How to Calculate the Vertex Form of a Quadratic Function

Oct 13, 2025 by miodeya.com 112 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 $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants. One of the important properties of a quadratic function is its vertex, which is the maximum or minimum point of the function. In this article, we will learn how to calculate the vertex form of a quadratic function.

What is the Vertex Form of a Quadratic Function?

The vertex form of a quadratic function is a way of expressing the function in the form $f(x) = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the function. The vertex form is useful because it allows us to easily identify the vertex of the function, which is the point where the function reaches its maximum or minimum value.

How to Calculate the Vertex Form of a Quadratic Function

To calculate the vertex form of a quadratic function, we need to complete the square. This involves rewriting the function in the form $f(x) = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the function.

Step 1: Identify the Coefficients of the Quadratic Function

The first step in calculating the vertex form of a quadratic function is to identify the coefficients of the function. In the function $f(x) = x^2 - x + 2$ , the coefficients are $a = 1$ , $b = -1$ , and $c = 2$ .

Step 2: Calculate the Value of $h$

The value of $h$ is calculated using the formula $h = -\frac{b}{2a}$ . Plugging in the values of $a$ and $b$ , we get $h = -\frac{-1}{2(1)} = \frac{1}{2}$ .

Step 3: Calculate the Value of $k$

The value of $k$ is calculated using the formula $k = f(h)$ . Plugging in the value of $h$ , we get $k = f\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^2 - \left(\frac{1}{2}\right) + 2 = \frac{1}{4} - \frac{1}{2} + 2 = \frac{7}{4}$ .

Step 4: Write the Vertex Form of the Quadratic Function

Now that we have calculated the values of $h$ and $k$ , we can write the vertex form of the quadratic function. The vertex form is $f(x) = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the function. Plugging in the values of $a$ , $h$ , and $k$ , we get $f(x) = 1\left(x - \frac{1}{2}\right)^2 + \frac{7}{4} = \left(x - \frac{1}{2}\right)^2 + \frac{7}{4}$ .

Conclusion

In this article, we learned how to calculate the vertex form of a quadratic function. We started by identifying the coefficients of the function, then calculated the value of $h$ using the formula $h = -\frac{b}{2a}$ . We then calculated the value of $k$ using the formula $k = f(h)$ . Finally, we wrote the vertex form of the quadratic function using the values of $a$ , $h$ , and $k$ . The vertex form of the quadratic function is $f(x) = \left(x - \frac{1}{2}\right)^2 + \frac{7}{4}$ .

Example

Let's consider another example. Suppose we have the quadratic function $f(x) = x^2 + 4x + 3$ . We can calculate the vertex form of this function using the same steps as before.

Step 1: Identify the Coefficients of the Quadratic Function

The coefficients of the function are $a = 1$ , $b = 4$ , and $c = 3$ .

Step 2: Calculate the Value of $h$

The value of $h$ is calculated using the formula $h = -\frac{b}{2a}$ . Plugging in the values of $a$ and $b$ , we get $h = -\frac{4}{2(1)} = -2$ .

Step 3: Calculate the Value of $k$

The value of $k$ is calculated using the formula $k = f(h)$ . Plugging in the value of $h$ , we get $k = f(-2) = (-2)^2 + 4(-2) + 3 = 4 - 8 + 3 = -1$ .

Step 4: Write the Vertex Form of the Quadratic Function

Discussion

The vertex form of a quadratic function is a useful way of expressing the function in a way that makes it easy to identify the vertex of the function. The vertex form is particularly useful in calculus, where it is used to find the maximum or minimum value of a function.

Conclusion

References

[1] "Quadratic Functions" by Math Open Reference
[2] "Vertex Form of a Quadratic Function" by Purplemath
[3] "Completing the Square" by Khan Academy

Final Answer

Introduction

In our previous article, we learned how to calculate the vertex form of a quadratic function. In this article, we will answer some frequently asked questions about calculating the vertex form of a quadratic function.

Q: What is the vertex form of a quadratic function?

A: The vertex form of a quadratic function is a way of expressing the function in the form $f(x) = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the function.

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

A: To calculate the vertex form of a quadratic function, you need to complete the square. This involves rewriting the function in the form $f(x) = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the function.

Q: What is the formula for calculating the value of $h$ ?

A: The formula for calculating the value of $h$ is $h = -\frac{b}{2a}$ .

Q: What is the formula for calculating the value of $k$ ?

A: The formula for calculating the value of $k$ is $k = f(h)$ .

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

A: To write the vertex form of a quadratic function, you need to plug in the values of $a$ , $h$ , and $k$ into the formula $f(x) = a(x - h)^2 + k$ .

Q: What is the significance of the vertex form of a quadratic function?

A: The vertex form of a quadratic function is useful because it allows us to easily identify the vertex of the function, which is the point where the function reaches its maximum or minimum value.

Q: Can I use the vertex form of a quadratic function to find the maximum or minimum value of a function?

A: Yes, you can use the vertex form of a quadratic function to find the maximum or minimum value of a function. The vertex form of a quadratic function is particularly useful in calculus, where it is used to find the maximum or minimum value of a function.

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

A: To determine whether a quadratic function is a maximum or minimum, you need to look at the value of $k$ . If $k$ is positive, the function is a minimum. If $k$ is negative, the function is a maximum.

Q: Can I use the vertex form of a quadratic function to graph a quadratic function?

A: Yes, you can use the vertex form of a quadratic function to graph a quadratic function. The vertex form of a quadratic function is particularly useful in graphing a quadratic function, as it allows us to easily identify the vertex of the function.

Q: How do I graph a quadratic function using the vertex form?

A: To graph a quadratic function using the vertex form, you need to plot the vertex of the function and then use the formula $f(x) = a(x - h)^2 + k$ to find the values of the function at different points.

Conclusion

In this article, we answered some frequently asked questions about calculating the vertex form of a quadratic function. We hope that this article has been helpful in understanding the concept of the vertex form of a quadratic function.

References

[1] "Quadratic Functions" by Math Open Reference
[2] "Vertex Form of a Quadratic Function" by Purplemath
[3] "Completing the Square" by Khan Academy

Final Answer

The final answer is $\boxed{\left(\frac{1}{2}, \frac{7}{4}\right)}$ .