Finding The 'a' Value Equation For A Parabola With Minimum And A Point

Hey guys! Let's dive into a super interesting problem involving parabolas. We're given some key information about a parabola – its minimum point and another point it passes through – and our mission is to figure out the equation that helps us find the 'a' value, which is crucial for defining the parabola's shape. So, grab your thinking caps, and let's get started!

Understanding Parabolas and Their Equations

Before we jump into solving the problem, let's quickly recap what parabolas are and the different forms their equations can take. This foundational knowledge will make the problem-solving process much smoother and clearer.

Parabolas are U-shaped curves that are super common in math and physics. You see them everywhere, from the path of a ball thrown in the air to the curves in satellite dishes. Mathematically, a parabola is defined as the set of all points that are equidistant to a fixed point (the focus) and a fixed line (the directrix). But for our purposes, we're more interested in the algebraic representation of a parabola.

The standard form of a parabola equation is y = ax² + bx + c, where a, b, and c are constants. The coefficient a plays a vital role – it determines whether the parabola opens upwards (if a > 0) or downwards (if a < 0) and how “wide” or “narrow” the parabola is. The larger the absolute value of a, the narrower the parabola.

However, for this problem, the vertex form of a parabola equation is our best friend. The vertex form is given by y = a(x - h)² + k, where (h, k) represents the vertex of the parabola. The vertex is the point where the parabola changes direction – it's either the minimum point (if the parabola opens upwards) or the maximum point (if the parabola opens downwards). Knowing the vertex makes writing the equation much simpler, as we'll see shortly.

In our problem, we're given that the minimum of the parabola is located at (-1, -3). This is our vertex! So, we know that h = -1 and k = -3. We also know that the point (0, 1) lies on the parabola. This gives us an x and a y value that we can plug into our equation. The big question is: How do we use this information to find the equation that helps us solve for a?

Think of the a value as the parabola's stretch factor. It tells us how much the parabola is vertically stretched or compressed compared to the basic parabola y = x². A larger a means a steeper, narrower parabola, while a smaller a means a wider, flatter parabola. If a is negative, the parabola is flipped upside down.

Setting Up the Equation: Using the Vertex Form

Okay, now we're ready to tackle the problem head-on! We know the vertex form of a parabola is y = a(x - h)² + k, and we know the vertex (h, k) is (-1, -3). Let's plug these values into the equation:

y = a(x - (-1))² + (-3)

Simplifying this, we get:

y = a(x + 1)² - 3

This equation is a great start! It represents a parabola with its vertex at (-1, -3). The only unknown here is a, which determines the shape and direction of the parabola. To find a, we need more information, and luckily, we have it! We know that the point (0, 1) lies on the parabola. This means that when x = 0, y = 1. We can substitute these values into our equation:

1 = a(0 + 1)² - 3

And there it is! This is the equation we can solve to determine the value of a. This equation perfectly captures the relationship between a and the given information about the parabola. It's like a mathematical puzzle piece that fits perfectly into the bigger picture of the parabola's equation.

Notice how we used the vertex form to our advantage. It allowed us to directly incorporate the vertex coordinates into the equation, making the setup much simpler. If we had started with the standard form y = ax² + bx + c, we would have had to do a lot more work to find the relationship between the coefficients and the given information.

Solving for 'a' and Completing the Parabola's Equation

While the question only asks for the equation to solve for a, let's go the extra mile and actually solve for a. This will give us a complete picture of the parabola's equation and further solidify our understanding.

We have the equation:

1 = a(0 + 1)² - 3

Let's simplify and solve for a:

1 = a(1)² - 3 1 = a - 3

Add 3 to both sides:

4 = a

So, we found that a = 4! This tells us that the parabola opens upwards (since a is positive) and is vertically stretched compared to the basic parabola y = x².

Now that we know a, we can write the complete equation of the parabola in vertex form:

y = 4(x + 1)² - 3

This is the equation of the parabola that has a minimum at (-1, -3) and passes through the point (0, 1). Pretty cool, huh?

We can also expand this equation to get the standard form, if we want:

y = 4(x² + 2x + 1) - 3 y = 4x² + 8x + 4 - 3 y = 4x² + 8x + 1

This is the same parabola, just written in a different form. Both forms are useful in different situations.

Why This Equation Works: A Deeper Look

Let's take a moment to really understand why the equation 1 = a(0 + 1)² - 3 works. It's not just a random jumble of symbols; it represents a fundamental relationship between the parabola's parameters and the points it passes through.

Remember, the equation y = a(x + 1)² - 3 describes all the points on the parabola. Any point (x, y) that satisfies this equation lies on the curve. We were given a specific point, (0, 1), that we know lies on the parabola. This means that the coordinates of this point must satisfy the equation.

By plugging in x = 0 and y = 1, we're essentially saying: