Mastering Parabolas Analyzing F(x) = Mx² + Nx + K For Upward Opening Curves
Hey guys! Let's dive into the fascinating world of parabolas, specifically those described by the quadratic function f(x) = mx² + nx + k. We're going to explore the conditions that make a parabola open upwards and how its vertex plays a crucial role. This is super important for understanding quadratic functions and their graphs, so buckle up and let's get started!
Understanding the Basics: Quadratic Functions and Parabolas
Before we tackle the specifics of our problem, let's refresh our understanding of quadratic functions and parabolas. A quadratic function is a polynomial function of degree two, generally expressed in the form f(x) = ax² + bx + c. When we graph a quadratic function, we get a U-shaped curve called a parabola. The parabola can open upwards or downwards, and it has a special point called the vertex, which is either the minimum or maximum point on the curve. The coefficients 'a', 'b', and 'c' in the quadratic function determine the shape and position of the parabola.
Delving Deeper into the Coefficient 'm': Now, in our specific function, f(x) = mx² + nx + k, the coefficient 'm' plays a pivotal role in determining the parabola's orientation. This is the key to understanding whether the parabola opens upwards or downwards. If 'm' is positive, the parabola opens upwards, resembling a smile. Conversely, if 'm' is negative, the parabola opens downwards, like a frown. Think of it this way: a positive 'm' means the quadratic term dominates for large values of x, making the function increase as x moves away from the vertex in either direction. A negative 'm' does the opposite. So, the sign of 'm' is our first clue in unlocking the secrets of this parabola. This is incredibly important for solving our problem! Remember this, guys! The coefficient 'm' is the gatekeeper to the parabola's direction. We need to keep this in mind as we explore the other conditions.
The Vertex: The Heart of the Parabola: The vertex of a parabola is its most defining feature. It's the turning point, the place where the parabola changes direction. For a parabola that opens upwards, the vertex represents the minimum value of the function. For a downward-opening parabola, it's the maximum value. The coordinates of the vertex are given by (-b/2a, f(-b/2a)). In our case, the x-coordinate of the vertex is -n/2m, and the y-coordinate is the function's value at that point. The problem tells us that the vertex is at (-2, s), where s > 0. This is crucial information! It tells us not only the x-coordinate of the vertex but also that the minimum value of the function is a positive number. This gives us a significant constraint to work with. Thinking about the vertex, guys, it's like the anchor point of the parabola. It determines the parabola's position in the coordinate plane, and knowing its coordinates gives us vital clues about the coefficients of the quadratic function.
The Interplay of Coefficients: A Sneak Peek: The coefficients 'm', 'n', and 'k' are not independent of each other. They are intricately linked and determine the parabola's shape and position. The vertex coordinates, the direction the parabola opens, and the y-intercept are all dictated by these coefficients. For instance, the y-intercept is simply the value of the function when x = 0, which is 'k'. The x-coordinate of the vertex is -n/2m, as we mentioned earlier. This relationship is key to solving problems involving parabolas. Understanding how these coefficients dance together is like learning the secret handshake of quadratic functions! It allows us to predict the behavior of the parabola based on the values of 'm', 'n', and 'k'. So, let's keep these connections in mind as we analyze the given statements.
Analyzing the Given Information: f(x) = mx² + nx + k
Now that we've laid the foundation, let's focus on the specific information given in the problem. We know that the function f(x) = mx² + nx + k represents a parabola that opens upwards and has its vertex at (-2, s), where s > 0. This seemingly simple statement packs a lot of punch! Let's unpack it piece by piece and see what truths we can uncover. This is where the real detective work begins, guys! We have clues, and now we need to use our knowledge of parabolas to decipher them.
The Upward-Opening Parabola: Unveiling the First Truth: The first key piece of information is that the parabola opens upwards. As we discussed earlier, this directly implies that the coefficient 'm' must be positive. Why? Because a positive 'm' ensures that the parabola curves upwards. This is statement (i) in the problem, and we can confidently say it MUST be true. This is a great start! We've already confirmed one of the statements. It's like finding the first piece of a puzzle – it gives us momentum and helps us see the bigger picture. So, let's keep going and see what else we can deduce.
The Vertex at (-2, s): A Goldmine of Information: The fact that the vertex is at (-2, s) with s > 0 is incredibly valuable. It gives us two pieces of information: the x-coordinate of the vertex is -2, and the y-coordinate (which is the minimum value of the function) is a positive number. This is like hitting the jackpot! Knowing the vertex allows us to establish relationships between the coefficients 'm', 'n', and 'k'. Remember, the x-coordinate of the vertex is given by -n/2m. So, we have -n/2m = -2. Let's simplify this equation. Multiplying both sides by -2m, we get n = 4m. Bingo! This confirms statement (ii) in the problem. Guys, we're on a roll! We've used the vertex information to deduce a crucial relationship between 'n' and 'm'. This is the power of understanding the properties of parabolas – we can use seemingly simple facts to uncover hidden truths.
The Positive y-coordinate 's': A Subtle but Powerful Clue: The fact that s > 0, meaning the y-coordinate of the vertex is positive, tells us that the minimum value of the function is positive. This means the parabola sits entirely above the x-axis. Now, the y-coordinate of the vertex is f(-2), which is m(-2)² + n(-2) + k = 4m - 2n + k. Since we know n = 4m, we can substitute this into the expression: 4m - 2(4m) + k = 4m - 8m + k = -4m + k. So, we have -4m + k = s, and since s > 0, we get -4m + k > 0, which implies k > 4m. This inequality is important, but it doesn't directly tell us whether k < 5. This is where we need to be careful! We can't jump to conclusions. We know k is greater than 4m, but we don't have enough information to definitively say it's less than 5. Statement (iii) says k < 5, but we cannot confirm this statement must be true.
Evaluating the Statements: Which Ones Must Be True?
We've dissected the problem, analyzed the given information, and derived relationships between the coefficients. Now, let's put it all together and determine which of the statements must be true. This is the final showdown! We've gathered our evidence, and now we need to present our case.
Statement (i): m > 0: We established that this statement must be true because the parabola opens upwards. A positive 'm' is the defining characteristic of an upward-opening parabola. So, we can confidently say that statement (i) is correct.
Statement (ii): n = 4m: We derived this relationship from the x-coordinate of the vertex. By substituting -n/2m = -2, we directly obtained n = 4m. Therefore, statement (ii) must also be true.
Statement (iii): k < 5: This is where things get tricky. We know that k > 4m, but we don't have any information that directly limits k to being less than 5. It's possible for k to be less than 5, but it's not a certainty. We need a MUST be true statement, guys, and this one doesn't cut it. So, statement (iii) is not necessarily true.
The Verdict: The Answer and Why
Based on our analysis, statements (i) and (ii) must be true, while statement (iii) is not necessarily true. Therefore, the correct answer is (a): (i) and (ii).
We did it! We successfully navigated the world of parabolas, deciphered the clues, and arrived at the correct answer. This problem highlights the importance of understanding the fundamental properties of quadratic functions and how the coefficients relate to the shape and position of the parabola. Remember, guys, practice makes perfect! The more you work with these concepts, the more comfortable you'll become. Keep exploring, keep questioning, and keep learning!
