Logarithmic Function Transformations Find The Correct Equation

Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of logarithmic functions and their transformations. We've got a question on the table that's all about figuring out the correct equation for a transformed logarithmic graph. So, let's put on our detective hats and get started!

The Question at Hand

We're given that $f(x)$ is a transformation of the graph of $g(x) = \log_2 x$. Our mission, should we choose to accept it, is to determine which of the following equations correctly represents $f(x)$:

A. $f(x) = -\log_2(x+2)$ B. $f(x) = \log_2(x-2) + 2$ C. $f(x) = -\log_2 x + 2$ D. $f(x) = \log_2 x - 2$

To crack this, we need to understand how different transformations affect the graph of a logarithmic function. So, let's break down the key transformations and see how they play out.

Understanding Logarithmic Transformations

Before we jump into analyzing the options, let's refresh our understanding of logarithmic transformations. The general form of a transformed logarithmic function is:

$f(x) = a \log_b(x - h) + k$

Where:

• a represents a vertical stretch or compression and reflection about the x-axis if a is negative.
• b is the base of the logarithm.
• h represents a horizontal translation (shift).
• k represents a vertical translation (shift).

With this in mind, let's break down how each transformation affects the graph:

Vertical Stretch/Compression and Reflection (a)

The coefficient a in front of the logarithmic function plays a crucial role in shaping the graph. If $|a| > 1$, the graph undergoes a vertical stretch, making it appear steeper. Conversely, if $0 < |a| < 1$, the graph experiences a vertical compression, causing it to flatten out. But that's not all! The sign of a also dictates whether the graph is reflected about the x-axis. A negative a flips the graph upside down, creating a mirror image across the x-axis. In essence, a acts like a sculptor, stretching, compressing, and reflecting the logarithmic curve.

For instance, consider $f(x) = 2\log_2(x)$. Here, a = 2, which means the graph of $\log_2(x)$ will be stretched vertically by a factor of 2. Every y-value on the original graph will be doubled, resulting in a steeper curve. On the other hand, if we have $f(x) = 0.5\log_2(x)$, the graph will be compressed vertically, making it appear flatter.

Now, let's bring in the reflection. If we have $f(x) = -\log_2(x)$, the negative sign in front of the logarithm indicates a reflection about the x-axis. The entire graph of $\log_2(x)$ will be flipped, with points above the x-axis now appearing below, and vice versa. This transformation is essential for understanding options like A and C in our original question, where a negative sign is present.

Horizontal Translation (h)

Horizontal translations are all about shifting the graph left or right along the x-axis. The value h inside the logarithm, in the form $(x - h)$, controls this movement. Now, here's a little twist to keep in mind: a positive h shifts the graph to the right, while a negative h shifts it to the left. It's like the opposite of what you might intuitively expect!

For example, let's consider $f(x) = \log_2(x - 3)$. Here, h = 3, which means the graph of $\log_2(x)$ will be shifted 3 units to the right. The vertical asymptote, which is normally at $x = 0$ for the basic $\log_2(x)$ function, will also shift 3 units to the right, now residing at $x = 3$. This shift is crucial for understanding how the domain of the function changes with horizontal translations.

Conversely, if we have $f(x) = \log_2(x + 2)$, then h = -2, and the graph will be shifted 2 units to the left. The asymptote will also move 2 units to the left, now sitting at $x = -2$. Understanding this inverse relationship between the sign of h and the direction of the shift is key to accurately interpreting logarithmic transformations.

Vertical Translation (k)

Vertical translations are perhaps the most straightforward of the logarithmic transformations. The value k added outside the logarithm simply shifts the entire graph up or down along the y-axis. A positive k moves the graph upwards, while a negative k moves it downwards. It's a direct and intuitive shift that affects all the y-values of the function.

For instance, let's take a look at $f(x) = \log_2(x) + 4$. Here, k = 4, which means the graph of $\log_2(x)$ will be shifted 4 units upwards. Every point on the original graph will move 4 units higher, resulting in a parallel shift of the entire curve. This transformation is easy to visualize and directly impacts the range of the function.

On the flip side, if we have $f(x) = \log_2(x) - 1$, then k = -1, and the graph will be shifted 1 unit downwards. All the y-values will decrease by 1, and the entire curve will slide down. Vertical translations are essential for adjusting the vertical position of the logarithmic graph and can be easily identified by the constant term added or subtracted from the function.

Analyzing the Options

Now that we've got a solid grasp of logarithmic transformations, let's dissect each option and see how it transforms the basic graph of $g(x) = \log_2 x$.

Option A: $f(x) = -\log_2(x+2)$

This equation packs a double punch in terms of transformations. First, the negative sign in front of the logarithm signals a reflection about the x-axis. This means the graph will be flipped upside down compared to the basic $\log_2 x$ graph. Second, the (x + 2) inside the logarithm indicates a horizontal translation. Remember, adding 2 inside the logarithm shifts the graph to the left by 2 units. So, this option represents a reflection over the x-axis and a shift 2 units to the left.

Option B: $f(x) = \log_2(x-2) + 2$

This option also involves two transformations, but this time they're a horizontal shift and a vertical shift. The (x - 2) inside the logarithm indicates a horizontal translation of 2 units to the right. The + 2 outside the logarithm represents a vertical translation of 2 units upwards. So, this graph is shifted 2 units right and 2 units up.

Option C: $f(x) = -\log_2 x + 2$

This equation features a reflection about the x-axis (due to the negative sign) and a vertical translation of 2 units upwards (due to the + 2). Notice that there's no horizontal translation in this option, as there's no term being added or subtracted inside the logarithm with x.

Option D: $f(x) = \log_2 x - 2$

This option is the simplest of the bunch, involving only a vertical translation. The - 2 outside the logarithm indicates a shift of 2 units downwards. There's no reflection or horizontal shift here, so the graph will simply move down along the y-axis.

Finding the Correct Equation

To pinpoint the correct equation, we'd ideally have a visual representation of the transformed graph $f(x)$. However, since we don't have a graph in front of us, we need to think critically about how these transformations would alter key features of the logarithmic function, such as its asymptote and general shape. We have to rely on our understanding of transformations and logic to deduce the answer.

Without a visual reference, it's challenging to definitively say which option is correct. However, if we had a specific point on the graph of $f(x)$ or knew the location of its asymptote, we could plug those values into each equation and see which one holds true. Alternatively, if we knew the graph was reflected over the x-axis and shifted upwards, we could narrow it down to option C. Or, if it was shifted to the right and upwards, option B would be the likely candidate.

Final Thoughts

Wrapping our heads around logarithmic transformations can feel like solving a puzzle, but once you understand the roles of a, h, and k, you'll be navigating these transformations like a pro. Remember, practice makes perfect, so keep exploring different logarithmic functions and their transformations to solidify your understanding.

In this case, without a visual aid, pinpointing the exact answer is tough. But remember, the key is to break down each transformation and understand how it affects the graph. Keep exploring, keep questioning, and you'll conquer those logarithmic transformations in no time!