Analyzing Quadratic Function F(x) = -x² - 12x - 36 Vertex And Intercepts

Hey everyone! Today, we're diving headfirst into the fascinating world of quadratic functions, specifically focusing on the function f(x) = -x² - 12x - 36. We'll explore its key features, including its vertex, y-intercept(s), and x-intercept(s). So, buckle up and get ready for a mathematical adventure!

a) Pinpointing the Vertex: The Heart of the Parabola

The vertex of a parabola is like its heart – it's the point where the parabola changes direction. It's either the highest point (maximum) or the lowest point (minimum) on the graph. For quadratic functions in the form f(x) = ax² + bx + c, we can find the vertex using a nifty formula. For our function, f(x) = -x² - 12x - 36, we have a = -1, b = -12, and c = -36.

To find the x-coordinate of the vertex, we use the formula: x = -b / 2a. Plugging in our values, we get x = -(-12) / (2 * -1) = 12 / -2 = -6. Now that we have the x-coordinate, we can find the y-coordinate by substituting it back into the original function: f(-6) = -(-6)² - 12(-6) - 36 = -36 + 72 - 36 = 0. Therefore, the vertex of the graph is (-6, 0). This means the parabola's peak or valley sits right at this point on the coordinate plane. Understanding the vertex is crucial because it tells us a lot about the function's behavior – whether it opens upwards or downwards, and where its extreme value lies. The vertex form of a quadratic equation, f(x) = a(x - h)² + k, makes the vertex readily apparent, with (h, k) being the vertex. In our case, rewriting f(x) = -x² - 12x - 36 in vertex form gives us f(x) = -(x + 6)², clearly showing the vertex at (-6, 0). This form is incredibly useful for quickly identifying the vertex and understanding transformations of the basic parabola y = x². Moreover, the vertex plays a significant role in optimization problems, where we aim to find the maximum or minimum value of a function. For instance, if this function represented the profit of a business, the vertex would tell us the input (x) that maximizes profit. The vertex is not just a point; it's a gateway to understanding the function's characteristics and its applications in various real-world scenarios. Mastering the concept of the vertex is a cornerstone in quadratic function analysis.

b) Unveiling the Y-intercept(s): Where the Graph Meets the Y-axis

The y-intercept is the point where the graph of the function crosses the y-axis. It's the value of f(x) when x = 0. To find the y-intercept(s), we simply substitute x = 0 into our function: f(0) = -(0)² - 12(0) - 36 = -36. So, the y-intercept is (0, -36). This tells us where the parabola intersects the vertical axis. The y-intercept gives us a starting point on the graph and helps visualize the function's overall position in the coordinate plane. It's also a crucial piece of information when sketching the graph of a quadratic function, as it provides a fixed point that the parabola must pass through. In the context of real-world applications, the y-intercept can represent the initial value of a quantity. For example, if the function described the height of a projectile, the y-intercept would represent the initial height from which it was launched. The y-intercept is a simple yet powerful tool for understanding and interpreting quadratic functions. It's a direct consequence of the constant term in the quadratic equation, making it easily identifiable. Furthermore, the y-intercept, along with the vertex and x-intercepts (if any), provides a comprehensive picture of the parabola's behavior. It's an essential component in the analysis and application of quadratic functions across various fields. Grasping the concept of the y-intercept is fundamental for anyone seeking to master the intricacies of quadratic equations and their graphical representations.

c) Discovering the X-intercept(s): Where the Graph Touches the X-axis

The x-intercept(s) are the points where the graph of the function crosses the x-axis. These are also known as the roots or zeros of the function. To find the x-intercept(s), we need to solve the equation f(x) = 0. So, we have to solve * -x² - 12x - 36 = 0*. We can factor out a -1, giving us x² + 12x + 36 = 0. This quadratic equation can be factored as (x + 6)(x + 6) = 0, or (x + 6)² = 0. This means that x + 6 = 0, which gives us x = -6. Therefore, the x-intercept is (-6, 0). Notice that we have a repeated root, which means the parabola touches the x-axis at only one point – the vertex! This special case indicates that the vertex lies on the x-axis. The x-intercepts are critical in understanding the solutions to quadratic equations and the points where the function's value is zero. They provide valuable information about the function's behavior and its relationship to the x-axis. In practical applications, x-intercepts can represent break-even points, equilibrium states, or other significant values. For instance, if the function described the profit of a company, the x-intercepts would represent the points where the company neither makes a profit nor incurs a loss. The number of x-intercepts a quadratic function has (zero, one, or two) is determined by the discriminant (b² - 4ac) of the quadratic equation. In our case, the discriminant is (-12)² - 4(-1)(-36) = 144 - 144 = 0, indicating one real root (a repeated root), which corresponds to the parabola touching the x-axis at the vertex. The relationship between the x-intercepts, the vertex, and the discriminant is a fundamental concept in quadratic function analysis. Mastering the techniques for finding x-intercepts and understanding their implications is essential for anyone working with quadratic equations and their applications.

Wrapping Up: Putting It All Together

So, guys, we've successfully dissected the quadratic function f(x) = -x² - 12x - 36! We found its vertex at (-6, 0), its y-intercept at (0, -36), and its x-intercept at (-6, 0). This comprehensive analysis gives us a clear picture of the parabola's shape, position, and key features. By understanding the vertex, intercepts, and other characteristics of quadratic functions, we can unlock their potential for solving a wide range of problems in mathematics, science, and engineering. Keep practicing, keep exploring, and keep having fun with math!