Quadratic equations are a fundamental part of algebra and are used to solve a wide range of problems in mathematics, physics, and engineering. One of the most important aspects of a quadratic equation is its turning point, which is the point where the curve changes direction from up to down or down to up. Finding the turning point of a quadratic equation is a crucial skill that can help you solve complex problems in various fields. In this article, we will discuss the steps to find the turning point of a quadratic equation.

What is a Quadratic Equation?

A quadratic equation is a second-degree polynomial equation that contains a variable raised to the power of two. The standard form of a quadratic equation is ax?? + bx + c = 0, where a, b, and c are constants, and x is the variable. The coefficient 'a' is the most crucial factor in a quadratic equation as it determines whether the curve opens up or down.

What is the Turning Point of a Quadratic Equation?

The turning point of a quadratic equation is the point where the curve changes direction from up to down or down to up. It is also known as the vertex of the parabola. The turning point is the minimum or maximum point of the quadratic equation, depending on whether the curve opens up or down, respectively. In other words, the turning point is the point where the quadratic equation has its highest or lowest value.

Steps to Find the Turning Point of a Quadratic Equation

Here are the steps to find the turning point of a quadratic equation:

Step 1: Identify the Coefficients

The first step in finding the turning point of a quadratic equation is to identify the coefficients a, b, and c. The coefficient 'a' is the most crucial factor in determining the direction of the curve. If 'a' is positive, the curve opens up, and if 'a' is negative, the curve opens down.

Step 2: Find the x-Coordinate of the Turning Point

The x-coordinate of the turning point can be found using the formula x = -b/2a. This formula gives the axis of symmetry of the parabola, which is the vertical line that divides the parabola into two equal halves.

Step 3: Find the y-Coordinate of the Turning Point

Once you have found the x-coordinate of the turning point, you can find the y-coordinate using the quadratic equation. Substituting the x-coordinate into the equation will give you the y-coordinate. The y-coordinate is the minimum or maximum point of the parabola.

Example of Finding the Turning Point of a Quadratic Equation

Let's take an example to understand how to find the turning point of a quadratic equation:

Find the turning point of the quadratic equation y = 2x?? - 4x + 3

Solution:

Step 1: Identify the Coefficients

a = 2, b = -4, c = 3

Step 2: Find the x-Coordinate of the Turning Point

x = -b/2a = -(-4)/2(2) = 1

Step 3: Find the y-Coordinate of the Turning Point

y = 2(1)?? - 4(1) + 3 = 1

Therefore, the turning point of the quadratic equation y = 2x?? - 4x + 3 is (1, 1).

Conclusion

Finding the turning point of a quadratic equation is a crucial skill that can help you solve complex problems in various fields. By following the steps mentioned above, you can easily find the turning point of any quadratic equation. It is essential to understand the coefficients of a quadratic equation, as they determine the direction of the curve. The turning point is the minimum or maximum point of the quadratic equation, depending on whether the curve opens up or down, respectively.

Related video of How to Find the Turning Point of a Quadratic Equation