Content

• Steps
• Method 1 of 2: Formula for Finding the Vertex
• Method 2 of 2: Completing the Square
• Tips
• Warnings
• What do you need

The vertex of a quadratic parabola is its highest or lowest point. To find the vertex of a parabola, you can use a special formula or the square's complement method. How to do this is described below.

Steps

Method 1 of 2: Formula for Finding the Vertex

1. 1 Find the quantities a, b, and c. In a quadratic equation, the coefficient at x = a, at x = b, constant (coefficient without variable) = c. For example, take the equation: y = x + 9x + 18. Here a = 1, b = 9, and c = 18.
2. 2 Use the formula to calculate the value for the x-coordinate of the vertex. The vertex is also the point of symmetry of the parabola. Formula for finding the x coordinate of a parabola: x = -b / 2a. Plug in the appropriate values ​​to calculate x.
   • x = -b / 2a
   • x = - (9) / (2) (1)
   • x = -9 / 2
3. 3 Plug the x-value you find into the original equation to calculate the y-value. Now that you know the value of x, simply plug it into the original equation to find y. Thus, the formula for finding the vertex of a parabola can be written as a function: (x, y) = [(-b / 2a), f (-b / 2a)]... This means that to find y, you must first find x using the formula, and then plug the value of x into the original equation. Here's how it's done:
   • y = x + 9x + 18
   • y = (-9/2) + 9 (-9/2) +18
   • y = 81/4 -81/2 + 18
   • y = 81/4 -162/4 + 72/4
   • y = (81 - 162 + 72) / 4
   • y = -9/4
4. 4 Write the x and y values ​​as a pair of coordinates. Now that you know that x = -9/2 and y = -9/4, write them down as coordinates in the form: (-9/2, -9/4). The vertex of the parabola is located at the coordinates (-9/2, -9/4). If you need to draw this parabola, then its vertex lies at the lowest point, since the coefficient of x is positive.

Method 2 of 2: Completing the Square

1. 1 Write down the equation. Complementing the square is another way to find the vertex of a parabola. By applying this method, you will find the x and y coordinates at once, without having to substitute x in the original equation. For example, given the equation: x + 4x + 1 = 0.
2. 2 Divide each coefficient by the coefficient at x. In our case, the coefficient at x is 1, so we can skip this step. Division by 1 will not change anything.
3. 3 Move the constant to the right side of the equation. Constant - coefficient without variable. Here it is 1... Move 1 to the right by subtracting 1 from both sides of the equation. Here's how to do it:
   • x + 4x + 1 = 0
   • x + 4x + 1 -1 = 0 - 1
   • x + 4x = - 1
4. 4 Complete the left side of the equation to a full square. To do this, just find (b / 2) and add the result to both sides of the equation. Substitute 4 instead of b, as 4x is the coefficient b of our equation.
   • (4/2) = 2 = 4. Now add 4 to both sides of the equation to get:
     • x + 4x + 4 = -1 + 4
     • x + 4x + 4 = 3
5. 5 Simplifying the left side of the equation. We see that x + 4x + 4 is a complete square. It can be written as: (x + 2) = 3
6. 6 Use it to find the x and y coordinates. You can find x by simply setting (x + 2) to 0. Now that (x + 2) = 0, calculate x: x = -2. The y coordinate is the constant on the right side of a complete square. So, y = 3. The vertex of the parabola of the equation x + 4x + 1 = (-2, 3)

Tips

• Define a, b, and c correctly.
• Record preliminary calculations. This will not only help in the process of work, but will also allow you to see where mistakes were made.
• Do not disturb the order of calculations.

Warnings

• Check your answer!
• Make sure you know how to determine the coefficients of a, b, and c. If you don't know, the answer will be wrong.
• Don't panic - solving such problems takes practice.

What do you need

• Paper or computer
• Calculator