Finding The Quadratic Function K(x) Step By Step Guide

Hey guys! Today, we're diving deep into the fascinating world of quadratic functions. We've got a real head-scratcher here, a continuous quadratic function named $k$ that's playing hide-and-seek with its equation. All we have are some ordered pairs in a table, and our mission, should we choose to accept it, is to figure out the value of $k(x)$. Buckle up, because we're about to embark on a mathematical adventure filled with plotting points, solving systems of equations, and ultimately, unveiling the secrets of this quadratic function.

Understanding Quadratic Functions

Before we even think about tackling our specific problem, let's take a moment to refresh our understanding of quadratic functions. At its heart, a quadratic function is a polynomial function of degree two. This means it can be written in the general form:

$$f(x) = ax^2 + bx + c$$

Where a, b, and c are constants, and a is not equal to zero (otherwise, it would just be a linear function!). The graph of a quadratic function is a parabola, a U-shaped curve that opens either upwards (if a is positive) or downwards (if a is negative). The key features of a parabola include its vertex (the minimum or maximum point), its axis of symmetry (a vertical line that divides the parabola into two symmetrical halves), and its intercepts (the points where the parabola crosses the x-axis and y-axis).

In our quest to decipher the function $k(x)$, we'll be leveraging the general form of a quadratic function. By substituting the given ordered pairs into this form, we'll be able to create a system of equations that we can then solve to find the values of a, b, and c. Once we have these values, we'll have the complete equation for $k(x)$, and we can calculate $k(x)$ for any value of x. Think of it like piecing together a puzzle – each ordered pair is a piece, and the general form of the quadratic function is the framework that holds it all together.

The Challenge: Finding $k(x)$ from Ordered Pairs

Our specific challenge revolves around the function $k$, which we know is a continuous quadratic function. What makes this tricky is that we don't have the equation for $k$ right away. Instead, we're given a table of ordered pairs: (-1, 5), (0, 8), (1, 5), (2, 0), (3, -7), and (4, -16). Each of these pairs represents a point on the parabola that is the graph of $k(x)$. Our goal is to use this information to determine the equation for $k(x)$.

Now, guys, remember the general form of a quadratic function: $f(x) = ax^2 + bx + c$. Since $k$ is a quadratic function, we can write it as:

$$k(x) = ax^2 + bx + c$$

Our mission is to find the values of a, b, and c. To do this, we're going to use the ordered pairs from the table. Each ordered pair gives us an x-value and its corresponding $k(x)$-value. We can plug these values into the equation above, and each pair will give us a linear equation in terms of a, b, and c. Because we have three unknowns (a, b, and c), we'll need at least three ordered pairs to create a system of three equations. Once we have this system, we can use techniques like substitution or elimination to solve for a, b, and c. This might sound a bit daunting, but trust me, we'll break it down step by step, and you'll see how it all comes together.

Setting Up the System of Equations

Okay, guys, let's get our hands dirty and start building our system of equations. We're going to take the ordered pairs from the table and plug them into the general form of our quadratic function, $k(x) = ax^2 + bx + c$. Remember, each ordered pair is in the form (x, k(x)), so we'll substitute the x-value for x and the $k(x)$-value for $k(x)$ in the equation.

Let's start with the ordered pair (-1, 5). Plugging these values into our equation, we get:

$$5 = a(-1)^2 + b(-1) + c$$

Simplifying, this becomes:

$$5 = a - b + c$$

That's our first equation! Now let's use the ordered pair (0, 8). Substituting these values, we have:

$$8 = a(0)^2 + b(0) + c$$

This simplifies to:

$$8 = c$$

Awesome! We've already found the value of c! This will make our lives much easier. Let's move on to the next ordered pair, (1, 5). Plugging in these values, we get:

$$5 = a(1)^2 + b(1) + c$$

Which simplifies to:

$$5 = a + b + c$$

Now we have three equations:

1. $$5 = a - b + c$$

2. $$8 = c$$

3. $$5 = a + b + c$$

This is our system of equations! Notice that we already know the value of c, which is 8. We can use this information to simplify our system further. In the next section, we'll use this system to solve for a and b.

Solving for a and b

Alright, guys, we've got our system of equations, and we're ready to solve for a and b. Remember, we already know that c = 8, which is a huge advantage. We can substitute this value into our other equations to make them simpler.

