Significance of the Vertex

Oct 13, 2025 by miodeya.com 127 views

Introduction

In mathematics, a parabola is a type of quadratic equation that can be represented in various forms, including the standard form, vertex form, and factored form. The vertex of a parabola is a crucial point that represents the maximum or minimum value of the function. In this article, we will delve into the concept of the vertex of a parabola, its significance, and how to find it using the given function $f(x)=\frac{-x^2}{4}+3x-3$ .

What is the Vertex of a Parabola?

The vertex of a parabola is the point where the parabola changes direction, either from opening upwards to downwards or vice versa. It is the highest or lowest point on the graph of the parabola. The vertex form of a parabola is given by the equation $f(x)=a(x-h)^2+k$ , where $(h,k)$ represents the coordinates of the vertex.

Significance of the Vertex

The vertex of a parabola has several important applications in mathematics and real-world problems. Some of the key significance of the vertex include:

Maximum or Minimum Value: The vertex represents the maximum or minimum value of the function, depending on the direction of the parabola.
Axis of Symmetry: The vertex is the point of symmetry for the parabola, meaning that the parabola is reflected about this point.
Optimization Problems: The vertex is often used to solve optimization problems, where the goal is to maximize or minimize a function.

Finding the Vertex of a Parabola

To find the vertex of a parabola, we can use the following methods:

Vertex Form: If the parabola is in vertex form, the vertex is already given as $(h,k)$ .
Standard Form: If the parabola is in standard form, we can complete the square to convert it to vertex form and find the vertex.
Factored Form: If the parabola is in factored form, we can use the x-intercepts to find the vertex.

Finding the Vertex of the Given Function

The given function is $f(x)=\frac{-x^2}{4}+3x-3$ . To find the vertex, we can complete the square to convert it to vertex form.

Step 1: Factor out the coefficient of $x^2$

The coefficient of $x^2$ is $-\frac{1}{4}$ . We can factor this out to get:

f(x)=-\frac{1}{4}(x^2-12x)+3

Step 2: Complete the Square

To complete the square, we need to add and subtract the square of half the coefficient of $x$ inside the parentheses:

f(x)=-\frac{1}{4}(x^2-12x+36-36)+3

Step 3: Simplify the Expression

Simplifying the expression, we get:

f(x)=-\frac{1}{4}(x-6)^2+3+9

f(x)=-\frac{1}{4}(x-6)^2+12

Step 4: Identify the Vertex

The vertex form of the parabola is $f(x)=-\frac{1}{4}(x-6)^2+12$ . The vertex is given as $(h,k)=(6,12)$ .

Conclusion

In conclusion, the vertex of a parabola is a crucial point that represents the maximum or minimum value of the function. We have discussed the significance of the vertex, how to find it using the given function, and the different methods to find the vertex of a parabola. The vertex form of a parabola is given by the equation $f(x)=a(x-h)^2+k$ , where $(h,k)$ represents the coordinates of the vertex.

Answer

The correct answer is:

C. ${$ (6,6)$}$ is incorrect
D. ${$ (6,24)$}$ is incorrect
A. ${$ (-6,-30)$}$ is incorrect
B. ${$ (-6,-12)$}$ is incorrect
The correct answer is C. ${$ (6,12)$}$
Vertex of a Parabola: Frequently Asked Questions =====================================================

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point where the parabola changes direction, either from opening upwards to downwards or vice versa. It is the highest or lowest point on the graph of the parabola.

Q: Why is the vertex important?

A: The vertex is important because it represents the maximum or minimum value of the function, depending on the direction of the parabola. It is also the point of symmetry for the parabola, meaning that the parabola is reflected about this point.

Q: How do I find the vertex of a parabola?

A: There are several methods to find the vertex of a parabola, including:

Vertex Form: If the parabola is in vertex form, the vertex is already given as $(h,k)$ .
Standard Form: If the parabola is in standard form, we can complete the square to convert it to vertex form and find the vertex.
Factored Form: If the parabola is in factored form, we can use the x-intercepts to find the vertex.

Q: What is the vertex form of a parabola?

A: The vertex form of a parabola is given by the equation $f(x)=a(x-h)^2+k$ , where $(h,k)$ represents the coordinates of the vertex.

Q: How do I complete the square to find the vertex?

A: To complete the square, we need to add and subtract the square of half the coefficient of $x$ inside the parentheses. This will allow us to rewrite the equation in vertex form.

Q: What is the significance of the axis of symmetry?

A: The axis of symmetry is the vertical line that passes through the vertex of the parabola. It is the line of symmetry for the parabola, meaning that the parabola is reflected about this line.

Q: How do I find the axis of symmetry?

A: To find the axis of symmetry, we need to find the x-coordinate of the vertex. This can be done by using the equation $x=-\frac{b}{2a}$ , where $a$ and $b$ are the coefficients of the quadratic equation.

Q: What is the relationship between the vertex and the axis of symmetry?

A: The vertex and the axis of symmetry are related in that the vertex is the point of symmetry for the parabola, and the axis of symmetry is the vertical line that passes through the vertex.

Q: Can the vertex be negative?

A: Yes, the vertex can be negative. This occurs when the parabola opens downwards and the vertex is below the x-axis.

Q: Can the vertex be a fraction?

A: Yes, the vertex can be a fraction. This occurs when the parabola opens upwards and the vertex is above the x-axis.

Q: How do I graph a parabola with a negative vertex?

A: To graph a parabola with a negative vertex, we need to use the vertex form of the equation and plot the vertex on the graph. We can then use the axis of symmetry to draw the parabola.

Q: How do I graph a parabola with a fractional vertex?

A: To graph a parabola with a fractional vertex, we need to use the vertex form of the equation and plot the vertex on the graph. We can then use the axis of symmetry to draw the parabola.