Parabola Challenge Identifying Upward Opening And Narrower Parabolas

Hey guys! Let's dive into the world of parabolas and figure out which one opens upward and appears narrower than the given parabola, $y = -3x^2 + 2x - 1$. This is a super fun topic, and by the end of this article, you'll be a parabola pro!

Understanding Parabolas

Before we jump into the specifics, let's refresh our understanding of parabolas. A parabola is a U-shaped curve that can open upwards or downwards, and its shape is determined by a simple quadratic equation in the form of $y = ax^2 + bx + c$. The coefficient 'a' plays a crucial role in defining the parabola's direction and width. So, keep that in mind as we proceed.

The Role of 'a' in Parabolas

The coefficient 'a' in the quadratic equation $y = ax^2 + bx + c$ is the key to understanding the parabola's behavior. It dictates two crucial aspects: the direction in which the parabola opens and how wide or narrow it appears. If 'a' is positive, the parabola opens upward, resembling a smiley face. If 'a' is negative, the parabola opens downward, like a frowny face. Now, let's talk about the width. The larger the absolute value of 'a', the narrower the parabola. Conversely, the smaller the absolute value of 'a', the wider the parabola. Think of it this way: a large 'a' pulls the parabola inward, making it skinny, while a small 'a' allows it to spread out. This understanding is crucial for comparing parabolas and identifying their unique characteristics.

For instance, consider $y = 2x^2$ and $y = 0.5x^2$. The first parabola, with $a = 2$, will be narrower than the second parabola, with $a = 0.5$. Similarly, $y = -3x^2$ opens downward and is narrower than $y = -x^2$, which also opens downward but is wider. Therefore, when we're looking for parabolas that open upward and are narrower, we need to focus on parabolas with a positive 'a' value that is larger in absolute terms than the 'a' value of our reference parabola. This principle will guide us as we analyze the given options and pinpoint the correct answer. Understanding this fundamental concept makes comparing parabolas much easier and helps in quickly identifying their key features.

Comparing Parabolas

When comparing parabolas, we mainly focus on the coefficient of the $x^2$ term, which is 'a'. This single value tells us whether the parabola opens upwards or downwards and how narrow or wide it is. Let’s break it down:

• Direction: If 'a' is positive, the parabola opens upwards. If 'a' is negative, it opens downwards.
• Width: The larger the absolute value of 'a', the narrower the parabola. The smaller the absolute value of 'a', the wider the parabola.

For example, if we have two parabolas, $y = 5x^2$ and $y = 2x^2$, both open upwards because their 'a' values are positive. However, $y = 5x^2$ is narrower because $|5| > |2|$. Conversely, if we compare $y = -3x^2$ and $y = -x^2$, both open downwards, but $y = -3x^2$ is narrower because $|-3| > |-1|$. This comparison technique is essential for solving problems involving parabolas, and it allows us to quickly assess the key characteristics of each parabola without needing to graph them. By focusing on the 'a' value, we can efficiently determine how parabolas relate to each other in terms of their direction and width, which is fundamental for understanding their behavior in various mathematical contexts.

Analyzing the Given Parabola

Our reference parabola is $y = -3x^2 + 2x - 1$. The coefficient of the $x^2$ term here is -3. This tells us two things: the parabola opens downwards (because -3 is negative), and it has a certain width determined by the absolute value of -3, which is 3. We're looking for a parabola that opens upwards and is narrower. So, we need a positive 'a' value that is greater than 3. Remember, the larger the absolute value of 'a', the narrower the parabola. Since our reference parabola has an 'a' value of -3, we need to find a parabola with an 'a' value that is positive and has an absolute value greater than 3 to ensure it's both upward-opening and narrower.

Breaking Down the Reference Parabola

Let's dive deeper into our reference parabola, $y = -3x^2 + 2x - 1$, to fully grasp its characteristics. The key element here is the coefficient of the $x^2$ term, which is -3. This negative sign immediately tells us that the parabola opens downwards, like a frown. The absolute value of this coefficient, $|-3| = 3$, gives us an indication of the parabola's width. A larger absolute value means a narrower parabola, and a smaller absolute value means a wider parabola. In this case, the parabola is relatively narrow due to the 3. Think of it as the larger the number, the steeper the curve. The other terms, $2x$ and -1, affect the parabola's position in the coordinate plane but don't change its basic shape or direction. The $2x$ term shifts the parabola horizontally, and the -1 term shifts it vertically. However, for our purposes of comparing direction and width, we can focus primarily on the $x^2$ coefficient. Understanding these nuances helps us quickly assess and compare parabolas without needing to graph them, making problem-solving much more efficient. This in-depth analysis provides a solid foundation for tackling the question at hand and identifying the correct option from the given choices.

Evaluating the Options

Now, let's examine each option and see which one fits our criteria: opening upwards and being narrower than $y = -3x^2 + 2x - 1$.

Option A: $y = 4x^2 - 2x - 1$

The coefficient of $x^2$ is 4. Since 4 is positive, this parabola opens upwards. The absolute value of 4 is 4, which is greater than 3. So, this parabola is narrower than our reference parabola. This option looks promising!

Option B: $y = -4x^2 + 2x - 1$

The coefficient of $x^2$ is -4. Since -4 is negative, this parabola opens downwards. We need a parabola that opens upwards, so this option is incorrect. Bye-bye, Option B!

Option C: $y = x^2 + 4x$

The coefficient of $x^2$ is 1. Since 1 is positive, this parabola opens upwards. The absolute value of 1 is 1, which is less than 3. This means the parabola is wider than our reference parabola, not narrower. So, this option is not the one we're looking for.

Option D: $y = -2x^2 + x + 3$

The coefficient of $x^2$ is -2. Since -2 is negative, this parabola opens downwards. Again, we need an upward-opening parabola, so this option is incorrect. Option D, you're out!

The Solution

After analyzing all the options, we found that only Option A, $y = 4x^2 - 2x - 1$, opens upward and is narrower than our reference parabola $y = -3x^2 + 2x - 1$. The coefficient 4 is positive, indicating it opens upwards, and its absolute value is greater than 3, making it narrower.

Final Answer

So, the correct answer is:

a. $y = 4x^2 - 2x - 1$

Awesome job, guys! You've successfully navigated the world of parabolas and found the one that opens upward and is narrower. Keep practicing, and you'll become parabola masters in no time!