Finding The Value Of K In Function Transformations A Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of function transformations, specifically focusing on vertical shifts. We've got a table of values for a function g(x), and we know that g(x) is related to another function f(x) through a simple addition – a constant k. Our mission? To find the elusive value of k. So, buckle up, and let's get started on this mathematical adventure!

Decoding the Function Relationship

In this section, we'll dissect the relationship between g(x) and f(x), laying the groundwork for finding k. Let's restate the problem for clarity: We are given that g(x) = f(x) + k. This equation is the key to unlocking the mystery. It tells us that the value of g(x) is simply the value of f(x) shifted vertically by k units. If k is positive, the graph of f(x) shifts upward, and if k is negative, it shifts downward.

Now, let's look at the table of values provided. We have x values and their corresponding g(x) values. To find k, we need to figure out how g(x) is different from f(x). But wait a minute! We don't have any direct information about f(x). This is where we need to get a little clever. The beauty of this problem is that we don't actually need to know the explicit form of f(x). We can use the fact that the same constant k is added to f(x) for every value of x to get g(x). This consistent shift is what we'll exploit to find our answer. Think of it like this: if we knew f(x) for a specific x, and we knew g(x) for the same x, we could simply subtract f(x) from g(x) to find k. Since we don't know f(x), we need a different approach, which we'll explore in the next section. We'll focus on identifying patterns and using the given information strategically to isolate k. This involves carefully examining the g(x) values and looking for clues about the underlying f(x) and the vertical shift. Remember, the key concept here is that k represents a constant vertical shift applied to the entire function f(x). So, let's dive deeper into the data and see what we can uncover!

Unraveling the Value of k

Okay, let's roll up our sleeves and dive into unraveling the value of k. Remember our core equation: g(x) = f(x) + k. Our mission is to isolate k, and we can do this by cleverly using the data provided in the table. The table gives us pairs of x and g(x) values, but we're missing f(x). So, how can we bypass this missing piece of the puzzle? Well, here's the trick: we can consider two different x values, let's call them x1 and x2. We know that: g(x1) = f(x1) + k and g(x2) = f(x2) + k. Now, let's subtract these two equations. This might seem like a random step, but watch the magic unfold! Subtracting the first equation from the second, we get: g(x2) - g(x1) = (f(x2) + k) - (f(x1) + k). Notice what happens to the k terms? They cancel each other out! This leaves us with: g(x2) - g(x1) = f(x2) - f(x1). This equation is incredibly useful because it relates the differences in g(x) values to the differences in f(x) values, without explicitly involving k. While this doesn't directly give us k, it helps us understand the relationship between the two functions.

However, this approach didn't directly give us the value of k. We need to think a bit differently. Let's revisit the original equation: g(x) = f(x) + k. If we could somehow determine the value of f(x) for a specific x, we could easily find k by rearranging the equation to: k = g(x) - f(x). This is where the answer choices can become our allies. The answer choices provide potential values for k. We can test each of these values to see if they fit the given data. Let's try a process of elimination, using the table of values and the equation g(x) = f(x) + k. By strategically analyzing the table and testing the answer choices, we can pinpoint the value of k that satisfies the given condition. This method of working backward, combined with our understanding of function transformations, will lead us to the correct solution. So, let's put on our detective hats and start testing those answer choices!

Cracking the Code with Answer Choices

Alright, it's time to put our detective hats on and crack the code using the answer choices! We have four potential values for k: -3, -2, 2, and 3. Our mission is to find the one that fits perfectly into the equation g(x) = f(x) + k. Remember, this equation tells us that g(x) is obtained by adding k to f(x). So, if we subtract k from g(x), we should get f(x). Let's start by picking a value from the table, say x = 0. From the table, we see that g(0) = 2. Now, let's test each answer choice as a potential value for k.

• If k = -3: Then f(0) = g(0) - k = 2 - (-3) = 5. So, if k is -3, f(0) would be 5.
• If k = -2: Then f(0) = g(0) - k = 2 - (-2) = 4. So, if k is -2, f(0) would be 4.
• If k = 2: Then f(0) = g(0) - k = 2 - 2 = 0. So, if k is 2, f(0) would be 0.
• If k = 3: Then f(0) = g(0) - k = 2 - 3 = -1. So, if k is 3, f(0) would be -1.

Now, we have potential values for f(0) based on each possible k. But how do we know which one is correct? This is where we need to pick another x value from the table and repeat the process. If our chosen k is correct, the resulting f(x) values should be consistent across different x values. Let's try x = 2. From the table, g(2) = 4. Let's calculate f(2) for each potential k:

• If k = -3: Then f(2) = g(2) - k = 4 - (-3) = 7.
• If k = -2: Then f(2) = g(2) - k = 4 - (-2) = 6.
• If k = 2: Then f(2) = g(2) - k = 4 - 2 = 2.
• If k = 3: Then f(2) = g(2) - k = 4 - 3 = 1.

Now, we need to think about the nature of the function f(x). While we don't know its explicit formula, we can infer some things. Notice the symmetry in the g(x) values: they are the same for x = -4 and x = 4, and also the same for x = -2 and x = 2. This suggests that f(x) might also have some symmetry. However, we don't need to make strong assumptions about f(x). The key is to look for inconsistencies. If a particular value of k leads to wildly different f(x) values for different x, it's likely incorrect. By carefully comparing the f(x) values we've calculated for x = 0 and x = 2, we can start to eliminate answer choices and narrow down the possibilities. So, let's put on our critical thinking caps and see which k value emerges as the winner!

The Grand Finale: Identifying the Correct Value of k

Okay, guys, the moment of truth has arrived! We've done the groundwork, tested the answer choices, and now it's time to identify the correct value of k. Let's recap what we've found so far. We tested each answer choice by plugging it into the equation f(x) = g(x) - k for two different x values: x = 0 and x = 2. Here's a summary of our findings:

• For x = 0 (g(0) = 2):
  • If k = -3, then f(0) = 5
  • If k = -2, then f(0) = 4
  • If k = 2, then f(0) = 0
  • If k = 3, then f(0) = -1
• For x = 2 (g(2) = 4):
  • If k = -3, then f(2) = 7
  • If k = -2, then f(2) = 6
  • If k = 2, then f(2) = 2
  • If k = 3, then f(2) = 1

Now, let's look for a k value that gives us a consistent relationship between the f(x) values. By consistent, I mean that the f(x) values should make sense in the context of a function transformation. Remember, k represents a vertical shift. So, if we find a k that works for one x value, it should work for all x values in the table. Let's examine the results closely. Notice anything interesting? Look at the case where k = 2. When k = 2, we found that f(0) = 0 and f(2) = 2. Now, let's test this k value with another x value from the table, say x = -2. From the table, g(-2) = 4. If k = 2, then f(-2) = g(-2) - k = 4 - 2 = 2. Aha! We're seeing a pattern. When k = 2, we have f(2) = 2 and f(-2) = 2. This symmetry is a good sign!

Let's try one more x value, just to be absolutely sure. Let's use x = 4. From the table, g(4) = 6. If k = 2, then f(4) = g(4) - k = 6 - 2 = 4. And for x = -4, g(-4) = 6, so f(-4) = g(-4) - k = 6 - 2 = 4. Again, we see consistency. The f(x) values seem to be behaving well when k = 2. On the other hand, if you try the other k values, you'll find that the resulting f(x) values don't exhibit this consistent behavior. They jump around in a way that doesn't align with a simple vertical shift. Therefore, based on our analysis and the process of elimination, we can confidently conclude that the correct value of k is 2. So, the answer is C! You did it! We successfully navigated the world of function transformations and found the missing piece of the puzzle. Give yourselves a pat on the back, and keep exploring the fascinating world of mathematics!