Identifying Functions With A Vertex At X = 0

Hey everyone! Let's dive into some cool math stuff today, specifically focusing on functions and their vertices. We're going to explore a few absolute value functions and figure out which ones have a vertex where the x-value is 0. Don't worry, it's not as scary as it sounds! We'll break it down step by step, and by the end, you'll be able to spot these functions like a pro. So, grab your pencils, and let's get started. Understanding the vertex of a function is crucial in mathematics, as it provides key information about the function's behavior, especially in quadratic and absolute value functions. The vertex represents either the minimum or maximum point of the function, which helps in sketching the graph and analyzing its properties. In this case, we are only focusing on those absolute value functions with the vertex located at x = 0. The importance lies in recognizing the relationship between the function's equation and the location of its vertex. This skill is fundamental for understanding transformations of functions, where the vertex acts as a reference point for shifts and stretches.

Let's consider the functions provided: $f(x)=|x|$, $f(x)=|x|+3$, $f(x)=|x+3|$, $f(x)=|x|-6$, and $f(x)=|x+3|-6$. Each of these functions is an absolute value function, which means their graphs will have a characteristic V-shape. The vertex of an absolute value function is the point where the direction of the graph changes, it's the tip of the V. Our task is to determine which of these functions have their vertex located at an x-value of 0. Remember that the vertex form of an absolute value function is generally written as $f(x) = a|x - h| + k$, where (h, k) is the vertex of the function. Therefore, the function's vertex can be easily determined once we recognize this form. When h is 0, the vertex's x-coordinate is also 0. So, we'll analyze each function to see if it fits this pattern, paying close attention to any horizontal shifts. The absolute value function is a fundamental concept in algebra, frequently used to model real-world scenarios, such as distance from a reference point or the magnitude of an error. Being able to quickly identify the vertex of an absolute value function is a very valuable skill, especially when dealing with applications or complex problems.

Analyzing the Functions

Alright, let's get down to the nitty-gritty and analyze each function individually. We'll look at each one, identify its vertex, and see if the x-value of the vertex is 0. Keep in mind that we're dealing with absolute value functions, which have a characteristic V-shape, and the vertex is the point where this V changes direction. This is a crucial step in understanding the function's graph and its transformations. By looking at how the function is written, we can determine its vertex. The vertex form of an absolute value function, as we mentioned earlier, is f(x) = a|x - h| + k, where (h, k) is the vertex. The constant a determines the function's stretching or compressing behavior, while h and k define the horizontal and vertical shifts. Recognizing this form makes it easier to pinpoint the vertex without having to graph the function. This is especially helpful in standardized tests and real-world applications where time is of the essence. Let's see what we can find, and which function has x=0 for its vertex. Let's do this!

Function 1: $f(x) = |x|$

This is the simplest absolute value function, and it's a great starting point. In the function $f(x) = |x|$, there are no added or subtracted values inside or outside the absolute value. This function is in the form f(x) = |x - h| + k where h = 0 and k = 0. So, we can directly determine that the vertex of this function is at (0, 0). The vertex is located at the origin. Since the x-value of the vertex is 0, this function is one of the ones we're looking for! This function serves as the base function for understanding transformations of absolute value functions. When other functions are compared to f(x) = |x|, their changes become clearer. So, knowing this function will help us with the other ones. The graph of $f(x) = |x|$ is a V-shape, symmetrical about the y-axis, with its lowest point (vertex) at the origin. Understanding how the vertex is affected by transformations will help us understand the behavior of more complex absolute value functions.

Function 2: $f(x) = |x| + 3$

Here, we have a vertical shift. The function $f(x) = |x| + 3$ can be rewritten as f(x) = |x - 0| + 3. In this case, h = 0 and k = 3, so the vertex is at (0, 3). Since the x-value of the vertex is 0, this function also meets our criteria. The addition of 3 outside the absolute value shifts the entire graph upwards by 3 units. Notice how the shape of the graph is exactly the same as the original, but just moved. This is a vertical transformation; only the y-value of the vertex changes. This function is an example of a vertical transformation of the parent function $f(x) = |x|$. The vertex is directly influenced by the constant outside the absolute value. Therefore, it's very easy to find the vertex here.

Function 3: $f(x) = |x + 3|$

Now, we have a horizontal shift. The function $f(x) = |x + 3|$ can be rewritten as f(x) = |x - (-3)| + 0. Thus, h = -3 and k = 0. This means the vertex is at (-3, 0). The x-value of the vertex is -3, which is not 0, so this function is not what we're looking for. The graph is shifted 3 units to the left compared to the graph of $f(x) = |x|$. Understanding this horizontal shift is key to determining the vertex's location. This function demonstrates a horizontal transformation, where the vertex moves along the x-axis. Since the x-value is not 0, we can skip it. This shows us how the position of the vertex changes when a value is added or subtracted inside the absolute value. The graph will look the same, but it will be moved horizontally.

Function 4: $f(x) = |x| - 6$

Again, we have a vertical shift. The function $f(x) = |x| - 6$ is like f(x) = |x - 0| - 6, where h = 0 and k = -6. So, the vertex is at (0, -6). The x-value of the vertex is 0, so this function fits our criteria. This is a downward shift of 6 units compared to the graph of $f(x) = |x|$. The shape of the graph stays the same. The change in the y-value indicates the vertical shift of the function, while the x-value remains unaffected. The graph will be moved downwards, but the x-value will remain at 0.

Function 5: $f(x) = |x + 3| - 6$

This function has both horizontal and vertical shifts. We can rewrite it as f(x) = |x - (-3)| - 6, which gives us h = -3 and k = -6. The vertex is located at (-3, -6). The x-value of the vertex is -3, so this is not a function that we are looking for. Here, the graph of $f(x) = |x|$ has been shifted both horizontally (3 units to the left) and vertically (6 units down). This shows how the vertex is affected by both horizontal and vertical transformations. To determine the vertex, we must consider both the terms inside and outside the absolute value. Since the x-value is not 0, we can exclude this function from our answer. This is a very useful example of how the h and k values change the overall appearance and position of the function's graph.

Conclusion: The Functions with a Vertex at x = 0

Alright guys, we've analyzed all the functions, and now we know which ones have a vertex with an x-value of 0. Based on our analysis, the functions are: $f(x) = |x|$, $f(x) = |x| + 3$, and $f(x) = |x| - 6$. We were able to identify them by recognizing the relationships between the function's equation and the vertex's coordinates. Understanding how transformations change the function is key. The vertex is essential in graphing the function. Great job sticking with it! Understanding how these transformations work helps build a strong foundation for more complex mathematical concepts.

So there you have it! You've successfully identified the functions that meet our criteria. Keep practicing, and you'll become a master of absolute value functions in no time! Keep an eye on the horizontal shifts, and you'll be set. I hope this was helpful! Good luck, and keep learning!