Finding The Vertex And Range Of Y = |x+3| + 2 Absolute Value Function
Determining the vertex and range of the absolute value function y = |x + 3| + 2 is a fundamental concept in algebra. This article aims to provide a comprehensive explanation of how to identify these key features, enhancing your understanding of absolute value functions and their graphical representations. We will dissect the function, explain the transformations involved, and arrive at the correct vertex and range. This understanding is crucial not only for academic success in mathematics but also for various applications in fields like physics and engineering where absolute value functions are used to model real-world scenarios.
Decoding the Absolute Value Function
At its core, the absolute value function, denoted as |x|, returns the non-negative magnitude of a number. In simpler terms, it gives the distance of a number from zero on the number line. For instance, |3| = 3 and |-3| = 3. Understanding this basic principle is crucial for analyzing more complex absolute value functions like the one presented in the question, y = |x + 3| + 2. This function is a transformation of the basic absolute value function, |x|, and these transformations are what determine its unique vertex and range.
Transformations of Absolute Value Functions
The function y = |x + 3| + 2 involves two key transformations applied to the basic absolute value function, y = |x|:
- Horizontal Shift: The term inside the absolute value, x + 3, causes a horizontal shift. Specifically, adding 3 to x shifts the graph 3 units to the left. This is because the function now reaches its minimum value (where the absolute value part becomes zero) when x = -3. Understanding horizontal shifts is essential as it directly impacts the x-coordinate of the vertex.
- Vertical Shift: The addition of 2 outside the absolute value, represented by the '+ 2' in the function, causes a vertical shift. This shifts the entire graph 2 units upwards. This vertical shift directly influences the y-coordinate of the vertex and the lower bound of the range.
By recognizing these transformations, we can start to visualize the graph of the function and anticipate the location of its vertex and the extent of its range. The ability to identify transformations is a powerful tool in analyzing various types of functions, not just absolute value functions.
Identifying the Vertex
The vertex of an absolute value function is the point where the graph changes direction, forming a 'V' shape. For the basic absolute value function y = |x|, the vertex is at the origin (0, 0). However, transformations shift this vertex, and understanding these shifts is key to finding the vertex of the given function, y = |x + 3| + 2. To find the vertex, we need to determine the point where the expression inside the absolute value becomes zero, as this is where the function reaches its minimum value.
Determining the x-coordinate
The x-coordinate of the vertex is found by setting the expression inside the absolute value to zero and solving for x:
- x + 3 = 0
- x = -3
This tells us that the vertex is shifted 3 units to the left from the origin, a direct result of the horizontal transformation.
Determining the y-coordinate
The y-coordinate of the vertex is the value of the function when x = -3. Substituting x = -3 into the function, we get:
- y = |-3 + 3| + 2
- y = |0| + 2
- y = 2
This indicates that the vertex is also shifted 2 units upwards due to the vertical transformation. Therefore, the vertex of the function y = |x + 3| + 2 is (-3, 2). This point is the lowest point on the graph of the function, a crucial piece of information for determining the range.
Determining the Range
The range of a function is the set of all possible output values (y-values) that the function can produce. For absolute value functions, the range is limited by the vertex, as the function will always produce values greater than or equal to the y-coordinate of the vertex. The absolute value ensures that the output is always non-negative, and the vertical shift further influences the range. Considering the vertex we found earlier, we can now deduce the range of the function y = |x + 3| + 2.
Understanding the Lower Bound
Since the absolute value of any number is always greater than or equal to zero, the minimum value of |x + 3| is 0. This occurs when x = -3. The '+ 2' in the function then shifts this minimum value upwards by 2 units. Therefore, the smallest possible value of y is 2. This forms the lower bound of the range.
Understanding the Upper Bound
As x moves away from -3 in either direction (either towards positive or negative infinity), the value of |x + 3| increases without bound. Consequently, the value of y also increases without bound. This means there is no upper limit to the possible y-values. The function can theoretically reach infinitely high values as x moves further away from -3.
Expressing the Range
Combining the lower and upper bounds, we can express the range of the function y = |x + 3| + 2 as y ≥ 2. In interval notation, this is written as [2, ∞). This notation signifies that the range includes all real numbers greater than or equal to 2. This understanding of range is crucial for various mathematical applications, including optimization problems and data analysis.
Conclusion
In summary, the vertex of the absolute value function y = |x + 3| + 2 is (-3, 2), and its range is 2 ≤ y < ∞. This was determined by understanding the transformations applied to the basic absolute value function and how these transformations affect the position of the vertex and the possible output values. By correctly identifying these features, we arrive at the answer C. (-3, 2); 2 ≤ y < ∞. Mastering the identification of the vertex and range of absolute value functions is a vital skill in mathematics, laying the foundation for understanding more complex functions and their applications in various fields. The key takeaways are the importance of recognizing horizontal and vertical shifts and how they directly impact the vertex and, consequently, the range of the function. With practice, these concepts become intuitive, allowing for quick and accurate analysis of absolute value functions.
By understanding the underlying principles of absolute value functions and their transformations, we can confidently tackle similar problems and apply this knowledge to more advanced mathematical concepts. This article has provided a detailed explanation, ensuring a solid understanding of the vertex and range of absolute value functions. This knowledge empowers you to analyze and interpret such functions in various contexts, further enhancing your mathematical proficiency.