Finding Vertex And Zeros Of Quadratic Function F(x) = X² - 6x + 5

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In the realm of mathematics, quadratic functions hold a significant position due to their frequent appearance in various applications, ranging from physics and engineering to economics and computer science. Understanding the properties of these functions, such as their vertex and zeros, is crucial for solving related problems and gaining deeper insights into their behavior. In this article, we will delve into a step-by-step guide on how to find the vertex and zeros of the quadratic function f(x) = x² - 6x + 5. This detailed exploration will equip you with the knowledge and skills to analyze and interpret quadratic functions effectively.

Understanding Quadratic Functions

Before we embark on the journey of finding the vertex and zeros, it's essential to establish a solid understanding of what quadratic functions are. A quadratic function is a polynomial function of degree two, meaning the highest power of the variable is two. The general form of a quadratic function is expressed as:

f(x) = ax² + bx + c

where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The graph of a quadratic function is a parabola, a symmetrical U-shaped curve. The vertex of the parabola is the point where the curve changes direction, and it represents either the minimum or maximum value of the function. The zeros of the quadratic function, also known as the roots or x-intercepts, are the points where the parabola intersects the x-axis. These points correspond to the values of x for which f(x) = 0.

Finding the Vertex

The vertex of a quadratic function is a critical point that provides valuable information about the function's behavior. It represents the point where the parabola reaches its minimum or maximum value. There are two primary methods for finding the vertex: the completing the square method and the vertex formula method. Let's explore each of these methods in detail.

1. Completing the Square Method

The completing the square method involves rewriting the quadratic function in vertex form, which is expressed as:

f(x) = a(x - h)² + k

where (h, k) represents the coordinates of the vertex. To complete the square, we manipulate the original quadratic function through a series of algebraic steps. Let's apply this method to our example function, f(x) = x² - 6x + 5:

  1. Group the x² and x terms: f(x) = (x² - 6x) + 5
  2. Complete the square inside the parentheses: To complete the square, we take half of the coefficient of the x term (-6), square it ((-3)² = 9), and add and subtract it inside the parentheses: f(x) = (x² - 6x + 9 - 9) + 5
  3. Rewrite the expression inside the parentheses as a squared term: f(x) = (x - 3)² - 9 + 5
  4. Simplify the expression: f(x) = (x - 3)² - 4

Now, the function is in vertex form, f(x) = (x - 3)² - 4. By comparing this to the general vertex form, we can identify the vertex as (h, k) = (3, -4). Therefore, the vertex of the quadratic function f(x) = x² - 6x + 5 is (3, -4).

2. Vertex Formula Method

The vertex formula method provides a direct approach to finding the vertex without the need for completing the square. The x-coordinate of the vertex (h) is given by the formula:

h = -b / 2a

where 'a' and 'b' are the coefficients of the quadratic function in the standard form, f(x) = ax² + bx + c. Once we find the x-coordinate (h), we can substitute it back into the original function to find the y-coordinate (k) of the vertex:

k = f(h)

Let's apply this method to our example function, f(x) = x² - 6x + 5. Here, a = 1, b = -6, and c = 5.

  1. Find the x-coordinate (h) of the vertex: h = -(-6) / (2 * 1) = 3
  2. Find the y-coordinate (k) of the vertex: k = f(3) = (3)² - 6(3) + 5 = 9 - 18 + 5 = -4

Thus, the vertex of the quadratic function f(x) = x² - 6x + 5 is (3, -4), which aligns with the result obtained using the completing the square method.

Finding the Zeros

The zeros of a quadratic function are the points where the parabola intersects the x-axis. These points represent the solutions to the quadratic equation f(x) = 0. There are several methods for finding the zeros, including factoring, using the quadratic formula, and completing the square. Let's explore each of these methods in detail.

1. Factoring

Factoring involves expressing the quadratic function as a product of two linear factors. If we can factor the quadratic function, we can easily find the zeros by setting each factor equal to zero and solving for x. Let's apply this method to our example function, f(x) = x² - 6x + 5:

  1. Factor the quadratic expression: x² - 6x + 5 = (x - 5)(x - 1)
  2. Set each factor equal to zero: x - 5 = 0 or x - 1 = 0
  3. Solve for x: x = 5 or x = 1

Therefore, the zeros of the quadratic function f(x) = x² - 6x + 5 are x = 5 and x = 1.

2. Quadratic Formula

The quadratic formula is a general formula that can be used to find the zeros of any quadratic function, regardless of whether it can be factored easily. The quadratic formula is expressed as:

x = (-b ± √(b² - 4ac)) / 2a

where 'a', 'b', and 'c' are the coefficients of the quadratic function in the standard form, f(x) = ax² + bx + c. Let's apply this method to our example function, f(x) = x² - 6x + 5. Here, a = 1, b = -6, and c = 5.

  1. Substitute the values of a, b, and c into the quadratic formula: x = (-(-6) ± √((-6)² - 4 * 1 * 5)) / (2 * 1)
  2. Simplify the expression: x = (6 ± √(36 - 20)) / 2 x = (6 ± √16) / 2 x = (6 ± 4) / 2
  3. Solve for the two possible values of x: x = (6 + 4) / 2 = 5 x = (6 - 4) / 2 = 1

Thus, the zeros of the quadratic function f(x) = x² - 6x + 5 are x = 5 and x = 1, which aligns with the result obtained using factoring.

3. Completing the Square

Completing the square can also be used to find the zeros of a quadratic function. After completing the square and rewriting the function in vertex form, we can set f(x) equal to zero and solve for x. Let's apply this method to our example function, f(x) = x² - 6x + 5.

We have already completed the square in the previous section and found the vertex form of the function to be:

f(x) = (x - 3)² - 4

  1. Set f(x) equal to zero: (x - 3)² - 4 = 0
  2. Isolate the squared term: (x - 3)² = 4
  3. Take the square root of both sides: x - 3 = ±√4 x - 3 = ±2
  4. Solve for x: x = 3 + 2 = 5 x = 3 - 2 = 1

Therefore, the zeros of the quadratic function f(x) = x² - 6x + 5 are x = 5 and x = 1, which aligns with the results obtained using factoring and the quadratic formula.

Conclusion

In this comprehensive guide, we have explored the methods for finding the vertex and zeros of the quadratic function f(x) = x² - 6x + 5. We discussed the completing the square method and the vertex formula method for finding the vertex, and we examined factoring, the quadratic formula, and completing the square for finding the zeros. By mastering these techniques, you can effectively analyze and interpret quadratic functions, which are fundamental in various mathematical and real-world applications. Understanding the vertex and zeros provides valuable insights into the behavior and properties of quadratic functions, enabling you to solve related problems and make informed decisions.