Unlock The Secrets: Analyzing F(x)=(x+4)(x-6) Graph

Hey guys! Today, we're diving deep into the world of quadratic functions, specifically the function f(x) = (x+4)(x-6). We've got a part of its graph shown, and our mission, should we choose to accept it, is to figure out which statements about this function are actually true. We've got some options to explore, and trust me, it's going to be a fun ride! This is a classic example of the kind of problem you might see in algebra, and understanding these concepts is super crucial for future math adventures.

Delving into the Vertex: Is it (1, -25) or (1, -24)?

Let's kick things off by talking about the vertex. Now, for those of you who might be a little rusty, the vertex is the turning point of a parabola, which is the U-shaped curve that represents a quadratic function. It's either the minimum point (if the parabola opens upwards) or the maximum point (if the parabola opens downwards). Figuring out the vertex is a key step in understanding the behavior of the function.

So, we're given two possible options for the vertex: (1, -25) and (1, -24). How do we figure out which one is the real deal? Well, there are a couple of ways we can tackle this. One way is to find the x-coordinate of the vertex first. Remember that a quadratic function in the form of f(x) = a(x - r)(x - s) has roots (or x-intercepts) at x = r and x = s. In our case, f(x) = (x + 4)(x - 6), so the roots are x = -4 and x = 6. The axis of symmetry, which runs right through the vertex, is exactly halfway between these roots. We can find the x-coordinate of the vertex by averaging the roots: (-4 + 6) / 2 = 1. Alright, so far so good! Both options agree that the x-coordinate of the vertex is indeed 1.

But what about the y-coordinate? This is where things get interesting. To find the y-coordinate, we simply plug the x-coordinate (which we know is 1) back into our original function: f(1) = (1 + 4)(1 - 6) = (5)(-5) = -25. Bam! There it is. The y-coordinate of the vertex is -25. This means the vertex of the function is (1, -25). Option A is looking pretty solid right now, and Option B is definitely out of the running. This process highlights the importance of understanding how the roots of a quadratic equation relate to its vertex and overall shape. By finding the roots and then calculating the average, we efficiently determined the x-coordinate of the vertex. Subsequently, plugging this value back into the original equation allowed us to pinpoint the exact y-coordinate, solidifying our understanding of the function's behavior at its turning point. Remember, the vertex is a critical feature of a parabola, as it dictates the function's minimum or maximum value and provides insight into its symmetry. Mastering the techniques to find the vertex is crucial for analyzing and interpreting quadratic functions effectively. It's not just about memorizing formulas, but about understanding the underlying concepts and how they connect to the graph's visual representation.

Decoding the Graph: Is it Above or Below the x-axis?

Now, let's shift our focus to another crucial aspect of understanding quadratic functions: the graph's position relative to the x-axis. Specifically, we want to determine whether the graph lies above or below the x-axis between its roots. This seemingly simple question holds valuable information about the function's behavior and the sign of its values within a specific interval.

Remember, our function is f(x) = (x + 4)(x - 6), and we've already established that its roots are x = -4 and x = 6. These roots are the points where the parabola intersects the x-axis. Now, the question is: what happens to the graph between these points? Does it dip below the x-axis, indicating negative function values, or does it stay above, implying positive values? To answer this, we can employ a powerful technique: test a point within the interval. Let's choose a convenient value, say x = 0, which lies comfortably between -4 and 6. Plugging this into our function, we get f(0) = (0 + 4)(0 - 6) = (4)(-6) = -24. Aha! The result is negative. This tells us that at x = 0, the graph is below the x-axis. And because parabolas are smooth, continuous curves, if the graph is below the x-axis at one point within the interval between the roots, it will be below the x-axis for the entire interval. This elegant method of testing a single point allows us to deduce the function's behavior across a broader range, showcasing the interconnectedness of different aspects of the function. The sign of the function's value within this interval reveals important information about the parabola's shape and orientation. If the value is negative, the parabola dips below the x-axis, forming a U-shape that opens upwards. Conversely, if the value is positive, the parabola remains above the x-axis, creating an inverted U-shape that opens downwards. This understanding is crucial for sketching the graph of a quadratic function accurately and for solving inequalities involving quadratic expressions.

Furthermore, this concept extends beyond simply determining whether the graph is above or below the x-axis. It allows us to analyze the intervals where the function is positive, negative, or zero, providing a comprehensive understanding of its behavior. By combining this knowledge with the vertex and roots, we can paint a complete picture of the quadratic function and its graphical representation. This holistic approach to analyzing quadratic functions is essential for tackling more complex problems and for applying these concepts in various mathematical and real-world scenarios.

Putting it All Together: Selecting the True Statements

Alright guys, we've dissected the function f(x) = (x + 4)(x - 6) piece by piece. We've pinpointed the vertex, explored the graph's relationship with the x-axis, and now it's time to put it all together and nail down those true statements! Remember, the original question asked us to select two options that accurately describe the function.

We've already established that the vertex of the function is indeed at (1, -25). This means Option A is a definite winner. Option B, with its (1, -24) vertex claim, is out. Now, let's think about what else we've learned. We know the graph intersects the x-axis at x = -4 and x = 6. We also discovered that the graph dips below the x-axis between these roots. This means the function's values are negative in this interval. This aligns perfectly with the fact that the vertex, (1, -25), has a negative y-coordinate, further confirming that the parabola opens upwards.

So, what's the big takeaway here? Understanding the key features of a quadratic function – the vertex, the roots, and the graph's behavior relative to the x-axis – is crucial for making accurate statements about it. By systematically analyzing these elements, we can confidently navigate through different options and identify the truths hidden within the function's equation. This isn't just about memorizing formulas; it's about developing a deep conceptual understanding of how quadratic functions work. This understanding empowers you to not only solve problems but also to appreciate the elegance and interconnectedness of mathematical concepts.

In the grand scheme of things, mastering quadratic functions is a fundamental step in your mathematical journey. They pop up in various applications, from physics to engineering to economics. So, the time and effort you invest in understanding them now will pay dividends down the road. Keep practicing, keep exploring, and keep asking questions. You've got this!