Identify The Vertex, The Axis Of Symmetry, The Maximum Or Minimum Value, And The Domain And Range Of The Function:${ F(x) = -(x-5)^2 - 23 }$What Is The Range Of The Function? Select The Correct Answer Below And, If Necessary, Fill In The

Quadratic functions are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. In this article, we will focus on identifying the vertex, axis of symmetry, maximum or minimum value, domain, and range of the function $f(x) = -(x-5)^2 - 23$.

The Vertex Form of a Quadratic Function

The given function $f(x) = -(x-5)^2 - 23$ is in the vertex form of a quadratic function, which is $f(x) = a(x-h)^2 + k$, where $(h,k)$ is the vertex of the parabola. In this case, the vertex is $(5,-23)$.

Identifying the Axis of Symmetry

The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the parabola. In this case, the axis of symmetry is the line $x=5$.

Determining the Maximum or Minimum Value

Since the coefficient of the squared term is negative, the parabola opens downward, and the vertex is the maximum point. Therefore, the maximum value of the function is $-23$.

Finding the Domain and Range

The domain of a quadratic function is the set of all possible input values for which the function is defined. Since the function is a polynomial, it is defined for all real numbers, and the domain is $(-\infty,\infty)$.

The range of a quadratic function is the set of all possible output values for which the function is defined. Since the parabola opens downward, the range is all real numbers less than or equal to the maximum value, which is $-23$. Therefore, the range is $(-\infty,-23]$.

Conclusion

In conclusion, the vertex of the function $f(x) = -(x-5)^2 - 23$ is $(5,-23)$, the axis of symmetry is the line $x=5$, the maximum value is $-23$, the domain is $(-\infty,\infty)$, and the range is $(-\infty,-23]$.

Discussion

The range of a quadratic function is an important concept in mathematics, and it has various applications in real-world problems. For example, in physics, the range of a projectile is an important factor in determining the trajectory of the projectile.

Final Answer

In this article, we will answer some frequently asked questions about quadratic functions, including the vertex form, axis of symmetry, maximum or minimum value, domain, and range.

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 identify the axis of symmetry of a quadratic function?

A: To identify the axis of symmetry of a quadratic function, you need to find the vertex of the parabola. The axis of symmetry is a vertical line that passes through the vertex.

Q: What is the maximum or minimum value of a quadratic function?

A: The maximum or minimum value of a quadratic function depends on the direction of the parabola. If the parabola opens upward, the vertex is the minimum point, and the maximum value is not defined. If the parabola opens downward, the vertex is the maximum point, and the minimum value is not defined.

Q: How do I find the domain and range of a quadratic function?

A: The domain of a quadratic function is the set of all possible input values for which the function is defined. Since the function is a polynomial, it is defined for all real numbers, and the domain is $(-\infty,\infty)$. The range of a quadratic function is the set of all possible output values for which the function is defined. The range depends on the direction of the parabola and the maximum or minimum value.

Q: What is the difference between the vertex form and the standard 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. The standard form of a quadratic function is $f(x) = ax^2 + bx + c$. The vertex form is more convenient for graphing and finding the axis of symmetry, while the standard form is more convenient for factoring and solving equations.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you need to find the vertex of the parabola and the axis of symmetry. Then, you can use the vertex and the axis of symmetry to draw the parabola.

Q: What are some real-world applications of quadratic functions?

A: Quadratic functions have many real-world applications, including physics, engineering, economics, and computer science. Some examples include:

• Projectile motion: Quadratic functions are used to model the trajectory of a projectile.
• Optimization: Quadratic functions are used to find the maximum or minimum value of a function.
• Data analysis: Quadratic functions are used to model data that follows a quadratic pattern.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you need to find the values of x that satisfy the equation. There are several methods to solve quadratic equations, including factoring, the quadratic formula, and graphing.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that is used to solve quadratic equations. It is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where a, b, and c are the coefficients of the quadratic equation.

Conclusion

In conclusion, quadratic functions are an important concept in mathematics, and they have many real-world applications. Understanding the properties of quadratic functions, including the vertex form, axis of symmetry, maximum or minimum value, domain, and range, is crucial for solving various mathematical problems.