Unveiling The Secrets: Analyzing A Parabola's Graph

Hey math enthusiasts! Let's dive into the fascinating world of parabolas. We're going to break down the function f(x) = -3x² + 36x - 119, which, when we rewrite it in vertex form, becomes f(x) = -3(x - 6)² - 11. Our mission? To uncover the truths about its graph. We will discuss its characteristics, and how to identify key features. So, buckle up, and let's unravel this mathematical mystery together! We'll be looking at some multiple-choice answers, so it's a bit like a math treasure hunt. Get ready to flex those brain muscles and have some fun with the curve!

Unmasking the Vertex: The Heart of the Parabola

First things first, let's talk about the vertex. The vertex is like the main headquarters of the parabola. It is the point where the parabola changes direction, the point of no return! In our vertex form equation, f(x) = -3(x - 6)² - 11, the vertex reveals itself quite nicely. Notice how it's structured: a(x - h)² + k. In this format, the vertex's coordinates are simply (h, k). Easy peasy! Looking at our equation, h is 6 and k is -11. So, the vertex is at the coordinates (6, -11), not (-6, -11). That is a bit of a trick, isn't it? Keep in mind the sign flips in the equation! Therefore, we can immediately discard the statement claiming the vertex is at (-6, -11). The vertex is the most crucial part of any parabola, so understanding this helps with other mathematical problems. This point is critical because it tells us the maximum or minimum value of the function, depending on whether the parabola opens downwards or upwards. The coordinates of the vertex also help with graphing the equation, as it is the turning point of the curve. Getting the vertex correct is like setting the foundation of a house. Without this key detail, the rest of the work will be more difficult to achieve, or even impossible. This also helps you discover the axis of symmetry! The axis of symmetry is a vertical line that passes through the vertex. Without knowing the vertex you will never find the axis of symmetry. The vertex also helps you find the domain and the range of the equation. So yeah, finding the vertex is pretty important, and you will use this concept in other fields as well.

Maximum or Minimum? The Parabola's Direction

Now, let's talk about whether our parabola is a happy face or a sad face. Does it have a minimum point, or a maximum point? This is where the leading coefficient (the number in front of the x²) comes into play. If the leading coefficient is negative (like our -3), the parabola opens downwards. Imagine a frown; it has a maximum point at the vertex. If the leading coefficient is positive, the parabola opens upwards, like a smile, and has a minimum point at the vertex. The sign of the leading coefficient is like the secret code telling us which direction the parabola opens. In our equation, the -3 tells us it opens downwards, meaning it has a maximum value. Therefore, it does not have a minimum. So, we can eliminate the statement claiming it has a minimum, as that's simply not true! Understanding whether a parabola has a maximum or a minimum is important in many real-world applications. For instance, if you're an engineer designing a bridge, you'll need to know the maximum load it can bear, which can be modeled using a parabola. Or if you are looking to find the distance of a thrown ball. These are very important applications that use this very simple concept. The sign of the leading coefficient is like a superpower, letting us instantly know the parabola's orientation. This concept is simple, but extremely useful, and you will use it many times over in the future. Now, we are one step closer to solving the puzzle and understanding the concept to the fullest.

Exploring the Parabola's Behavior

Let's keep going and look at some more aspects of the graph. We already know the vertex is at (6, -11) and that the parabola opens downwards. This means the parabola will have a maximum value at the vertex. Because of this, we know that the range will be all values less than or equal to -11. This also affects the axis of symmetry. It is always a vertical line that passes through the vertex. We can easily find the axis of symmetry, because we know the x-coordinate of the vertex! The axis of symmetry is the line x = 6. Understanding the axis of symmetry is vital for sketching the parabola and understanding its symmetrical nature. We also need to understand the zeros of the function, which are the points where the parabola crosses the x-axis. To find these, we would need to set f(x) = 0 and solve for x. However, due to the position of the vertex and the direction the parabola opens, we know that it does not cross the x-axis, and therefore it has no real zeros. This all gives us a complete picture of the behavior of the parabola and helps with many types of math problems. We know that the domain of this function is all real numbers, because any value of x can be used in the function. Also, the function is continuous. This means there are no breaks or holes in the curve. That also helps us easily visualize and analyze the function. All these characteristics describe the function and help us better understand it.

Conclusion: Selecting the Correct Statements

Based on our analysis, we can select the correct statements about the graph of f(x) = -3(x - 6)² - 11: the vertex is at (6, -11). Also, the parabola has a maximum. The other statements are incorrect. This parabola's behavior is dictated by its vertex, its direction (downwards), and its position on the coordinate plane. It has a maximum value and does not cross the x-axis. Using this information, we have solved the problem! Congratulations! Keep practicing and you will get better. Math can be tricky, but if you approach it systematically, you can solve any problem. By understanding these concepts, you're not just answering a question; you're building a solid foundation in algebra. Keep up the excellent work, and never stop exploring the fascinating world of mathematics! Understanding these fundamental concepts will give you the tools to solve all types of math problems. Remember that practice is key, and with each problem you solve, you are getting better and more confident. If you have questions, never be afraid to ask, as there are many resources available to assist you. Have fun, and enjoy the journey!