Finding The Maximum Value Of F(x) = -x² + 4x - 5
Hey guys! Today, we're diving into the world of quadratic functions and tackling a super common problem: finding the maximum value of a function. Specifically, we're going to be working with the function f(x) = -x² + 4x - 5. This might seem a little daunting at first, but trust me, we'll break it down step-by-step, and you'll be a pro in no time! So, grab your thinking caps, and let's get started!
Understanding Quadratic Functions
Before we jump into the specifics of our function, let's quickly recap what quadratic functions are all about. At their core, quadratic functions are polynomial functions with the highest degree being 2. This means they generally take the form of 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, a U-shaped curve. Now, here's where it gets interesting: if the coefficient 'a' is positive, the parabola opens upwards, forming a valley. If 'a' is negative, like in our case (a = -1), the parabola opens downwards, forming a peak. This peak is what we call the vertex, and it represents either the minimum (if the parabola opens upwards) or the maximum (if the parabola opens downwards) value of the function.
Knowing this about parabolas is crucial because our goal is to find the maximum value, which corresponds to the y-coordinate of the vertex when the parabola opens downwards. There are a couple of ways we can find this vertex, and we'll explore both to give you a complete understanding. One method involves completing the square, which transforms the quadratic function into vertex form, making the vertex easily identifiable. The other method utilizes a formula to directly calculate the x-coordinate of the vertex. We’ll go through both methods to make sure you're comfortable with each. Remember, understanding the underlying concepts of quadratic functions and their graphical representation is key to solving these kinds of problems efficiently. We'll make sure to highlight the key takeaways as we go along so you can easily apply these techniques to other quadratic functions you might encounter. Think of it like unlocking a new superpower in your math toolkit! So, let's keep exploring and see how we can pinpoint the maximum value of our function.
Method 1: Completing the Square
One of the most powerful techniques for analyzing quadratic functions is completing the square. This method allows us to rewrite the function in a form that directly reveals the vertex of the parabola. The vertex, as we discussed, is the point where the function reaches its maximum or minimum value. Let's apply this to our function, f(x) = -x² + 4x - 5.
Step 1: Factor out the coefficient of x² from the first two terms.
In our case, the coefficient of x² is -1. Factoring this out, we get:
f(x) = -(x² - 4x) - 5
Notice how we've only factored out the -1 from the terms containing 'x'. The constant term, -5, remains outside the parentheses. This is an important step in the process.
Step 2: Complete the square inside the parentheses.
To complete the square, we need to add and subtract a specific value inside the parentheses. This value is calculated as (b/2)², where 'b' is the coefficient of the x term inside the parentheses. In our case, b = -4, so (b/2)² = (-4/2)² = (-2)² = 4. We add and subtract this value inside the parentheses:
f(x) = -(x² - 4x + 4 - 4) - 5
Step 3: Rewrite the expression inside the parentheses as a squared term.
The first three terms inside the parentheses (x² - 4x + 4) now form a perfect square trinomial, which can be rewritten as (x - 2)²:
f(x) = -((x - 2)² - 4) - 5
Step 4: Distribute the negative sign and simplify.
Distribute the negative sign outside the parentheses to both terms inside:
f(x) = -(x - 2)² + 4 - 5
Now, simplify the constant terms:
f(x) = -(x - 2)² - 1
Step 5: Identify the vertex.
The function is now in vertex form, which is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. Comparing this to our function, f(x) = -(x - 2)² - 1, we can see that h = 2 and k = -1. Therefore, the vertex of the parabola is (2, -1).
Since the coefficient of the (x - 2)² term is negative (-1), the parabola opens downwards, and the vertex represents the maximum point. This means the maximum value of the function is -1, and it occurs at x = 2. Completing the square gives us a clear visual of the parabola's structure, highlighting the vertex and helping us determine the maximum or minimum value. This method is especially useful for understanding how transformations affect the graph of a quadratic function.
Method 2: Using the Vertex Formula
Another efficient way to find the maximum or minimum value of a quadratic function is by using the vertex formula. This formula directly calculates the x-coordinate of the vertex, which we can then use to find the corresponding y-coordinate (the maximum or minimum value). Let's see how this works with our function, f(x) = -x² + 4x - 5.
The Vertex Formula:
For a quadratic function in the form f(x) = ax² + bx + c, the x-coordinate of the vertex (often denoted as xᵥ) is given by:
xᵥ = -b / 2a
Step 1: Identify the coefficients a, b, and c.
In our function, f(x) = -x² + 4x - 5, we have:
a = -1 b = 4 c = -5
Step 2: Apply the vertex formula to find the x-coordinate of the vertex.
Substitute the values of 'a' and 'b' into the formula:
xᵥ = -4 / (2 * -1) xᵥ = -4 / -2 xᵥ = 2
So, the x-coordinate of the vertex is 2.
Step 3: Find the y-coordinate of the vertex (the maximum value).
To find the y-coordinate, we substitute the x-coordinate (xᵥ = 2) back into the original function:
f(2) = -(2)² + 4(2) - 5 f(2) = -4 + 8 - 5 f(2) = -1
Therefore, the y-coordinate of the vertex is -1. This is the maximum value of the function.
Step 4: State the vertex and the maximum value.
The vertex of the parabola is (2, -1), and the maximum value of the function is -1, which occurs at x = 2. Using the vertex formula provides a quick and direct method for finding the vertex of a parabola. It avoids the algebraic manipulation involved in completing the square and is particularly useful when you only need to find the maximum or minimum value without needing the vertex form of the equation. This formula is a staple in quadratic function analysis and is worth memorizing for its efficiency and accuracy.
Conclusion
Alright, guys! We've successfully determined the maximum value of the function f(x) = -x² + 4x - 5 using two different methods: completing the square and the vertex formula. We found that the maximum value is -1, and it occurs at the point x = 2. This means the vertex of the parabola is at (2, -1).
Both methods have their advantages. Completing the square provides a deeper understanding of how the function transforms and reveals the vertex form, while the vertex formula offers a quicker way to calculate the vertex directly. The best method to use often depends on the specific problem and what information you need to find. For instance, if you need to sketch the graph of the parabola, completing the square might be more helpful as it gives you the vertex form, which is ideal for graphing. However, if you only need to find the maximum or minimum value, the vertex formula might be more efficient.
Understanding these concepts is super important for tackling a wide range of problems in algebra and calculus. Whether you're optimizing a business process, modeling projectile motion, or just trying to ace your math test, knowing how to find the maximum or minimum of a quadratic function is a valuable skill. Keep practicing these techniques, and you'll be well-equipped to handle any quadratic function that comes your way! Remember, math is like building with blocks; each concept builds upon the previous one, so a strong foundation in quadratic functions will pave the way for more advanced topics. Keep up the great work, and happy problem-solving!