Finding Functions With A Single X-Intercept At (-6,0)

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Hey guys! Today, we're diving into the fascinating world of functions and their x-intercepts. Specifically, we're on a quest to identify the function that boasts a single, solitary x-intercept nestled snugly at the point (-6, 0). This might sound like a daunting task, but fear not! We'll break it down step-by-step, making sure you grasp the underlying concepts and emerge with a solid understanding. So, buckle up, and let's embark on this mathematical adventure!

Understanding X-Intercepts

Before we jump into the specific functions, let's quickly recap what x-intercepts actually are. Think of the x-intercept as the point where a function's graph crosses the x-axis. At this special point, the function's value, often denoted as f(x) or y, is precisely zero. In simpler terms, to find the x-intercepts, we need to solve the equation f(x) = 0. The solutions to this equation will be the x-coordinates of the points where the graph intersects the x-axis. Each of these points represents an x-intercept. Understanding this fundamental concept is crucial for tackling the problem at hand. So, remember, x-intercepts are where the function's value dips down to zero, marking the spot where the graph kisses the x-axis. Now, with this core idea firmly in place, we can confidently move forward and explore the functions presented to us, carefully analyzing their behavior to pinpoint the one with the unique property of having only one x-intercept at (-6, 0).

Now that we have a solid grasp of what x-intercepts are, let's consider how we find them. As mentioned earlier, we find x-intercepts by setting the function, f(x), equal to zero and solving for x. This makes sense because at the x-intercept, the y-value (which is the same as f(x)) is always zero. Think about the coordinate plane – the x-axis is defined as the line where y = 0. So, any point that lies on the x-axis will have a y-coordinate of 0, and this is exactly what an x-intercept represents. Once we solve the equation f(x) = 0, we'll get the x-values where the function crosses the x-axis. These x-values are the x-coordinates of our x-intercepts. We can then write the x-intercepts as ordered pairs in the form (x, 0), since the y-coordinate is always zero at an x-intercept. This method provides us with a clear and straightforward way to determine the x-intercepts of any function, which is essential for our quest to find the function with a lone x-intercept at (-6, 0).

Why are x-intercepts important, you might ask? Well, they tell us a great deal about the behavior of a function and its graph. X-intercepts can represent key points in real-world scenarios modeled by functions. For instance, in a physics context, an x-intercept might represent the time when a projectile hits the ground. In business, it could represent the break-even point where costs equal revenue. Mathematically, x-intercepts are also closely related to the roots or zeros of a function, which are the values of x that make the function equal to zero. Finding these roots is a fundamental task in algebra and calculus, and understanding x-intercepts provides a visual and intuitive way to approach this task. Furthermore, the number and location of x-intercepts can give us clues about the shape and characteristics of a function's graph. For example, a quadratic function (a function of the form f(x) = ax² + bx + c) can have zero, one, or two x-intercepts, depending on the discriminant (b² - 4ac). This information helps us visualize the parabola and its relationship to the x-axis. So, x-intercepts aren't just abstract mathematical concepts; they're powerful tools that help us understand and interpret functions and their applications in various fields.

Analyzing the Functions

Now, let's roll up our sleeves and dive into the given functions. We have four contenders vying for the title of the function with a single x-intercept at (-6, 0). We'll analyze each one methodically, using our understanding of x-intercepts and how to find them. Remember, our goal is to set each function equal to zero and solve for x. The solutions we obtain will be the x-coordinates of the x-intercepts. We'll be paying close attention to both the number of x-intercepts and their location. This meticulous approach will allow us to identify the function that uniquely satisfies our criteria – one x-intercept, and that x-intercept must be at (-6, 0). Let's get started!

Function 1: f(x) = x(x - 6)

Let's begin with the first function: f(x) = x(x - 6). To find the x-intercepts, we set f(x) equal to zero: x(x - 6) = 0. This equation is already factored, which makes our job much easier! According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. So, we have two possibilities: either x = 0 or x - 6 = 0. Solving the second equation for x, we get x = 6. Therefore, this function has two x-intercepts: one at x = 0 (corresponding to the point (0, 0)) and another at x = 6 (corresponding to the point (6, 0)). Since this function has two x-intercepts, it doesn't fit our criteria of having only one x-intercept at (-6, 0). So, we can eliminate this one from our list. Remember, we're looking for a function that has a unique x-intercept located precisely at (-6, 0). This function, with its two distinct x-intercepts, doesn't match our desired profile. But don't worry, we still have three more functions to investigate, and one of them might just be the perfect fit!

Function 2: f(x) = (x - 6)(x - 6)

Next up, we have the function f(x) = (x - 6)(x - 6), which can also be written as f(x) = (x - 6)². Again, we set f(x) equal to zero to find the x-intercepts: (x - 6)(x - 6) = 0. This equation has a repeated factor of (x - 6). Setting this factor equal to zero, we get x - 6 = 0, which gives us x = 6. However, since this factor is repeated, we say that x = 6 is a root of multiplicity 2. This means that the graph of the function touches the x-axis at x = 6 but doesn't cross it. In terms of x-intercepts, we still only have one x-intercept, located at x = 6 (corresponding to the point (6, 0)). While this function has only one x-intercept, it's not at the location we're looking for. We need an x-intercept at (-6, 0), and this function's sole x-intercept is at (6, 0). So, we can eliminate this function as well. We're getting closer, but we haven't found our match yet. Let's keep going! The key takeaway here is the concept of multiplicity. A repeated root, like in this case, indicates that the graph touches the x-axis at that point but doesn't pass through it. This distinction is important when analyzing the behavior of functions and their graphs.

