the result by -4

Now, we need to multiply the result by -4.

-4(64) = -256

Step 4: Add 2 to the result

Finally, we need to add 2 to the result.

-256 + 2 = -254

Conclusion

Therefore, the value of p(7) is -254.

Example Use Cases

Evaluating a quadratic function at a given value of x has numerous applications in various fields. Here are a few example use cases:

  • Physics: In physics, quadratic functions are used to model the motion of objects under the influence of gravity. For example, the height of a projectile as a function of time can be modeled using a quadratic function.
  • Engineering: In engineering, quadratic functions are used to design and optimize systems. For example, the stress on a beam as a function of its length can be modeled using a quadratic function.
  • Economics: In economics, quadratic functions are used to model the behavior of economic systems. For example, the demand for a product as a function of its price can be modeled using a quadratic function.

Tips and Tricks

Here are a few tips and tricks for evaluating quadratic functions:

  • Use the vertex form: The vertex form of a quadratic function is f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. This form can make it easier to evaluate the function at a given value of x.
  • Use the factoring method: The factoring method involves factoring the quadratic expression into the product of two binomials. This method can make it easier to evaluate the function at a given value of x.
  • Use the quadratic formula: The quadratic formula is a formula that can be used to find the roots of a quadratic equation. This formula can be used to evaluate the function at a given value of x.

Conclusion

In conclusion, evaluating a quadratic function at a given value of x is a fundamental concept in mathematics. By understanding the properties of quadratic functions and using the appropriate techniques, we can evaluate these functions with ease. In this article, we evaluated the function p(x) = -4(x - 15)^2 + 2 at x = 7 and found that the value of p(7) is -254. We also discussed the importance of quadratic functions in various fields and provided tips and tricks for evaluating these functions.

Introduction

Evaluating quadratic functions is a fundamental concept in mathematics, and it has numerous applications in various fields. In our previous article, we discussed how to evaluate a quadratic function at a given value of x, specifically the function p(x) = -4(x - 15)^2 + 2 at x = 7. In this article, we will answer some frequently asked questions about quadratic function evaluation.

Q&A

Q: What is the difference between a quadratic function and a linear function?

A: A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. A linear function, on the other hand, is a polynomial function of degree one, which means the highest power of the variable is one.

Q: How do I know if a function is quadratic or linear?

A: To determine if a function is quadratic or linear, you need to look at the highest power of the variable. If the highest power is two, the function is quadratic. If the highest power is one, the function is linear.

Q: What is the vertex form of a quadratic function?

A: The vertex form of a quadratic function is f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.

Q: How do I evaluate a quadratic function at a given value of x?

A: To evaluate a quadratic function at a given value of x, you need to substitute x into the function and simplify the expression.

Q: What is the importance of quadratic functions in physics?

A: Quadratic functions are used to model the motion of objects under the influence of gravity. For example, the height of a projectile as a function of time can be modeled using a quadratic function.

Q: What is the importance of quadratic functions in engineering?

A: Quadratic functions are used to design and optimize systems. For example, the stress on a beam as a function of its length can be modeled using a quadratic function.

Q: What is the importance of quadratic functions in economics?

A: Quadratic functions are used to model the behavior of economic systems. For example, the demand for a product as a function of its price can be modeled using a quadratic function.

Q: How do I use the factoring method to evaluate a quadratic function?

A: The factoring method involves factoring the quadratic expression into the product of two binomials. This method can make it easier to evaluate the function at a given value of x.

Q: How do I use the quadratic formula to evaluate a quadratic function?

A: The quadratic formula is a formula that can be used to find the roots of a quadratic equation. This formula can be used to evaluate the function at a given value of x.

Tips and Tricks

Here are a few tips and tricks for evaluating quadratic functions:

  • Use the vertex form: The vertex form of a quadratic function is f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. This form can make it easier to evaluate the function at a given value of x.
  • Use the factoring method: The factoring method involves factoring the quadratic expression into the product of two binomials. This method can make it easier to evaluate the function at a given value of x.
  • Use the quadratic formula: The quadratic formula is a formula that can be used to find the roots of a quadratic equation. This formula can be used to evaluate the function at a given value of x.

Conclusion

In conclusion, evaluating quadratic functions is a fundamental concept in mathematics, and it has numerous applications in various fields. By understanding the properties of quadratic functions and using the appropriate techniques, we can evaluate these functions with ease. In this article, we answered some frequently asked questions about quadratic function evaluation and provided tips and tricks for evaluating these functions.

Example Use Cases

Evaluating quadratic functions has numerous applications in various fields. Here are a few example use cases:

  • Physics: In physics, quadratic functions are used to model the motion of objects under the influence of gravity. For example, the height of a projectile as a function of time can be modeled using a quadratic function.
  • Engineering: In engineering, quadratic functions are used to design and optimize systems. For example, the stress on a beam as a function of its length can be modeled using a quadratic function.
  • Economics: In economics, quadratic functions are used to model the behavior of economic systems. For example, the demand for a product as a function of its price can be modeled using a quadratic function.

Further Reading

If you want to learn more about quadratic functions and their applications, here are a few resources you can consult:

  • Textbooks: There are many textbooks available on quadratic functions and their applications. Some popular textbooks include "Calculus" by Michael Spivak and "Linear Algebra and Its Applications" by Gilbert Strang.
  • Online Resources: There are many online resources available on quadratic functions and their applications. Some popular online resources include Khan Academy, MIT OpenCourseWare, and Wolfram Alpha.
  • Research Papers: There are many research papers available on quadratic functions and their applications. Some popular research papers include "Quadratic Functions and Their Applications" by David M. Bressoud and "Quadratic Functions in Physics" by John R. Taylor.