Let's rewrite our first and third equations with c = 8:

1. $$5 = a - b + 8$$

2. $$5 = a + b + 8$$

Now, let's subtract 8 from both sides of each equation:

1. $$-3 = a - b$$

2. $$-3 = a + b$$

Now we have a system of two equations with two unknowns, a and b. This is something we can definitely handle! There are a couple of ways we can solve this system. One common method is elimination. Notice that the b terms in the two equations have opposite signs. This means that if we add the two equations together, the b terms will cancel out.

Let's add the two equations:

$$(-3) + (-3) = (a - b) + (a + b)$$

This simplifies to:

$$-6 = 2a$$

Now, we can solve for a by dividing both sides by 2:

$$a = -3$$

We've found a! Now we can plug this value back into either of our two-variable equations to solve for b. Let's use the second equation, $-3 = a + b$:

$$-3 = -3 + b$$

Adding 3 to both sides, we get:

$$b = 0$$

And there you have it! We've solved for a, b, and c. We know that a = -3, b = 0, and c = 8. Now we can finally write the equation for our quadratic function, $k(x)$.

The Grand Reveal: The Equation for $k(x)$

After all our hard work, the moment of truth has arrived! We've successfully found the values of a, b, and c, and now we can piece them together to form the equation for our quadratic function, $k(x)$.

Remember the general form of a quadratic function:

$$k(x) = ax^2 + bx + c$$

We found that a = -3, b = 0, and c = 8. So, we simply substitute these values into the equation:

$$k(x) = -3x^2 + 0x + 8$$

Simplifying, we get:

$$k(x) = -3x^2 + 8$$

And there it is! This is the equation for the continuous quadratic function $k(x)$ that passes through the ordered pairs given in the table. The negative coefficient of the $x^2$ term tells us that the parabola opens downwards, and the constant term of 8 tells us that the y-intercept is at the point (0, 8).

Now that we have the equation for $k(x)$, we can use it to find the value of $k(x)$ for any value of x. We can also graph the function to visualize the parabola. It's amazing how we were able to uncover the secrets of this function just by using a few ordered pairs and our knowledge of quadratic functions.

Putting it to the Test: Verifying Our Solution

Before we pop the champagne and declare victory, it's always a good idea to double-check our work. We want to make sure that the equation we found for $k(x)$, which is $k(x) = -3x^2 + 8$, actually produces the $k(x)$-values in our original table for the given x-values. Think of it like quality control – we want to be 100% confident in our solution.

Let's take each x-value from the table and plug it into our equation to see if we get the corresponding $k(x)$-value. We'll go through each ordered pair one by one:

• For x = -1: $k(-1) = -3(-1)^2 + 8 = -3(1) + 8 = 5$. This matches the table!
• For x = 0: $k(0) = -3(0)^2 + 8 = 0 + 8 = 8$. This also matches!
• For x = 1: $k(1) = -3(1)^2 + 8 = -3(1) + 8 = 5$. Another match!
• For x = 2: $k(2) = -3(2)^2 + 8 = -3(4) + 8 = -12 + 8 = 0$. Yep, this one checks out too!
• For x = 3: $k(3) = -3(3)^2 + 8 = -3(9) + 8 = -27 + 8 = -19$. Hmm, hold on a second! The table says $k(3) = -7$, but our equation gives us $k(3) = -19$. We've found a discrepancy!

It looks like we made a mistake somewhere in our calculations. This is a great reminder that even when we think we've got the answer, it's crucial to verify our work. Let's go back and carefully review our steps to see where we went wrong. This is all part of the problem-solving process, guys, and it's how we learn and improve our skills.

Tracing Back Our Steps

Okay, guys, we've hit a snag, but that's no reason to panic! The fact that our equation didn't match one of the ordered pairs in the table is actually super valuable information. It tells us that there's an error in our calculations, and now we have a specific place to focus our attention. We're going to methodically retrace our steps, starting from the beginning, to pinpoint where the mistake occurred.

