Finding The Vertex Of F(x) = X² + 10x - 3 A Step-by-Step Guide

by Sharif Sakr 63 views

Hey guys! Today, we're diving deep into the world of quadratic functions, specifically focusing on how to find the vertex of the function f(x) = x² + 10x - 3. This is a fundamental concept in algebra, and mastering it will not only help you ace your math exams but also provide a solid foundation for more advanced topics. So, let's break it down step by step, making sure everyone understands the ins and outs of finding the vertex. We'll explore different methods, explain the underlying principles, and provide plenty of examples to solidify your understanding.

Understanding Quadratic Functions

Before we jump into finding the vertex, let's quickly recap what a quadratic function is. A quadratic function is a polynomial function of the second degree, meaning the highest power of the variable (usually x) is 2. The general form of a quadratic function is:

f(x) = ax² + bx + c

Where a, b, and c are constants, and a is not equal to 0. The graph of a quadratic function is a parabola, which is a U-shaped curve. This parabola can open upwards (if a > 0) or downwards (if a < 0). The vertex is a crucial point on this parabola – it's either the minimum point (if the parabola opens upwards) or the maximum point (if the parabola opens downwards).

In our specific example, f(x) = x² + 10x - 3, we can see that a = 1, b = 10, and c = -3. Since a is positive (1 > 0), the parabola opens upwards, meaning the vertex represents the minimum point of the function.

Why is the vertex so important? Well, it gives us key information about the quadratic function's behavior. It tells us the function's minimum or maximum value, and it helps us understand the symmetry of the parabola. The vertex is located at the axis of symmetry, which is a vertical line that divides the parabola into two symmetrical halves. Knowing the vertex allows us to easily sketch the graph of the quadratic function and analyze its properties.

Methods to Find the Vertex

There are several methods to find the coordinates of the vertex of a quadratic function. We'll explore three popular approaches:

  1. Using the Vertex Formula: This is the most direct method, using a formula derived from completing the square.
  2. Completing the Square: This method involves rewriting the quadratic function in vertex form, which directly reveals the vertex coordinates.
  3. Using Calculus (Finding the Derivative): This method uses calculus concepts to find the critical point, which corresponds to the vertex.

Let's dive into each method in detail.

1. Using the Vertex Formula

The vertex formula is a powerful tool for quickly finding the vertex of a quadratic function. It directly provides the x-coordinate of the vertex, which can then be used to find the y-coordinate. The vertex formula is given by:

x = -b / (2a)

Where a and b are the coefficients from the standard form of the quadratic function, f(x) = ax² + bx + c. Once you find the x-coordinate, you can plug it back into the original function to find the y-coordinate of the vertex.

Let's apply this formula to our example function, f(x) = x² + 10x - 3. Here, a = 1 and b = 10. Plugging these values into the vertex formula, we get:

x = -10 / (2 * 1) = -10 / 2 = -5

So, the x-coordinate of the vertex is -5. Now, to find the y-coordinate, we substitute x = -5 back into the function:

f(-5) = (-5)² + 10(-5) - 3 = 25 - 50 - 3 = -28

Therefore, the vertex of the function f(x) = x² + 10x - 3 is (-5, -28). This means the minimum point of the parabola is at x = -5, and the minimum value of the function is -28.

The vertex formula is incredibly efficient and straightforward, making it a go-to method for finding the vertex. It's particularly useful when you just need the vertex coordinates and don't need to rewrite the function in a different form. Remember to identify the coefficients a and b correctly, and you'll be well on your way to finding the vertex!

2. Completing the Square

Completing the square is another powerful technique for finding the vertex of a quadratic function. This method involves rewriting the quadratic function in vertex form, which directly reveals the vertex coordinates. The vertex form of a quadratic function is:

f(x) = a(x - h)² + k

Where (h, k) are the coordinates of the vertex. Notice how the vertex coordinates are explicitly present in this form, making it easy to identify them once the function is rewritten.

Let's apply completing the square to our example function, f(x) = x² + 10x - 3. Here's how the process works:

  1. Group the x² and x terms: f(x) = (x² + 10x) - 3
  2. Complete the square inside the parentheses: To complete the square, we need to add and subtract a constant term that will make the expression inside the parentheses a perfect square trinomial. This constant is calculated as (b/2)², where b is the coefficient of the x term. In our case, b = 10, so (b/2)² = (10/2)² = 5² = 25. We add and subtract 25 inside the parentheses: f(x) = (x² + 10x + 25 - 25) - 3
  3. Rewrite the perfect square trinomial as a squared term: The expression x² + 10x + 25 is a perfect square trinomial and can be rewritten as (x + 5)²: f(x) = ((x + 5)²) - 25 - 3
  4. Simplify the expression: Combine the constant terms: f(x) = (x + 5)² - 28

Now, the function is in vertex form, f(x) = (x - h)² + k, where a = 1, h = -5, and k = -28. Therefore, the vertex is (-5, -28). Notice that h has the opposite sign inside the parentheses.

Completing the square is a versatile method that not only helps you find the vertex but also provides valuable insights into the function's behavior. It's a bit more involved than the vertex formula, but it reinforces your understanding of algebraic manipulation and the structure of quadratic functions. Guys, remember that practice makes perfect with completing the square, so try it out with different quadratic functions to master the technique!

3. Using Calculus (Finding the Derivative)

For those familiar with calculus, finding the derivative of a quadratic function offers another elegant method to determine the vertex. The vertex corresponds to the critical point of the function, which is where the derivative equals zero. This is because at the vertex, the slope of the tangent line to the parabola is horizontal (zero slope).

Let's apply this method to our example function, f(x) = x² + 10x - 3. Here's the process:

  1. Find the derivative of the function: The derivative of f(x) = x² + 10x - 3 with respect to x is: f'(x) = 2x + 10
  2. Set the derivative equal to zero and solve for x: To find the critical point, we set f'(x) = 0: 2x + 10 = 0 2x = -10 x = -5

So, the x-coordinate of the vertex is -5. This is the same result we obtained using the vertex formula and completing the square.

  1. Find the y-coordinate by substituting x back into the original function: Substitute x = -5 into f(x) = x² + 10x - 3: f(-5) = (-5)² + 10(-5) - 3 = 25 - 50 - 3 = -28

Therefore, the vertex of the function f(x) = x² + 10x - 3 is (-5, -28).

Using calculus to find the vertex provides a powerful connection between algebra and calculus concepts. It reinforces the idea that the derivative represents the slope of the tangent line and that critical points correspond to maximum or minimum values of a function. This method is particularly useful for more complex functions where completing the square might be challenging. For all the calculus enthusiasts out there, this method offers a sophisticated approach to finding the vertex!

Conclusion

Finding the vertex of a quadratic function is a fundamental skill in algebra, and we've explored three different methods to achieve this: using the vertex formula, completing the square, and using calculus. Each method offers a unique perspective and reinforces different mathematical concepts. For the function f(x) = x² + 10x - 3, we found that the vertex is (-5, -28) using all three approaches.

Whether you prefer the directness of the vertex formula, the algebraic elegance of completing the square, or the calculus-based approach of finding the derivative, mastering these methods will significantly enhance your understanding of quadratic functions. Guys, keep practicing, and you'll become vertex-finding pros in no time! Remember to choose the method that best suits your understanding and the specific problem you're tackling. With a solid grasp of these techniques, you'll be well-equipped to tackle more complex mathematical challenges in the future. So, go forth and conquer those parabolas!