How to Find the Vertex of a Quadratic Equation

The vertex of the graph of a quadratic function is the highest or lowest possible output for that function. If you’re looking at a graph, the vertex would be the highest or lowest point on the parabola. The easiest way to find the vertex is to use the vertex formula. If your equation is in the form ax^2 + bx + c = y, you can find the x-value of the vertex by using the formula x = -b/2a. So, if you’re working with the equation 2x^2 + 4x + 9 = y, a = 2, b = 4, and c = 9. Plug the a and b values into the vertex formula to find the x value for the vertex, or the number you’d have to input into the equation to get the highest or lowest possible y. In this example, x = -4/2(2), or -1. Once you have the x value of the vertex, plug it into the original equation to find the y value. In our example, 2(-1)^2 + 4(-1) + 9 = 3. So, the x-value of the vertex is -1, and the y-value is 3. Describe the vertex by writing it down as an ordered pair in parentheses, or (-1, 3). For this particular equation, the vertex is the lowest point, since the a-value is greater than 0. Parabolas with a negative a-value open downward, so the vertex would be the highest point instead of the lowest. You can also figure out the vertex using the method of completing the square. This involves re-expressing the equation in the form of a perfect square plus a constant, then finding which x value would make the squared term equal to 0. For example, say you are trying to find the vertex of 3x^2 + 6x – 2 = y. Add 2 to both sides to get the constant out of the way. This will give you 3x^2 + 6x = y + 2. Then, factor out the coefficient of the first term to get 3(x^2 + 2x) = y + 2. Now, plug the coefficient of the b-term into the formula (b/2)^2. In this case, (2/2)^2 = 1. Multiply the result by the coefficient of the a-term and add the product to the right side of the equation. Also add the result to the inside of the parentheses on the left side. In our example, this will give you 3(x^2 + 2x + 1) = y + 2 + 3(1), which you can simplify to 3(x^2 + 2x + 1) = y + 5. Finally, factor the left side of the equation to get 3(x + 1)^2 = y + 5. Subtract 5 from both sides of the equation to get 3(x + 1)^2 – 5 = y. You can now reformat your quadratic equation into a new formula, a(x + h)^2 + k = y. To find the vertex, set x = -h so that the squared term is equal to 0, and set y = k. In this particular case, you would write 3(x + 1)^2 + (-5) = y. To make x = -h, input -1 as the x value. Then you’ll get 3(-1 + 1)^2 – 5 = y, which simplifies to 3(0) – 5 = y, or -5=y. Write the vertex as (-1, -5). If you want to learn how to find the vertex of the equation by completing the square, keep reading the article!