Find Min/Max Of F(x) = X^2 - 8x + 10 By Completing The Square

Let's dive into the world of quadratic functions and explore how to find their minimum or maximum values using a technique called "completing the square." This method is super useful and will give you a solid understanding of how these functions behave. In this article, we'll break down the process step-by-step using the example function f(x) = x^2 - 8x + 10. By the end, you'll be able to tackle similar problems with confidence and ease.

Understanding Quadratic Functions

Before we jump into the nitty-gritty, let's quickly recap what quadratic functions are all about. 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) = ax^2 + bx + c

Where 'a', 'b', and 'c' are constants. The graph of a quadratic function is a parabola, which is a U-shaped curve. This parabola can open upwards or downwards, depending on the sign of the coefficient 'a'. If 'a' is positive, the parabola opens upwards, and the function has a minimum value. If 'a' is negative, the parabola opens downwards, and the function has a maximum value. For our example function, f(x) = x^2 - 8x + 10, 'a' is 1 (positive), so we know we're looking for a minimum value.

The Power of Completing the Square

Completing the square is a technique that allows us to rewrite a quadratic expression in a form that makes it easy to identify the vertex of the parabola. The vertex is the point where the parabola changes direction – it's either the minimum or maximum point of the function. The vertex form of a quadratic function is:

f(x) = a(x - h)^2 + k

In this form, the vertex of the parabola is at the point (h, k). If 'a' is positive, 'k' is the minimum value of the function. If 'a' is negative, 'k' is the maximum value. So, our goal is to transform our given function, f(x) = x^2 - 8x + 10, into this vertex form. This is where completing the square comes in handy, allowing us to rewrite the quadratic expression by manipulating it algebraically without changing its inherent value.

Step-by-Step: Completing the Square for f(x) = x^2 - 8x + 10

Let's walk through the process of completing the square for our function, f(x) = x^2 - 8x + 10. We'll break it down into manageable steps so you can follow along easily.

Step 1: Focus on the x^2 and x terms

First, we'll isolate the terms containing x^2 and x. In our case, these are x^2 and -8x. We can rewrite the function as:

f(x) = (x^2 - 8x) + 10

We're essentially grouping these terms together, setting the stage for the next crucial step.

Step 2: Complete the Square

This is where the magic happens! To complete the square, we need to add and subtract a specific value inside the parenthesis. This value is calculated by taking half of the coefficient of the 'x' term, squaring it, and then adding and subtracting it. The coefficient of our 'x' term is -8. Half of -8 is -4, and squaring -4 gives us 16. So, we'll add and subtract 16 inside the parenthesis:

f(x) = (x^2 - 8x + 16 - 16) + 10

Notice that we're adding and subtracting the same value, so we're not actually changing the overall value of the function. We're just rewriting it in a more useful form.

Step 3: Rewrite as a Perfect Square

The first three terms inside the parenthesis (x^2 - 8x + 16) now form a perfect square trinomial. This means they can be factored into the square of a binomial. In this case:

x^2 - 8x + 16 = (x - 4)^2

So, we can rewrite our function as:

f(x) = ((x - 4)^2 - 16) + 10

This step is the heart of the completing the square method, turning a complex expression into a neat, manageable square.

Step 4: Simplify

Now, let's simplify the expression by removing the extra parenthesis and combining the constant terms:

f(x) = (x - 4)^2 - 16 + 10 f(x) = (x - 4)^2 - 6

We've successfully transformed our function into vertex form: f(x) = a(x - h)^2 + k, where a = 1, h = 4, and k = -6.

Identifying the Minimum Value

Now that our function is in vertex form, f(x) = (x - 4)^2 - 6, identifying the minimum value is a breeze. Remember that the vertex of the parabola is at the point (h, k), which in our case is (4, -6). Since 'a' is positive (a = 1), the parabola opens upwards, meaning the vertex represents the minimum point of the function. Therefore, the minimum value of the function is the y-coordinate of the vertex, which is k = -6.

So, the minimum value of f(x) = x^2 - 8x + 10 is -6, and it occurs when x = 4.

Conclusion

Completing the square is a powerful technique for finding the minimum or maximum value of a quadratic function. By rewriting the function in vertex form, we can easily identify the vertex of the parabola, which tells us the function's extreme value. In our example, we successfully transformed f(x) = x^2 - 8x + 10 into vertex form, f(x) = (x - 4)^2 - 6, and determined that the minimum value of the function is -6. Practice this method with different quadratic functions, and you'll become a pro at finding their minimum and maximum values! For further reading and to enhance your understanding, consider exploring resources like Khan Academy's Algebra section on quadratic functions. This external link provides additional explanations, examples, and practice problems that can solidify your grasp of this important mathematical concept.