Function 3: f(x) = (x + 6)(x - 6)

Moving on to the third function, f(x) = (x + 6)(x - 6). To find the x-intercepts, we set f(x) equal to zero: (x + 6)(x - 6) = 0. This equation is also factored, making our task straightforward. Using the zero-product property, we set each factor equal to zero: x + 6 = 0 and x - 6 = 0. Solving these equations, we get x = -6 and x = 6. This means that the function has two x-intercepts: one at x = -6 (corresponding to the point (-6, 0)) and another at x = 6 (corresponding to the point (6, 0)). While this function does have an x-intercept at (-6, 0), it also has another x-intercept at (6, 0). Since we're looking for a function with only one x-intercept at (-6, 0), this function doesn't fit the bill. We're narrowing down our options, though! We have just one function left to analyze. Remember, the zero-product property is a powerful tool for solving equations where a product of factors is equal to zero. It allows us to break down the equation into simpler parts, making it easier to find the solutions.

Function 4: f(x) = (x + 6)(x + 6)

Finally, we arrive at the last contender: f(x) = (x + 6)(x + 6), which can also be written as f(x) = (x + 6)². To find the x-intercepts, we set f(x) equal to zero: (x + 6)(x + 6) = 0. This equation has a repeated factor of (x + 6). Setting this factor equal to zero, we get x + 6 = 0, which gives us x = -6. Since this factor is repeated, x = -6 is a root of multiplicity 2, just like we saw in the second function. This means that the graph touches the x-axis at x = -6 but doesn't cross it. In terms of x-intercepts, we have only one x-intercept, and it's located at x = -6 (corresponding to the point (-6, 0)). This is exactly what we've been searching for! This function has only one x-intercept, and that x-intercept is at (-6, 0). We've found our winner! Congratulations! We successfully identified the function that meets the specified criteria.

The Verdict

After carefully analyzing each function, we've arrived at a definitive answer. The function that has only one x-intercept at (-6, 0) is:

f(x) = (x + 6)(x + 6) or f(x) = (x + 6)²

We successfully navigated through the functions, applying our understanding of x-intercepts and the zero-product property. Remember, the key to solving this problem was recognizing that a repeated factor leads to a single x-intercept, and the value within the factor determines the location of that intercept. This exercise highlights the importance of a methodical approach and a solid grasp of fundamental mathematical concepts. Great job, guys! You've conquered the x-intercept challenge!

Key Takeaways

Let's solidify our understanding by summarizing the key concepts we've learned today:

  • X-intercepts: X-intercepts are the points where a function's graph crosses the x-axis. At these points, the value of the function, f(x), is zero.
  • Finding X-intercepts: To find the x-intercepts of a function, set f(x) = 0 and solve for x. The solutions are the x-coordinates of the x-intercepts.
  • Zero-Product Property: The zero-product property states that if the product of two or more factors is zero, then at least one of the factors must be zero. This property is crucial for solving factored equations.
  • Multiplicity: A repeated root (a factor that appears multiple times) indicates that the graph touches the x-axis at that point but doesn't cross it. This results in a single x-intercept at that location.
  • The Function: The function with only one x-intercept at (-6, 0) is f(x) = (x + 6)(x + 6) or f(x) = (x + 6)².

By mastering these concepts, you'll be well-equipped to tackle a wide range of problems involving functions and their graphs. Keep practicing, and you'll become a true x-intercept expert!

Practice Problems

To further hone your skills, try solving these practice problems:

  1. Which function has only one x-intercept at (2, 0)?

    • f(x) = x(x - 2)
    • f(x) = (x - 2)(x - 2)
    • f(x) = (x + 2)(x - 2)
    • f(x) = (x + 2)(x + 2)

  2. Find the x-intercepts of the function f(x) = x² - 4x + 3.

  3. Does the function f(x) = x² + 2x + 1 have one x-intercept, two x-intercepts, or no x-intercepts? Explain your reasoning.

By working through these problems, you'll deepen your understanding of x-intercepts and solidify your problem-solving abilities. Remember, practice makes perfect! So, grab a pencil and paper, and get ready to conquer these challenges!

Conclusion

We've reached the end of our journey to find the function with a single x-intercept at (-6, 0). Through careful analysis and a solid understanding of key concepts, we successfully identified the winner: f(x) = (x + 6)(x + 6) or f(x) = (x + 6)². I hope this exploration has shed light on the fascinating world of functions and their x-intercepts. Remember, mathematics is not just about memorizing formulas; it's about understanding the underlying principles and applying them to solve problems. So, keep exploring, keep questioning, and keep learning! Until next time, happy problem-solving, guys!