How to Write a Quadratic Equation in Vertex Form
Quick Answer
To write a quadratic equation in vertex form, use the format y = a(x - h)² + k. Complete the square if necessary, or identify perfect square trinomials directly.
Writing a quadratic equation in vertex form can help you understand its graph better, as it clearly shows the vertex of the parabola. The vertex form of a quadratic equation is expressed as y = a(x - h)² + k, where (h, k) represents the vertex of the parabola. This form is particularly useful because it makes it easy to identify the maximum or minimum point of the parabola, which is a key feature in graphing.
To convert the standard form of a quadratic equation, which is typically in the format y = ax² + bx + c, into vertex form, you may need to complete the square. Let's look at the example equation: y = x² + 10x + 25.
First, notice that this equation can actually be rewritten as a perfect square trinomial. A perfect square trinomial takes the form (x + p)², where p is a number that, when squared, gives you the constant term at the end of the trinomial. In this case, we identify the components of the trinomial:
- The x² term is straightforward.
- The 10x term can be thought of as 2 times a number (which we will find).
- The constant term is 25.
To find the number that completes the square, we can ask: "What number squared equals 25?" The answer is 5, since 5² = 25. Now, we check the linear term: 2 times this number should equal 10, and indeed, 2 × 5 = 10.
Therefore, we can rewrite the original equation as follows:
- y = x² + 10x + 25
- y = (x + 5)².
Now we have written the quadratic equation in vertex form. The vertex, which is the point (h, k), is (−5, 0) in this case. Knowing the vertex allows you to sketch the graph of the parabola accurately. Additionally, understanding the vertex form helps in analyzing the behavior of the quadratic function, such as determining whether it opens upwards or downwards (based on the value of 'a').
In conclusion, converting quadratic equations to vertex form not only simplifies the graphing process but also enhances your understanding of the function’s key characteristics. So, the next time you encounter a quadratic equation, look for opportunities to express it in vertex form. It will make your math experience much clearer and more enjoyable!
Was this answer helpful?