Finding the Vertex of the Given Quadratic Function

Oct 13, 2025 by thesweetdesigns.com 85 views

Introduction

In mathematics, a quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is given by $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants. The vertex of a quadratic function is the maximum or minimum point on the graph of the function. In this article, we will discuss how to find the vertex of a quadratic function, using the given quadratic function $g(x) = x^2 - 6x + 14$ as an example.

What is the Vertex of a Quadratic Function?

The vertex of a quadratic function is the point on the graph where the function changes from decreasing to increasing or vice versa. It is also the minimum or maximum point on the graph, depending on the direction of the parabola. The vertex can be found using the formula $x = -\frac{b}{2a}$ , where $a$ and $b$ are the coefficients of the quadratic function.

Finding the Vertex of the Given Quadratic Function

To find the vertex of the given quadratic function $g(x) = x^2 - 6x + 14$ , we need to identify the values of $a$ and $b$ . In this case, $a = 1$ and $b = -6$ . Now, we can use the formula $x = -\frac{b}{2a}$ to find the $x$ -coordinate of the vertex.

Calculating the x-coordinate of the vertex

$x = -\frac{b}{2a}$ $x = -\frac{-6}{2(1)}$ $x = -\frac{-6}{2}$ $x = 3$

Calculating the y-coordinate of the vertex

Now that we have the $x$ -coordinate of the vertex, we can find the $y$ -coordinate by plugging the value of $x$ into the quadratic function.

$y = g(x)$ $y = (3)^2 - 6(3) + 14$ $y = 9 - 18 + 14$ $y = 5$

Conclusion

In this article, we discussed how to find the vertex of a quadratic function using the formula $x = -\frac{b}{2a}$ . We used the given quadratic function $g(x) = x^2 - 6x + 14$ as an example and found the vertex to be at the point $(3, 5)$ . The vertex of a quadratic function is an important concept in mathematics, as it represents the maximum or minimum point on the graph of the function.

Applications of the Vertex of a Quadratic Function

The vertex of a quadratic function has many real-world applications. For example, in physics, the vertex of a quadratic function can represent the maximum or minimum height of an object in motion. In economics, the vertex of a quadratic function can represent the maximum or minimum profit of a business. In engineering, the vertex of a quadratic function can represent the maximum or minimum stress on a material.

Examples of Quadratic Functions with Different Vertices

Here are a few examples of quadratic functions with different vertices:

$f(x) = x^2 + 4x + 4$ has a vertex at $(x, y) = (-2, 0)$
$g(x) = x^2 - 2x + 1$ has a vertex at $(x, y) = (1, 0)$
$h(x) = x^2 + 3x + 2$ has a vertex at $(x, y) = (-1, 2)$

Tips for Finding the Vertex of a Quadratic Function

Here are a few tips for finding the vertex of a quadratic function:

Make sure to identify the values of $a$ and $b$ in the quadratic function.
Use the formula $x = -\frac{b}{2a}$ to find the $x$ -coordinate of the vertex.
Plug the value of $x$ into the quadratic function to find the $y$ -coordinate of the vertex.
Check your work by plugging the values of $x$ and $y$ back into the quadratic function to make sure they satisfy the equation.

Conclusion

In conclusion, the vertex of a quadratic function is an important concept in mathematics that represents the maximum or minimum point on the graph of the function. We discussed how to find the vertex of a quadratic function using the formula $x = -\frac{b}{2a}$ and used the given quadratic function $g(x) = x^2 - 6x + 14$ as an example. We also discussed the applications of the vertex of a quadratic function and provided a few examples of quadratic functions with different vertices. Finally, we provided a few tips for finding the vertex of a quadratic function.

Introduction

In our previous article, we discussed how to find the vertex of a quadratic function using the formula $x = -\frac{b}{2a}$ . We also provided a few examples of quadratic functions with different vertices and discussed the applications of the vertex of a quadratic function. In this article, we will answer some frequently asked questions (FAQs) about the vertex of a quadratic function.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the maximum or minimum point on the graph of the function. It is also the point where the function changes from decreasing to increasing or vice versa.

Q: How do I find the vertex of a quadratic function?

A: To find the vertex of a quadratic function, you need to identify the values of $a$ and $b$ in the function. Then, use the formula $x = -\frac{b}{2a}$ to find the $x$ -coordinate of the vertex. Finally, plug the value of $x$ into the function to find the $y$ -coordinate of the vertex.

Q: What is the formula for finding the vertex of a quadratic function?

A: The formula for finding the vertex of a quadratic function is $x = -\frac{b}{2a}$ , where $a$ and $b$ are the coefficients of the function.

Q: Can the vertex of a quadratic function be a maximum or a minimum?

A: Yes, the vertex of a quadratic function can be either a maximum or a minimum point on the graph of the function. This depends on the direction of the parabola.

Q: How do I determine whether the vertex of a quadratic function is a maximum or a minimum?

A: To determine whether the vertex of a quadratic function is a maximum or a minimum, you need to look at the direction of the parabola. If the parabola opens upwards, the vertex is a minimum point. If the parabola opens downwards, the vertex is a maximum point.

Q: Can the vertex of a quadratic function be a point of inflection?

A: No, the vertex of a quadratic function cannot be a point of inflection. A point of inflection is a point on the graph of a function where the concavity changes.

Q: How do I find the point of inflection of a quadratic function?

A: To find the point of inflection of a quadratic function, you need to find the second derivative of the function and set it equal to zero. Then, solve for $x$ to find the $x$ -coordinate of the point of inflection.

Q: Can the vertex of a quadratic function be a point of tangency?

A: No, the vertex of a quadratic function cannot be a point of tangency. A point of tangency is a point on the graph of a function where the function touches a line.

Q: How do I find the point of tangency of a quadratic function?

A: To find the point of tangency of a quadratic function, you need to find the equation of the line that is tangent to the function at a given point. Then, solve for the point of tangency.

Q: Can the vertex of a quadratic function be a point of discontinuity?

A: No, the vertex of a quadratic function cannot be a point of discontinuity. A point of discontinuity is a point on the graph of a function where the function is not defined.

Q: How do I find the point of discontinuity of a quadratic function?

A: To find the point of discontinuity of a quadratic function, you need to look for any points on the graph of the function where the function is not defined.

Conclusion

In this article, we answered some frequently asked questions (FAQs) about the vertex of a quadratic function. We discussed the definition of the vertex of a quadratic function, how to find the vertex, and some common misconceptions about the vertex. We also provided some tips for finding the vertex of a quadratic function and discussed some real-world applications of the vertex of a quadratic function.