First, let's revisit the system of equations we set up. This is a crucial step because if there's an error in the system, everything that follows will be incorrect. Here's the system we had:

1. $$5 = a - b + c$$

2. $$8 = c$$

3. $$5 = a + b + c$$

These equations were derived by plugging the ordered pairs (-1, 5), (0, 8), and (1, 5) into the general form of a quadratic function, $k(x) = ax^2 + bx + c$. Let's double-check these substitutions to make sure we didn't make any errors there. Plugging in (-1, 5):

5 = a(-1)^2 + b(-1) + c$ which simplifies to $5 = a - b + c$. This looks correct. Plugging in (0, 8): $8 = a(0)^2 + b(0) + c$ which simplifies to $8 = c$. This also looks good. Plugging in (1, 5): $5 = a(1)^2 + b(1) + c$ which simplifies to $5 = a + b + c$. This checks out as well. So far, so good! Our system of equations seems to be set up correctly. The next step is where we substituted *c* = 8 into the first and third equations. Let's re-examine that: Substituting *c* = 8 into $5 = a - b + c$ gives us $5 = a - b + 8$, which simplifies to $-3 = a - b$. This is correct. Substituting *c* = 8 into $5 = a + b + c$ gives us $5 = a + b + 8$, which simplifies to $-3 = a + b$. This also looks correct. We're still on the right track! Now, let's look at how we solved for *a* and *b*. We used the elimination method, adding the two equations together: $(-3) + (-3) = (a - b) + (a + b)$ which simplifies to $-6 = 2a$. Dividing both sides by 2, we get $a = -3$. This is still correct. Then, we plugged *a* = -3 into the equation $-3 = a + b$ to solve for *b*: $-3 = -3 + b$, which gives us $b = 0$. This also appears to be correct. We've meticulously gone through each step, and so far, we haven't found any errors. This is a bit perplexing! It means the mistake must be somewhere else. Perhaps the error isn't in our calculations, but in the problem itself, or in how we're interpreting the information. In the next section, we'll consider the possibility of a typo in the table or explore alternative approaches to solving the problem. This is what makes math so fascinating – it's a puzzle that sometimes requires us to think outside the box! ## Reassessing the Problem and Exploring Alternatives Okay, guys, we've done a thorough review of our calculations, and we haven't been able to find any mistakes. This suggests that the issue might not be in our math, but perhaps in the information we were given. Let's take a step back and consider the possibilities. One possibility is that there might be a typo in the table of ordered pairs. It's not uncommon for errors to slip in, and it's something we should consider. If there's a typo, the ordered pair that's most likely to be incorrect is the one that didn't match our equation: (3, -7). Our equation, $k(x) = -3x^2 + 8$, gave us $k(3) = -19$, while the table stated $k(3) = -7$. That's a significant difference. So, let's entertain the idea that the correct ordered pair might be (3, -19) instead of (3, -7). Another approach we can take is to consider whether a quadratic function is truly the best fit for the given data. While we were told that $k$ is a **continuous quadratic function**, it's worth exploring if there might be another type of function that could fit the data more accurately, especially if the ordered pairs are slightly off due to some kind of error or approximation. However, for the sake of this exercise, let's stick with the assumption that $k$ is indeed a quadratic function and that the discrepancy is likely due to a typo in the table. If we assume the correct ordered pair is (3, -19), then our equation, $k(x) = -3x^2 + 8$, would perfectly match all the given data points. This would be a much more satisfying outcome! But, guys, let's not jump to conclusions just yet. Before we definitively say that (3, -7) is a typo, let's try a different approach to solving the problem, just to be absolutely sure. We initially used the ordered pairs (-1, 5), (0, 8), and (1, 5) to create our system of equations. What if we used a different set of three ordered pairs? Would we arrive at the same equation, or would we uncover a different result? This is a great way to test the consistency of the data and our solution method. In the next section, we'll create a new system of equations using a different set of ordered pairs from the table and see if we can confirm our equation or uncover a different one. This will give us even more confidence in our final answer. ## Creating a New System of Equations for Verification Alright, guys, we're on a mission to be absolutely certain about our solution. We've identified a potential issue with one of the ordered pairs, but before we declare it a typo, let's try a different approach. We're going to create a new system of equations using a different set of three ordered pairs from the table. If we arrive at the same equation for $k(x)$, then we can be much more confident in our answer. If we get a different equation, then we know there's definitely something amiss, and we'll need to investigate further. This time, let's use the ordered pairs (0, 8), (2, 0), and (3, -7). We've already used (0, 8) in our previous system, but it's a nice, simple ordered pair that includes *c* directly, so it's a good one to keep. Remember, we'll plug each ordered pair into the general form of a quadratic function, $k(x) = ax^2 + bx + c$, to create our equations. We already know that using (0, 8) gives us the equation $8 = c$. Let's move on to (2, 0). Plugging these values in, we get: $0 = a(2)^2 + b(2) + c

