How to Find the Vertex of a Quadratic Function

Understanding the Definition of a Vertex

In mathematics, a vertex is the highest or lowest point on the graph of a quadratic function. It is also known as the maximum or minimum point, depending on the direction of the parabola. The vertex is a critical point on the graph, and it provides essential information about the behavior of the function.

Geometrically, the vertex represents the point where the parabola changes its direction, either from going upwards to downwards or from going downwards to upwards. Algebraically, the vertex is the point where the derivative of the function is equal to zero.

It is essential to understand the concept of the vertex because it helps in analyzing the behavior of a quadratic function. The vertex provides information about the maximum or minimum value of the function, the direction of the parabola, and the axis of symmetry. By finding the vertex, one can also determine the domain and range of the function and solve real-life problems related to quadratic equations.

Using the Formula to Find the Vertex

One way to find the vertex of a quadratic function is by using the formula. The formula for finding the vertex of a quadratic function in standard form is as follows:

Vertex = (-b/2a, f(-b/2a))

Where ‘a’, ‘b’, and ‘c’ are the coefficients of the quadratic function in the form of ax² + bx + c. ‘f’ represents the value of the function at the given x-coordinate.

To use the formula, one needs to substitute the values of ‘a’, ‘b’, and ‘c’ into the formula and simplify the expression. The x-coordinate of the vertex can be found by dividing -b by 2a. Once the x-coordinate is known, the y-coordinate can be found by substituting the x-coordinate into the function and solving for ‘f’.

For example, consider the quadratic function f(x) = 2x² + 4x – 3. To find the vertex, one can use the formula as follows:

Vertex = (-b/2a, f(-b/2a))
Vertex = (-4/(22), f(-4/(22)))
Vertex = (-1, f(-1))

To find the value of f(-1), substitute x = -1 into the function f(x):
f(-1) = 2(-1)² + 4(-1) – 3
f(-1) = -1

Therefore, the vertex of the quadratic function f(x) = 2x² + 4x – 3 is (-1,-1).

Applying Completing the Square Method

Another method to find the vertex of a quadratic function is by using the completing the square method. The completing the square method involves transforming a quadratic function from standard form into vertex form. Vertex form is a more convenient form for finding the vertex because the x-coordinate of the vertex is simply the opposite of the constant term, and the y-coordinate is the value of the function at the vertex.

To use the completing the square method, follow these steps:

1. Rewrite the quadratic function in the form of ax² + bx + c.
2. Complete the square by adding and subtracting (b/2a)² inside the parentheses.
3. Rewrite the expression as a perfect square and simplify the constant term.
4. Write the function in vertex form by factoring the perfect square expression.
5. Identify the vertex as (-b/2a, c – (b/2a)²).

For example, consider the quadratic function f(x) = x² + 6x – 1. To find the vertex using the completing the square method, follow these steps:

1. Rewrite the function in standard form: f(x) = x² + 6x – 1.
2. Complete the square: f(x) = (x² + 6x + 9) – 10.
3. Rewrite the expression as a perfect square and simplify the constant term: f(x) = (x + 3)² – 10.
4. Write the function in vertex form: f(x) = (x + 3)² – 10.
5. Identify the vertex as (-3, -10).

Therefore, the vertex of the quadratic function f(x) = x² + 6x – 1 is (-3, -10).

Graphical Method to Locate the Vertex

Another method to locate the vertex of a quadratic function is by using the graphical method. This method involves graphing the quadratic function and finding the coordinates of the vertex by observing the properties of the graph.

To use the graphical method, follow these steps:

1. Graph the quadratic function using a graphing calculator or software.
2. Observe the direction of the parabola. If the coefficient of x² is positive, the parabola opens upwards, and the vertex is the lowest point on the graph. If the coefficient of x² is negative, the parabola opens downwards, and the vertex is the highest point on the graph.
3. Locate the axis of symmetry. The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two symmetric halves. The equation of the axis of symmetry is x = -b/2a, where ‘a’ and ‘b’ are the coefficients of the quadratic function.
4. Find the x-coordinate of the vertex. The x-coordinate of the vertex is the same as the x-coordinate of the axis of symmetry.
5. Find the y-coordinate of the vertex. Substitute the x-coordinate of the vertex into the quadratic function and solve for ‘y’.

For example, consider the quadratic function f(x) = -2x² + 8x + 3. To locate the vertex using the graphical method, follow these steps:

1. Graph the function using a graphing calculator or software. The graph shows that the parabola opens downwards, and the axis of symmetry is x = 2.
2. The x-coordinate of the vertex is 2, which is the same as the x-coordinate of the axis of symmetry.
3. Substitute x = 2 into the quadratic function: f(2) = -2(2)² + 8(2) + 3 = 7.
4. Therefore, the vertex of the quadratic function f(x) = -2x² + 8x + 3 is (2, 7).

Using the graphical method is an excellent way to verify the results obtained from other methods and can also provide insight into the behavior of the quadratic function.

Real-life Applications of Finding the Vertex

Finding the vertex of a quadratic function has several real-life applications, especially in fields such as physics, engineering, and economics. The vertex provides information about the maximum or minimum value of the function, which is essential in optimizing certain situations.

One example of a real-life application of finding the vertex is in projectile motion problems. In these problems, a projectile is launched into the air, and its trajectory can be modeled by a quadratic function. The vertex of the function represents the maximum height of the projectile, and the x-coordinate of the vertex represents the time it takes for the projectile to reach its maximum height.

Another example is in manufacturing and production processes. Quadratic functions can be used to model the cost or revenue of producing a certain number of items. The vertex of the function represents the optimal point at which the cost is minimized or the revenue is maximized.

In the field of economics, quadratic functions can be used to model supply and demand curves. The vertex of the function represents the point at which the price and quantity of a good or service are at equilibrium.

In summary, finding the vertex of a quadratic function is a crucial step in solving various real-life problems. The vertex provides information about the maximum or minimum value of the function, which is essential in optimizing certain situations in physics, engineering, economics, and other fields.