Writing the Quadratic Function in Vertex Form

Oct 13, 2025 by miodeya.com 79 views

Understanding the Basics of Quadratic Functions

Quadratic functions are a type of polynomial function that can be written in the form of ${f(x) = ax^2 + bx + c}$ , where ${a}$ , ${b}$ , and ${c}$ are constants. The vertex of a quadratic function is the highest or lowest point on the graph of the function, and it is represented by the coordinates ${(h, k)}$ . The y-intercept of a quadratic function is the point where the graph of the function intersects the y-axis, and it is represented by the coordinates ${(0, c)}$ .

Given Information

In this problem, we are given that the vertex of the quadratic function is located at ${(1, 4)}$ , and the y-intercept of the function is located at ${(0, 1)}$ . This means that the coordinates of the vertex are ${(h, k) = (1, 4)}$ , and the coordinates of the y-intercept are ${(0, c) = (0, 1)}$ .

Writing the Quadratic Function in Vertex Form

The vertex form of a quadratic function is written as ${f(x) = a(x - h)^2 + k}$ , where ${(h, k)}$ is the vertex of the function. Since we are given that the vertex is located at ${(1, 4)}$ , we can substitute these values into the vertex form of the function to get:

${f(x) = a(x - 1)^2 + 4}$

Finding the Value of a

We are also given that the y-intercept of the function is located at ${(0, 1)}$ . This means that when ${x = 0}$ , the value of the function is ${f(0) = 1}$ . We can substitute these values into the equation above to get:

${1 = a(0 - 1)^2 + 4}$

Simplifying the equation, we get:

${1 = a(-1)^2 + 4}$

${1 = a + 4}$

Subtracting 4 from both sides of the equation, we get:

${-3 = a}$

Therefore, the value of ${a}$ is ${-3}$ .

Conclusion

In this problem, we were given that the vertex of a quadratic function is located at ${(1, 4)}$ , and the y-intercept of the function is located at ${(0, 1)}$ . We used this information to write the quadratic function in vertex form and then found the value of ${a}$ by substituting the coordinates of the y-intercept into the equation. The value of ${a}$ is ${-3}$ .

Example Use Case

This problem can be used as an example of how to find the value of ${a}$ in a quadratic function when the vertex and y-intercept are given. It can also be used to demonstrate how to write a quadratic function in vertex form and how to substitute values into the equation to solve for unknown variables.

Key Takeaways

The vertex form of a quadratic function is written as ${f(x) = a(x - h)^2 + k}$ , where ${(h, k)}$ is the vertex of the function.
The y-intercept of a quadratic function is the point where the graph of the function intersects the y-axis, and it is represented by the coordinates ${(0, c)}$ .
The value of ${a}$ can be found by substituting the coordinates of the y-intercept into the equation and solving for ${a}$ .

Q: What is a quadratic function?

A: A quadratic function is a type of polynomial function that can be written in the form of ${f(x) = ax^2 + bx + c}$ , where ${a}$ , ${b}$ , and ${c}$ are constants.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the highest or lowest point on the graph of the function, and it is represented by the coordinates ${(h, k)}$ .

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

A: To find the vertex of a quadratic function, you can use the formula ${h = -\frac{b}{2a}}$ and ${k = f(h)}$ .

Q: What is the y-intercept of a quadratic function?

A: The y-intercept of a quadratic function is the point where the graph of the function intersects the y-axis, and it is represented by the coordinates ${(0, c)}$ .

Q: How do I find the y-intercept of a quadratic function?

A: To find the y-intercept of a quadratic function, you can substitute ${x = 0}$ into the equation and solve for ${y}$ .

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

A: The vertex form of a quadratic function is written as ${f(x) = a(x - h)^2 + k}$ , where ${(h, k)}$ is the vertex of the function.

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

A: To write a quadratic function in vertex form, you can use the formula ${f(x) = a(x - h)^2 + k}$ , where ${(h, k)}$ is the vertex of the function.

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

A: To find the value of ${a}$ in a quadratic function, you can use the formula ${a = \frac{f(x) - k}{(x - h)^2}}$ , where ${(h, k)}$ is the vertex of the function.

Q: What is the significance of the value of a in a quadratic function?

A: The value of ${a}$ in a quadratic function determines the direction and width of the parabola. If ${a > 0}$ , the parabola opens upward. If ${a < 0}$ , the parabola opens downward.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you can use the following steps:

Find the vertex of the function.
Find the y-intercept of the function.
Plot the vertex and y-intercept on a coordinate plane.
Use the vertex form of the function to find the equation of the parabola.
Plot the parabola on the coordinate plane.

Q: What are some common applications of quadratic functions?

A: Quadratic functions have many real-world applications, including:

Modeling the trajectory of a projectile.
Finding the maximum or minimum value of a function.
Solving problems involving optimization.
Modeling population growth or decline.
Solving problems involving physics and engineering.

Q: What are some common mistakes to avoid when working with quadratic functions?

A: Some common mistakes to avoid when working with quadratic functions include:

Not using the correct formula for the vertex or y-intercept.
Not checking the domain and range of the function.
Not using the correct method for graphing the function.
Not checking for extraneous solutions.
Not using the correct method for solving systems of equations.

Conclusion

Quadratic functions are an important topic in mathematics, and they have many real-world applications. By understanding the basics of quadratic functions, including the vertex and y-intercept, you can solve problems involving optimization, physics, and engineering. Remember to use the correct formula for the vertex and y-intercept, and to check the domain and range of the function. With practice and patience, you can become proficient in working with quadratic functions.