Type The Correct Answer In Each Box. Use Numerals Instead Of Words. If Necessary, Use / For The Fraction Bar(s).The Function, \[$ F(x) \$\], Describes The Height Of A Dome On Top Of A Building, Where \[$ F(x) \$\] Is The Height From The

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Introduction

In mathematics, functions are used to describe the relationship between variables. In this case, we are given a function { f(x) $}$ that describes the height of a dome on top of a building. The height of the dome is a function of the variable { x $}$, which represents the distance from the center of the dome. In this article, we will explore the function { f(x) $}$ and determine its correct form.

The Function

The function { f(x) $}$ is given by:

{ f(x) = \frac{1}{2}x^2 + 3x + 2 $}$

This function describes the height of the dome as a function of the distance { x $}$ from the center of the dome. The function is a quadratic function, which means that it has a parabolic shape.

Graphing the Function

To understand the behavior of the function, we can graph it. The graph of the function { f(x) $}$ is a parabola that opens upwards. The vertex of the parabola is at the point { (x, f(x)) $}$, where { x $}$ is the value of the variable that minimizes the function.

Finding the Vertex

To find the vertex of the parabola, we can use the formula:

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

where { a $}$ and { b $}$ are the coefficients of the quadratic function. In this case, { a = \frac{1}{2} $}$ and { b = 3 $}$. Plugging these values into the formula, we get:

{ x = -\frac{3}{2(\frac{1}{2})} = -3 $}$

This means that the vertex of the parabola is at the point { (-3, f(-3)) $}$.

Evaluating the Function

To evaluate the function at the point { (-3, f(-3)) $}$, we can plug { x = -3 $}$ into the function:

{ f(-3) = \frac{1}{2}(-3)^2 + 3(-3) + 2 $}$

Simplifying the expression, we get:

{ f(-3) = \frac{1}{2}(9) - 9 + 2 $}$

{ f(-3) = \frac{9}{2} - 9 + 2 $}$

{ f(-3) = \frac{9}{2} - 7 $}$

{ f(-3) = \frac{9}{2} - \frac{14}{2} $}$

{ f(-3) = -\frac{5}{2} $}$

This means that the height of the dome at the point { (-3, f(-3)) $}$ is { -\frac{5}{2} $}$.

Conclusion

In this article, we explored the function { f(x) $}$ that describes the height of a dome on top of a building. We graphed the function, found the vertex of the parabola, and evaluated the function at the point { (-3, f(-3)) $}$. The height of the dome at this point is { -\frac{5}{2} $}$.

Key Takeaways

  • The function { f(x) $}$ describes the height of a dome on top of a building.
  • The function is a quadratic function, which means that it has a parabolic shape.
  • The vertex of the parabola is at the point { (-3, f(-3)) $}$.
  • The height of the dome at the point { (-3, f(-3)) $}$ is { -\frac{5}{2} $}$.

Practice Problems

  1. Find the vertex of the parabola described by the function { f(x) = x^2 + 4x + 3 $}$.
  2. Evaluate the function { f(x) = 2x^2 - 5x + 1 $}$ at the point { (2, f(2)) $}$.
  3. Graph the function { f(x) = x^2 - 2x - 3 $}$ and find its vertex.

Answers

  1. The vertex of the parabola is at the point { (-2, f(-2)) $}$.
  2. The value of the function at the point { (2, f(2)) $}$ is { f(2) = 2(2)^2 - 5(2) + 1 = 8 - 10 + 1 = -1 $}$.
  3. The graph of the function is a parabola that opens upwards. The vertex of the parabola is at the point { (-1, f(-1)) $}$.
    Q&A: Understanding the Function of a Dome's Height =====================================================

Introduction

In our previous article, we explored the function { f(x) $}$ that describes the height of a dome on top of a building. We graphed the function, found the vertex of the parabola, and evaluated the function at the point { (-3, f(-3)) $}$. In this article, we will answer some frequently asked questions about the function { f(x) $}$.

Q: What is the purpose of the function { f(x) $}$?

A: The function { f(x) $}$ is used to describe the height of a dome on top of a building. It is a mathematical model that helps us understand the relationship between the distance from the center of the dome and the height of the dome.

Q: What type of function is { f(x) $}$?

A: The function { f(x) $}$ is a quadratic function, which means that it has a parabolic shape.

Q: How do I graph the function { f(x) $}$?

A: To graph the function { f(x) $}$, you can use a graphing calculator or a computer program. You can also use a piece of graph paper and a pencil to draw the graph by hand.

Q: How do I find the vertex of the parabola described by the function { f(x) $}$?

A: To find the vertex of the parabola, you can use the formula:

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

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

Q: How do I evaluate the function { f(x) $}$ at a given point?

A: To evaluate the function { f(x) $}$ at a given point, you can plug the value of the variable into the function and simplify the expression.

Q: What is the height of the dome at the point { (-3, f(-3)) $}$?

A: The height of the dome at the point { (-3, f(-3)) $}$ is { -\frac{5}{2} $}$.

Q: Can I use the function { f(x) $}$ to model other types of domes?

A: Yes, you can use the function { f(x) $}$ to model other types of domes. However, you may need to adjust the coefficients of the function to fit the specific shape of the dome.

Q: How do I determine the coefficients of the function { f(x) $}$?

A: To determine the coefficients of the function { f(x) $}$, you can use data from the dome, such as the height and distance from the center of the dome.

Q: Can I use the function { f(x) $}$ to model other types of objects?

A: Yes, you can use the function { f(x) $}$ to model other types of objects, such as parabolas and ellipses.

Conclusion

In this article, we answered some frequently asked questions about the function { f(x) $}$. We hope that this article has been helpful in understanding the function and its applications.

Key Takeaways

  • The function { f(x) $}$ describes the height of a dome on top of a building.
  • The function is a quadratic function, which means that it has a parabolic shape.
  • The vertex of the parabola is at the point { (-3, f(-3)) $}$.
  • The height of the dome at the point { (-3, f(-3)) $}$ is { -\frac{5}{2} $}$.

Practice Problems

  1. Find the vertex of the parabola described by the function { f(x) = x^2 + 4x + 3 $}$.
  2. Evaluate the function { f(x) = 2x^2 - 5x + 1 $}$ at the point { (2, f(2)) $}$.
  3. Graph the function { f(x) = x^2 - 2x - 3 $}$ and find its vertex.

Answers

  1. The vertex of the parabola is at the point { (-2, f(-2)) $}$.
  2. The value of the function at the point { (2, f(2)) $}$ is { f(2) = 2(2)^2 - 5(2) + 1 = 8 - 10 + 1 = -1 $}$.
  3. The graph of the function is a parabola that opens upwards. The vertex of the parabola is at the point { (-1, f(-1)) $}$.