Find The Function With X-Intercepts At (0,0) And (4,0)

Hey guys! Let's dive into a fun math problem where we need to figure out which function crosses the x-axis at specific points. We're looking for a function that has x-intercepts at both (0,0) and (4,0). What does that actually mean, though? Well, an x-intercept is simply the point where the graph of the function crosses the x-axis. At these points, the y-value (or f(x) value) is always zero. So, we're essentially searching for a function that equals zero when x is 0 and when x is 4. This understanding is the key to unlocking this problem. Now, let's take a look at the options we have and break them down one by one. We'll explore how each function behaves at x = 0 and x = 4 to determine if it fits the criteria. It's like a detective game, but with math! We will use our knowledge of factored form to make quick work of identifying the correct x-intercepts. Remember, the factored form of a quadratic equation makes it super easy to spot the roots, which are the same as the x-intercepts. So, buckle up, and let's get started on this mathematical adventure!

Analyzing the Functions

Okay, let's get started by meticulously analyzing each function to pinpoint the one that gracefully intersects the x-axis at (0,0) and (4,0). Our mission, should we choose to accept it (and we do!), is to substitute x = 0 and x = 4 into each function and observe whether the output, f(x), obediently becomes zero. If it does, we've potentially found our culprit! Let's begin with the first function, f(x) = x(x - 4). When x = 0, we have f(0) = 0(0 - 4) = 0. Bingo! It passes the first test. Now, let's try x = 4: f(4) = 4(4 - 4) = 4(0) = 0. Double bingo! This function looks incredibly promising. But hold your horses, we can’t declare it the winner just yet. We need to rigorously examine the other options to be absolutely certain. Next up, we have f(x) = x(x + 4). Plugging in x = 0, we get f(0) = 0(0 + 4) = 0. Check! It hits the x-axis at (0,0). However, when we substitute x = 4, we find f(4) = 4(4 + 4) = 4(8) = 32. Oops! That's definitely not zero. So, this function is out of the running. Let's move on to the third contender, f(x) = (x - 4)(x - 4). If we set x = 0, then f(0) = (0 - 4)(0 - 4) = (-4)(-4) = 16. Strike one! This function doesn't pass through (0,0). Although we could continue to test x=4, we already know this function is not the one we are looking for. Finally, we have f(x) = (x + 4)(x + 4). Substituting x = 0, we get f(0) = (0 + 4)(0 + 4) = (4)(4) = 16. Another strike! This one also fails to cross the x-axis at (0,0). So, after carefully scrutinizing all the options, it's crystal clear that the function f(x) = x(x - 4) is the only one that satisfies our conditions. It gracefully dances through the x-axis at both (0,0) and (4,0). Mission accomplished!

Why f(x) = x(x-4) is the Answer

So, why is f(x) = x(x - 4) the undisputed champion in our quest for the function with x-intercepts at (0,0) and (4,0)? Let's break it down a bit further, guys. The magic lies in the factored form of the equation. When a quadratic function is written in factored form, like our f(x) = x(x - 4), the roots (or x-intercepts) are staring right back at us! Remember, roots are the values of x that make the function equal to zero. In this case, we have two factors: 'x' and '(x - 4)'. Setting each factor to zero gives us our roots. If x = 0, then the whole function becomes zero. That's our first x-intercept, (0,0). And if (x - 4) = 0, then x = 4, giving us our second x-intercept, (4,0). See how neatly it works? The factored form directly reveals where the function crosses the x-axis. This is super useful because it lets us quickly identify x-intercepts without having to graph the function or do any complicated calculations. Now, let's contrast this with the other options. Functions like f(x) = x(x + 4) have a root at x = 0 (which is good), but the other root is at x = -4, not x = 4. The functions f(x) = (x - 4)(x - 4) and f(x) = (x + 4)(x + 4) are a slightly different breed. These are perfect square trinomials, which means they only have one x-intercept. They touch the x-axis at one point and then bounce back, rather than crossing through. In the case of f(x) = (x - 4)(x - 4), the x-intercept is at (4,0), but it doesn't pass through (0,0). And f(x) = (x + 4)(x + 4) has an x-intercept at (-4,0), missing both our target points. So, the factored form of f(x) = x(x - 4) not only makes it easy to spot the x-intercepts but also highlights why the other functions don't fit the bill. It's all about understanding how the structure of the equation dictates the behavior of the graph!

Graphing the Function

To really solidify our understanding, let's take a moment to visualize the function f(x) = x(x - 4). Graphing this function provides a fantastic visual confirmation of our algebraic findings. When you graph f(x) = x(x - 4), you'll immediately see a parabola, a U-shaped curve, that's characteristic of quadratic functions. The most important thing to notice is that this parabola intersects the x-axis precisely at (0,0) and (4,0). It's like a visual high-five for our solution! The parabola opens upwards, which makes sense because the coefficient of the x² term (if we were to expand the function) is positive. This upward opening tells us that the function has a minimum value somewhere between our x-intercepts. The vertex of the parabola, which is the point where it changes direction, lies exactly halfway between the x-intercepts. In this case, the x-coordinate of the vertex is (0 + 4) / 2 = 2. To find the y-coordinate of the vertex, we plug x = 2 back into our function: f(2) = 2(2 - 4) = 2(-2) = -4. So, the vertex is at (2, -4), which is the lowest point on the graph. Graphing the function also gives us a broader sense of its behavior. We can see how the function is negative between x = 0 and x = 4 (the part of the parabola below the x-axis) and positive everywhere else (the parts above the x-axis). This visual representation is incredibly helpful in understanding the relationship between the equation, its roots, and the overall shape of the graph. It's like seeing the function come to life! By visualizing f(x) = x(x - 4), we not only confirm our solution but also gain a deeper appreciation for the beauty and interconnectedness of mathematics.

Key Takeaways

Alright, guys, we've journeyed through this problem, dissected the functions, and even visualized the solution. Now, let's distill the key takeaways from our mathematical adventure. Firstly, and perhaps most importantly, understanding the definition of an x-intercept is crucial. Remember, an x-intercept is simply the point where a graph crosses the x-axis, and at these points, the y-value (or f(x) value) is always zero. This understanding forms the bedrock for solving problems like this. Secondly, the factored form of a quadratic equation is your best friend when it comes to identifying x-intercepts. When a function is expressed as a product of factors, the roots (which are the same as the x-intercepts) are readily apparent. Just set each factor equal to zero and solve for x! This technique saves you a ton of time and effort compared to graphing or using the quadratic formula. Thirdly, don't be afraid to test the options! In this case, we systematically substituted x = 0 and x = 4 into each function to see if the output was zero. This methodical approach, while seemingly simple, is a powerful problem-solving strategy. It allows you to eliminate incorrect choices and zero in on the correct answer. Fourthly, visualizing the function through graphing can provide invaluable insights. A graph can confirm your algebraic solution and give you a more intuitive understanding of the function's behavior. Finally, remember that quadratic functions, with their characteristic parabolic shape, have either two, one, or zero x-intercepts. The number of x-intercepts depends on the discriminant of the quadratic equation, but we don't need to delve into that here. The main thing is to grasp the connection between the equation, its factored form, its roots, and its graph. By keeping these key takeaways in mind, you'll be well-equipped to tackle similar problems with confidence and flair! So keep practicing, keep exploring, and most importantly, keep enjoying the fascinating world of mathematics!