Finding Minimum Or Maximum Value Of F(x) = X² + 12x + 8 In Interval Notation

Hey guys! Today, we're diving into the fascinating world of quadratic functions. Specifically, we're going to figure out how to determine whether a quadratic function like f(x) = x² + 12x + 8 has a minimum or maximum value, and then we'll actually find that value. This is a super important skill in algebra and calculus, and it has tons of real-world applications, from optimizing business profits to figuring out the trajectory of a ball you throw. So, let's jump right in!

Understanding Quadratic Functions

Before we get to the nitty-gritty, let's make sure we're all on the same page about quadratic functions. A quadratic function is a polynomial function of degree two, meaning the highest power of 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 U-shape can either open upwards or downwards, and that's what determines whether we have a minimum or maximum value.

Now, the key to understanding whether a parabola opens upwards or downwards lies in the coefficient a. If a is positive, the parabola opens upwards, and we have a minimum value at the vertex of the parabola. Think of it like a smiley face – the lowest point of the smile is the minimum value. On the other hand, if a is negative, the parabola opens downwards, and we have a maximum value at the vertex. Imagine a frowny face – the highest point of the frown is the maximum value. In our specific function, f(x) = x² + 12x + 8, the coefficient a is 1 (since there's an implied 1 in front of the x² term). Since 1 is positive, we know our parabola opens upwards and we're looking for a minimum value. This is a crucial first step because it tells us what kind of value to expect. Getting this right saves us time and prevents confusion later on. Remember, identifying the sign of a is like setting the direction for our treasure hunt – it guides us straight to the prize!

Determining Minimum or Maximum Value

Okay, so we've established that our function f(x) = x² + 12x + 8 has a minimum value because the coefficient of x² is positive. But how do we actually find that minimum value? Well, the minimum or maximum value of a quadratic function occurs at the vertex of the parabola. The vertex is the turning point of the parabola – it's where the parabola changes direction. There are a couple of ways to find the vertex, but the most common and efficient method is using the vertex formula.

The vertex formula gives us the x-coordinate of the vertex, which we'll call h. The formula is: h = -b / 2a. Remember those a and b coefficients from the general form f(x) = ax² + bx + c? That's what we're using here! In our function, f(x) = x² + 12x + 8, a is 1 and b is 12. Plugging these values into the vertex formula, we get:

h = -12 / (2 * 1) = -12 / 2 = -6

So, the x-coordinate of the vertex is -6. This is super helpful because it tells us where the minimum value occurs. But we're not quite done yet – we need to find the actual minimum value, which is the y-coordinate of the vertex. To do this, we simply plug the x-coordinate we just found (h = -6) back into our original function:

f(-6) = (-6)² + 12(-6) + 8 = 36 - 72 + 8 = -28

Therefore, the minimum value of the function f(x) = x² + 12x + 8 is -28. This value occurs at the vertex of the parabola, specifically at the point (-6, -28). To summarize, we've used the vertex formula to find the x-coordinate of the vertex and then plugged that value back into the function to find the corresponding y-coordinate, which gives us the minimum value. This process is essential for understanding how quadratic functions behave and is a key skill in various mathematical contexts.

Expressing the Value in Interval Notation

Now, the final part of our task is to express the minimum value in interval notation. Interval notation is a way of representing a range of values using brackets and parentheses. Since we're dealing with a minimum value, we're interested in the range of y-values that the function can take. We know that the minimum value is -28, and since the parabola opens upwards, the function's y-values will range from -28 upwards to infinity.

In interval notation, we represent this as [-28, ∞). Let's break down what this means: The square bracket [ indicates that -28 is included in the interval, because -28 is the actual minimum value of the function. The parenthesis ) indicates that infinity is not included in the interval, because infinity is not a specific number – it's a concept representing a value that goes on without end. Understanding the difference between square brackets and parentheses is crucial for accurately representing intervals. A common mistake is to use a square bracket for infinity, but remember, infinity is always represented with a parenthesis.

Therefore, the minimum value of f(x) = x² + 12x + 8 expressed in interval notation is [-28, ∞). This notation concisely communicates the range of possible y-values for the function, which is a fundamental concept in understanding the behavior of quadratic functions and their graphs. By mastering interval notation, you'll be able to clearly and accurately communicate the range of solutions in various mathematical problems.

Alright guys, we've successfully determined that the quadratic function f(x) = x² + 12x + 8 has a minimum value, and we found that minimum value to be -28. We then expressed this value in interval notation as [-28, ∞). This whole process involved understanding the shape of a parabola, using the vertex formula, and correctly applying interval notation. These are all fundamental skills in algebra and will serve you well as you continue your mathematical journey. Keep practicing, and you'll become a pro at handling quadratic functions in no time! Remember, practice makes perfect, and the more you work with these concepts, the more natural they'll become. So keep up the great work!