Which simplifies to:

$$0 = 4a + 2b + c$$

Now, let's use the ordered pair (3, -7). Plugging these values in, we get:

$$-7 = a(3)^2 + b(3) + c$$

Which simplifies to:

$$-7 = 9a + 3b + c$$

So, our new system of equations is:

1. $$8 = c$$

2. $$0 = 4a + 2b + c$$

3. $$-7 = 9a + 3b + c$$

Now, let's substitute c = 8 into the second and third equations:

1. $$0 = 4a + 2b + 8$$

2. $$-7 = 9a + 3b + 8$$

Let's simplify these equations by subtracting 8 from both sides:

1. $$-8 = 4a + 2b$$

2. $$-15 = 9a + 3b$$

We can further simplify the first equation by dividing both sides by 2:

$$-4 = 2a + b$$

Now we have a system of two equations with two unknowns:

1. $$-4 = 2a + b$$

2. $$-15 = 9a + 3b$$

We can solve this system using either substitution or elimination. Let's use elimination. We can multiply the first equation by -3 to eliminate the b term:

$$12 = -6a - 3b$$

Now, let's add this modified equation to the second equation:

$$12 + (-15) = (-6a - 3b) + (9a + 3b)$$

Which simplifies to:

$$-3 = 3a$$

Dividing both sides by 3, we get:

$$a = -1$$

Now, let's plug a = -1 into the equation $-4 = 2a + b$ to solve for b:

$$-4 = 2(-1) + b$$

$$-4 = -2 + b$$

Adding 2 to both sides, we get:

$$b = -2$$

So, we've found a = -1, b = -2, and c = 8. Plugging these values into the general form of a quadratic function, we get:

$$k(x) = -1x^2 - 2x + 8$$

This is a different equation than the one we found earlier! This confirms that there is indeed an issue with the original data or the assumption that all the points fit a single quadratic function. The discrepancy likely stems from the ordered pair (3, -7), which doesn't seem to align with the other points on a smooth parabolic curve. In the next section, we'll discuss the implications of this finding and what it means for our problem-solving journey.

Conclusion: Embracing the Twists and Turns of Problem Solving

Well, guys, this has been quite a journey! We started with a seemingly straightforward problem – finding the equation of a continuous quadratic function given a set of ordered pairs. We confidently set up a system of equations, solved for the coefficients, and arrived at an equation. However, our initial solution hit a snag when we tried to verify it against all the given data points. This led us to a fascinating detour, where we questioned our assumptions, retraced our steps, and explored alternative approaches.

Our careful re-examination of our calculations didn't reveal any errors, which prompted us to consider the possibility of a typo in the given data. To further investigate, we created a new system of equations using a different set of ordered pairs. This led us to a different equation for $k(x)$, confirming that there is an inconsistency in the original data. The ordered pair (3, -7) appears to be the culprit, as it doesn't fit the parabolic trend defined by the other points.

So, what's the takeaway from all of this? First and foremost, it highlights the importance of verification in problem-solving. It's not enough to simply find an answer; we need to rigorously test it to ensure its accuracy. Second, it demonstrates the value of flexibility and critical thinking. When our initial approach didn't pan out, we didn't give up. We adapted, questioned our assumptions, and explored alternative methods.

This experience also underscores the fact that real-world problems are often messy and imperfect. Data may contain errors, and assumptions may not always hold true. Learning to navigate these complexities is a crucial skill in mathematics and beyond. While we weren't able to find a single quadratic function that perfectly fits all the given data points, we learned a great deal about the process of problem-solving, the importance of verification, and the need to be adaptable in the face of challenges.

So, next time you encounter a problem that throws you a curveball, remember this adventure. Embrace the twists and turns, question your assumptions, and never be afraid to explore new avenues. And most importantly, guys, always double-check your work!