Fill In The Equation For This Function:${ F(x) = (x-4)^2 + [?] }$

Mar 10, 2025 by ADMIN 67 views

**Solving the Equation for the Function f(x)**

Introduction to the Function

The given function is $f(x) = (x-4)^2 + [?]$ . This function represents a quadratic equation in the form of $f(x) = a(x-h)^2 + k$ , where $(h,k)$ is the vertex of the parabola. In this case, the vertex is $(4,0)$ , which means that the function has a minimum value of $0$ at $x=4$ . However, the function is incomplete, and we need to fill in the equation to make it a complete quadratic function.

Understanding the Structure of the Function

The given function is in the form of $f(x) = (x-4)^2 + [?]$ . This means that the function has a squared term, which is $(x-4)^2$ . The squared term represents a parabola that opens upwards, and its vertex is at $(4,0)$ . The squared term is always non-negative, which means that it will always be greater than or equal to $0$ . Therefore, the function will always be greater than or equal to $0$ .

Filling in the Equation

To fill in the equation, we need to determine the value of the constant term, which is represented by $[?]$ . Since the function has a minimum value of $0$ at $x=4$ , the constant term must be equal to $0$ . Therefore, the complete equation for the function is $f(x) = (x-4)^2 + 0$ .

Simplifying the Equation

The equation $f(x) = (x-4)^2 + 0$ can be simplified to $f(x) = (x-4)^2$ . This is because the constant term $0$ does not affect the value of the function. Therefore, the simplified equation for the function is $f(x) = (x-4)^2$ .

Graphing the Function

The function $f(x) = (x-4)^2$ is a quadratic function that represents a parabola. The parabola opens upwards, and its vertex is at $(4,0)$ . The graph of the function is a U-shaped curve that is symmetric about the vertical line $x=4$ .

Key Features of the Function

The function $f(x) = (x-4)^2$ has several key features that are important to understand. These features include:

Vertex: The vertex of the parabola is at $(4,0)$ .
Axis of Symmetry: The axis of symmetry is the vertical line $x=4$ .
Minimum Value: The minimum value of the function is $0$ , which occurs at $x=4$ .
Maximum Value: The maximum value of the function is undefined, which means that the function has no maximum value.

Real-World Applications of the Function

The function $f(x) = (x-4)^2$ has several real-world applications. These applications include:

Physics: The function can be used to model the motion of an object that is subject to a constant force.
Engineering: The function can be used to design the shape of a parabolic reflector.
Economics: The function can be used to model the demand for a product.

Conclusion

In conclusion, the function $f(x) = (x-4)^2 + [?]$ is a quadratic function that represents a parabola. The function has a vertex at $(4,0)$ and an axis of symmetry at $x=4$ . The function has a minimum value of $0$ at $x=4$ and no maximum value. The function has several real-world applications, including physics, engineering, and economics.

Final Answer

The final answer to the equation is $f(x) = (x-4)^2$ .

Q: What is the vertex of the parabola represented by the function f(x)?

A: The vertex of the parabola represented by the function f(x) is at (4,0).

Q: What is the axis of symmetry of the parabola represented by the function f(x)?

A: The axis of symmetry of the parabola represented by the function f(x) is the vertical line x=4.

Q: What is the minimum value of the function f(x)?

A: The minimum value of the function f(x) is 0, which occurs at x=4.

Q: What is the maximum value of the function f(x)?

A: The maximum value of the function f(x) is undefined, which means that the function has no maximum value.

Q: How can the function f(x) be used in real-world applications?

A: The function f(x) can be used in several real-world applications, including physics, engineering, and economics. For example, it can be used to model the motion of an object that is subject to a constant force, design the shape of a parabolic reflector, or model the demand for a product.

Q: What is the simplified equation for the function f(x)?

A: The simplified equation for the function f(x) is f(x) = (x-4)^2.

Q: What is the graph of the function f(x)?

A: The graph of the function f(x) is a U-shaped curve that is symmetric about the vertical line x=4.

Q: What are the key features of the function f(x)?

A: The key features of the function f(x) include:

Vertex: The vertex of the parabola is at (4,0).
Axis of Symmetry: The axis of symmetry is the vertical line x=4.
Minimum Value: The minimum value of the function is 0, which occurs at x=4.
Maximum Value: The maximum value of the function is undefined, which means that the function has no maximum value.

Q: Can the function f(x) be used to model any real-world phenomenon?

A: Yes, the function f(x) can be used to model any real-world phenomenon that can be represented by a quadratic equation. For example, it can be used to model the motion of an object that is subject to a constant force, design the shape of a parabolic reflector, or model the demand for a product.

Q: What are some common applications of the function f(x) in physics?

A: Some common applications of the function f(x) in physics include:

Modeling the motion of an object that is subject to a constant force.
Designing the shape of a parabolic reflector.
Modeling the trajectory of a projectile.

Q: What are some common applications of the function f(x) in engineering?

A: Some common applications of the function f(x) in engineering include:

Designing the shape of a parabolic reflector.
Modeling the stress on a beam.
Modeling the vibration of a system.

Q: What are some common applications of the function f(x) in economics?

A: Some common applications of the function f(x) in economics include:

Modeling the demand for a product.
Modeling the supply of a product.
Modeling the price of a product.

Conclusion

In conclusion, the function f(x) is a quadratic function that represents a parabola. The function has a vertex at (4,0) and an axis of symmetry at x=4. The function has a minimum value of 0 at x=4 and no maximum value. The function has several real-world applications, including physics, engineering, and economics.