Finding The Vertex And Classifying Minima Or Maxima Of F(x) = X² + 2x - 15

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Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of quadratic functions. Specifically, we're going to dissect the function f(x) = x² + 2x - 15 to pinpoint its vertex and classify it as either a minimum or maximum point. Buckle up, because we're about to embark on a mathematical adventure that's both insightful and, dare I say, fun!

Understanding Quadratic Functions

Before we jump into the specifics of our function, let's take a moment to appreciate the beauty and versatility of quadratic functions. These functions, characterized by the general form f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants and 'a' is not zero, are the cornerstones of many mathematical and real-world applications. Their graphs, graceful curves known as parabolas, hold a wealth of information about the function's behavior. The parabola's vertex, the point where it changes direction, is of particular interest, as it represents either the minimum or maximum value of the function. The coefficient 'a' plays a crucial role in determining the parabola's orientation: if 'a' is positive, the parabola opens upwards, indicating a minimum point; if 'a' is negative, the parabola opens downwards, signifying a maximum point.

Quadratic functions are not just abstract mathematical concepts; they are powerful tools for modeling real-world phenomena. They can describe the trajectory of a projectile, the shape of a suspension bridge, or the optimal dimensions for maximizing the area of a garden. Understanding the properties of quadratic functions, such as finding the vertex and determining its nature, allows us to make predictions, solve optimization problems, and gain a deeper understanding of the world around us. In our case, we have the quadratic function f(x) = x² + 2x - 15, where a = 1, b = 2, and c = -15. Since 'a' is positive (1 > 0), we know that the parabola opens upwards, and the vertex will be a minimum point. But how do we find the exact coordinates of this minimum point? Let's delve into the methods for finding the vertex.

Methods to Determine the Vertex

There are several elegant methods to determine the vertex of a quadratic function, each with its own unique charm and applicability. Let's explore two popular approaches: completing the square and using the vertex formula.

Method 1: Completing the Square

Completing the square is a powerful algebraic technique that transforms a quadratic expression into a perfect square trinomial, plus a constant term. This form reveals the vertex coordinates directly. Let's apply this method to our function f(x) = x² + 2x - 15. The first step involves focusing on the x² and x terms (x² + 2x) and identifying the constant term needed to complete the square. We take half of the coefficient of the x term (which is 2), square it ((2/2)² = 1), and add and subtract it within the expression:

f(x) = x² + 2x + 1 - 1 - 15

Notice that we've added and subtracted 1, so we haven't changed the function's value, only its form. Now, the first three terms form a perfect square trinomial: (x + 1)². We can rewrite the function as:

f(x) = (x + 1)² - 16

This is the vertex form of the quadratic function, f(x) = a(x - h)² + k, where (h, k) is the vertex. Comparing our transformed function to the vertex form, we see that h = -1 and k = -16. Therefore, the vertex of the parabola is (-1, -16). Completing the square is not just about finding the vertex; it's a fundamental technique that can be used to solve quadratic equations, derive the quadratic formula, and simplify many algebraic expressions.

Method 2: Vertex Formula

The vertex formula provides a direct way to calculate the coordinates of the vertex using the coefficients 'a', 'b', and 'c' from the standard form of the quadratic function, f(x) = ax² + bx + c. The x-coordinate of the vertex, often denoted as 'h', is given by the formula:

h = -b / 2a

In our case, f(x) = x² + 2x - 15, so a = 1 and b = 2. Plugging these values into the formula, we get:

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

To find the y-coordinate of the vertex, often denoted as 'k', we substitute the value of 'h' back into the original function:

k = f(h) = f(-1) = (-1)² + 2(-1) - 15 = 1 - 2 - 15 = -16

Thus, the vertex is (-1, -16), which confirms our result from completing the square. The vertex formula is a convenient and efficient tool, especially when the standard form of the quadratic function is readily available. It highlights the relationship between the coefficients of the quadratic function and the location of its vertex. By understanding this formula, we can quickly determine the vertex of any quadratic function, without the need for completing the square.

Classifying the Vertex: Minimum or Maximum?

Now that we've successfully located the vertex at (-1, -16), the next crucial step is to classify it as either a minimum or maximum point. Remember, the coefficient 'a' in the quadratic function f(x) = ax² + bx + c holds the key to this classification. If 'a' is positive, the parabola opens upwards, and the vertex represents the minimum value of the function. Conversely, if 'a' is negative, the parabola opens downwards, and the vertex represents the maximum value of the function.

In our case, f(x) = x² + 2x - 15, the coefficient 'a' is 1, which is clearly positive. This tells us that the parabola opens upwards, like a smile, and the vertex at (-1, -16) is indeed the minimum point of the function. This means that the function reaches its lowest value, -16, at x = -1. No other point on the parabola will have a y-coordinate lower than -16. Visualizing the graph of the function can further solidify this understanding. Imagine a parabola opening upwards, with its lowest point nestled at (-1, -16). This visual representation reinforces the concept of a minimum point.

The classification of the vertex as a minimum or maximum is not just a mathematical exercise; it has practical implications in optimization problems. For example, if this function represented the cost of production, the vertex would indicate the production level that minimizes the cost. Similarly, if the function represented the height of a projectile, the vertex would indicate the maximum height reached. Understanding the nature of the vertex allows us to make informed decisions and optimize various real-world scenarios.

Conclusion

In this exploration, we've successfully navigated the quadratic function f(x) = x² + 2x - 15 to determine its vertex and classify it as a minimum point. We employed two powerful techniques: completing the square and the vertex formula, each offering a unique perspective on the function's behavior. By completing the square, we transformed the function into vertex form, directly revealing the coordinates of the vertex. The vertex formula provided a more direct approach, utilizing the coefficients of the quadratic function to calculate the vertex coordinates. Both methods led us to the same conclusion: the vertex is located at (-1, -16).

Furthermore, we classified the vertex as a minimum point based on the positive coefficient of the x² term, indicating that the parabola opens upwards. This means that the function attains its minimum value, -16, at x = -1. This understanding is crucial for various applications, from optimization problems to modeling real-world phenomena. Mastering the techniques for finding and classifying the vertex of a quadratic function empowers us to analyze and interpret these functions effectively.

So, there you have it, folks! We've conquered the vertex and emerged victorious with a deeper understanding of quadratic functions. Keep exploring, keep questioning, and keep the mathematical flame burning bright!

Keywords: quadratic function, vertex, minimum point, completing the square, vertex formula