Finding a Quadratic Function Given the Vertex and a Point
In the realm of algebra, particularly in the study of quadratic functions, it's often necessary to find the function's equation given specific points. One common scenario involves finding the quadratic function when you know the vertex and another point on the parabola. This article guides you through this process, making it easy to understand and apply.
Understanding Quadratic Functions and Their Vertex Form
Quadratic functions are represented in the vertex form as f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. Here, 'a' is the coefficient that determines the direction and width of the parabola, 'h' is the x-coordinate of the vertex, and 'k' is the y-coordinate of the vertex.
Given the Vertex: Finding the Quadratic Function
When you're given the vertex (h, k), you can easily write the quadratic function in vertex form. For instance, if the vertex is at (3, -2), the function in vertex form would be:
f(x) = a(x - 3)² - 2
Now, you need to find the value of 'a' to complete the function. This is where the given point comes into play.
Using a Point to Determine the Value of 'a'
Let's say you're given the point (x₁, y₁) on the parabola. You can substitute these values into the vertex form equation to find 'a'. Here's how:
y₁ = a(x₁ - h)² + k
Plug in the given vertex (h, k) and the point (x₁, y₁):
y₁ = a(x₁ - h)² + k
Now, solve for 'a':
a = (y₁ - k) / (x₁ - h)²
Example: Finding the Quadratic Function
Let's find the quadratic function given the vertex (2, 1) and the point (4, 5).
1. Write the function in vertex form: f(x) = a(x - 2)² + 1
2. Plug in the point (4, 5): 5 = a(4 - 2)² + 1
3. Solve for 'a': a = (5 - 1) / (4 - 2)² = 4 / 4 = 1
4. Write the complete quadratic function: f(x) = (x - 2)² + 1
Practical Application: A Table of Common Quadratic Functions
Here's a table of common quadratic functions, their vertex forms, and the given points used to find them:
| Quadratic Function | Vertex Form | Given Point (x₁, y₁) |
|---|---|---|
| f(x) = (x - 3)² + 2 | f(x) = a(x - 3)² + 2 | (1, 4) |
| f(x) = -(x + 2)² + 3 | f(x) = a(x + 2)² - 3 | (-4, 11) |
