Analyzing F(x) = -2/3|x+4| - 6: True Or False?

Hey guys! Today, we're diving deep into the fascinating world of absolute value functions, specifically the function f(x) = -2/3|x+4| - 6. We'll dissect this equation piece by piece to uncover its key features and determine which statement about it rings true. Buckle up, because we're about to embark on a mathematical adventure! Let's analyze each aspect of the function and the related statements step by step.

Understanding the Parent Function: The Foundation of Our Exploration

Before we can truly understand our given function, f(x) = -2/3|x+4| - 6, we need to grasp the concept of the parent function. The parent function for absolute value functions is simply f(x) = |x|. This function forms a 'V' shape, with its vertex (the pointy bottom) sitting neatly at the origin (0, 0). The two arms of the 'V' extend outwards at a 45-degree angle, creating a symmetrical form. Think of it as the basic blueprint upon which all other absolute value functions are built. We need to understand the behavior of this parent function because all transformations applied to it will result in our final graph. So, focusing on the parent function, the simplest form of an absolute value function, is crucial for understanding more complex transformations. This base knowledge gives us a standard against which to compare the transformations we'll see in our specific function, f(x) = -2/3|x+4| - 6. This understanding of the parent function is crucial because the transformations—shifts, stretches, reflections—all build upon this foundational form. For example, consider the parent function f(x) = |x| and how the absolute value ensures that any input, whether positive or negative, produces a non-negative output. This results in the distinctive 'V' shape. The left side mirrors the right side, creating the symmetry characteristic of absolute value functions. Now, as we delve into the transformations, we'll see how each component of our given function manipulates this basic 'V' shape. Think of the parent function as the original mold, and the transformations as the adjustments that reshape and refine it into something new. It’s like starting with a basic clay figure and then molding it into a detailed sculpture. Without understanding the original form, it's much harder to appreciate the artistry of the final piece. In essence, by mastering the characteristics of the parent function, we equip ourselves with the necessary tools to decode the intricacies of more complex absolute value functions and confidently analyze their properties.

Decoding the Transformations: A Step-by-Step Analysis of f(x) = -2/3|x+4| - 6

Now, let's break down our function, f(x) = -2/3|x+4| - 6, and see how it transforms the parent function. This is where things get really interesting! We have three key components to consider: the -2/3 coefficient, the +4 inside the absolute value, and the -6 constant term. Each of these plays a distinct role in shaping the final graph. First, let's tackle the -2/3 coefficient. The negative sign is a big clue: it tells us that the graph will be reflected across the x-axis. Imagine flipping the 'V' shape upside down – that's what this negative sign does. The 2/3 part of the coefficient affects the vertical stretch or compression. Since 2/3 is between 0 and 1, it causes a vertical compression, making the 'V' shape wider than the parent function. So, the coefficient -2/3 combines a reflection and a compression. It's like taking the original 'V' shape, mirroring it, and then squishing it down a bit. Next, we have the +4 inside the absolute value. This is a horizontal shift. Remember, transformations inside the absolute value (affecting the 'x' value) work in the opposite direction of what you might expect. So, +4 actually shifts the graph 4 units to the left. This is a crucial point to remember – the sign inside the absolute value dictates the direction of the horizontal shift. It’s like moving the entire 'V' shape sideways along the x-axis. Finally, we have the -6 constant term. This is a vertical shift. It moves the entire graph down by 6 units along the y-axis. This is a straightforward shift, moving the whole function up or down. In summary, f(x) = -2/3|x+4| - 6 takes the parent function, reflects it, compresses it vertically, shifts it 4 units left, and shifts it 6 units down. By understanding these transformations, we can visualize the graph's final form and deduce its key properties. These transformations are the key to understanding the behavior of any absolute value function. So, mastering this process of decoding each component is essential for confidently analyzing and graphing these types of functions.

Evaluating the Statements: Which One Holds True?

Okay, now that we've dissected the function f(x) = -2/3|x+4| - 6, let's revisit the statements and see which one accurately describes it. This is where we put our newfound knowledge to the test! We'll go through each statement, armed with our understanding of transformations, and determine its validity.

Statement 1: The graph of f(x) has a vertex of (-4, 6).

Remember how we discussed the horizontal and vertical shifts? The +4 inside the absolute value shifts the graph 4 units to the left, and the -6 shifts it 6 units down. The vertex of the parent function is at (0, 0). Therefore, these shifts will move the vertex. A shift of 4 units left means the x-coordinate of the vertex will be -4. A shift of 6 units down means the y-coordinate will be -6. However, the reflection across the x-axis doesn't change the x-coordinate of the vertex, but it does impact the y-coordinate. So, the vertex will be at (-4, -6), not (-4, 6). Therefore, this statement is FALSE. It's crucial to remember that the vertex is the point that's directly affected by the shifts and reflections. By carefully tracing the impact of each transformation, we can accurately determine the vertex's new location.

Statement 2: The graph of f(x) is a horizontal stretch of the graph of the parent function.

We know that the coefficient -2/3 causes a vertical compression and a reflection. It affects the vertical aspect of the graph, making it wider or narrower vertically. A horizontal stretch or compression would be caused by a coefficient inside the absolute value, affecting the 'x' value. Since there's no such coefficient directly multiplying the 'x' inside the absolute value, there's no horizontal stretch. Therefore, this statement is FALSE. It's important to distinguish between vertical and horizontal stretches and compressions. A vertical stretch changes the height of the graph, while a horizontal stretch changes its width. In this case, the coefficient outside the absolute value only affects the vertical aspect.

Statement 3: The graph of f(x) opens upward.

The negative sign in front of the 2/3 coefficient is the key here. It indicates a reflection across the x-axis. The parent function opens upward, forming a 'V' shape pointing upwards. Reflecting it flips it upside down, so it opens downward. Therefore, this statement is FALSE. The sign of the coefficient outside the absolute value is the direct determinant of whether the graph opens upward or downward. A positive coefficient means it opens upward, while a negative coefficient means it opens downward.

Statement 4: The graph of f(x) has a domain

The domain of a function is the set of all possible input values (x-values) for which the function is defined. Absolute value functions, like our f(x) = -2/3|x+4| - 6, are defined for all real numbers. You can plug in any value for 'x', and you'll get a valid output. There are no restrictions, like division by zero or taking the square root of a negative number, that would limit the possible input values. Therefore, the domain is all real numbers, often written as (-∞, ∞). This statement needs to be completed to be evaluated. However, if the statement says