How to Find the Vertex of a Function Step-by-Step
Hi there, readers!
Have you ever wondered how to find the vertex of a function? It’s the turning point of the graph, and it can tell you a lot about the function’s behavior. In this article, we’ll guide you through the different methods for finding the vertex, so you can master this valuable skill. Let’s dive in!
Section 1: Vertex Formula
The vertex of a quadratic function can be found using the vertex formula:
Vertex = (-b/2a, f(-b/2a))
Where:
- a, b, and c are the coefficients of the quadratic equation in the form ax²+bx+c = 0
- f(-b/2a) is the y-value of the vertex
Example: Find the vertex of the function f(x) = x² + 4x + 3
Solution:
- a = 1, b = 4, c = 3
- Vertex = (-4/2, f(-4/2)) = (-2, f(-2))
- f(-2) = (-2)² + 4(-2) + 3 = -1
- Therefore, the vertex of the function is (-2, -1)
Section 2: Completing the Square
Completing the square is another method for finding the vertex. It involves manipulating the quadratic equation into the form:
(x + h)² + k = 0
Where:
- h = -b/2a
- k = c – b²/4a
Example: Find the vertex of the function f(x) = x² + 4x + 3 using completing the square
Solution:
- h = -4/2 * 1 = -2
- k = 3 – 4²/4 * 1 = -1
- Therefore, the vertex of the function is (-2, -1)
Section 3: Graphing the Function
If you can graph the function, you can find the vertex by identifying the turning point. The vertex will be the point where the graph changes direction.
Example: Find the vertex of the function f(x) = x² + 4x + 3 by graphing
Solution:
- Graph the function f(x) = x² + 4x + 3
- The graph is a parabola that opens upwards
- The vertex is the turning point of the parabola
- Therefore, the vertex of the function is (-2, -1)
Section 4: Summary Table of Methods
| Method | Formula | How to use | Example |
|---|---|---|---|
| Vertex Formula | Vertex = (-b/2a, f(-b/2a)) | Plug in the coefficients of the quadratic equation | f(x) = x² + 4x + 3, Vertex = (-2, -1) |
| Completing the Square | (x + h)² + k = 0 | Convert the quadratic equation into the form (x + h)² + k | f(x) = x² + 4x + 3, Vertex = (-2, -1) |
| Graphing | Identify the turning point of the graph | Plot the graph of the quadratic equation | f(x) = x² + 4x + 3, Vertex = (-2, -1) |
Section 5: Conclusion
Finding the vertex of a function is a fundamental skill in algebra. By understanding the vertex formula, completing the square, and graphing functions, you’ll be equipped to tackle this task with confidence. Don’t forget to explore our other articles on quadratic functions to deepen your knowledge.
FAQ about Finding the Vertex
What is the vertex of a parabola?
The vertex is the highest or lowest point on a parabola.
How do I find the x-coordinate of the vertex?
The x-coordinate of the vertex is: x = -b / 2a where a and b are the coefficients of the quadratic equation in the form ax^2 + bx + c.
How do I find the y-coordinate of the vertex?
The y-coordinate of the vertex can be found by plugging the x-coordinate into the original quadratic equation.
What if the parabola opens downward?
The vertex will be the lowest point on the parabola. Find the x-coordinate using the same formula and plug it in to find the y-coordinate.
How do I find the vertex if the equation is not in vertex form?
You can complete the square or use the quadratic formula to rewrite the equation in vertex form.
What is the axis of symmetry?
The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two symmetrical halves.
What is the equation of the axis of symmetry?
The equation of the axis of symmetry is x = -b / 2a.
How do I find the domain and range of the parabola based on the vertex?
The domain of the parabola is all real numbers, and the range depends on whether the parabola opens up or down. If it opens up, the range is [y-coord of vertex, ∞). If it opens down, the range is (-∞, y-coord of vertex].
Can there be more than one vertex in a parabola?
No, a parabola has only one vertex.
What is the significance of the vertex in graphing a parabola?
The vertex helps determine the shape, direction, and position of the parabola on a graph.
