How To Find The Vertex And Focus Of The Parabola Y² - 4y + 4x + 12 = 0 Step By Step

Jul 28, 2025 by ADMIN 84 views

Decoding the Parabola Find the Vertex and Focus of y^2-4y+4x+12=0

Hey guys! Today, we're diving deep into the fascinating world of parabolas. Parabolas aren't just cool curves; they pop up everywhere, from satellite dishes to the trajectory of a baseball. Mastering parabolas is crucial, especially if you're tackling mathematics, physics, or engineering. We're going to break down a specific problem that will help you understand how to find the vertex and focus of a parabola. So, let’s get started and unravel the mystery behind these curves!

Understanding the Parabola Equation

Before we jump into the problem, let's have a quick refresher on the standard forms of a parabola's equation. Understanding these forms is key to identifying the vertex and focus quickly. There are two main orientations for parabolas: those that open horizontally and those that open vertically.

Vertical Parabolas

The standard form for a parabola that opens vertically (upwards or downwards) is:

(x - h)^2 = 4p(y - k)

Where:

(h, k) is the vertex of the parabola.
p is the distance from the vertex to the focus and from the vertex to the directrix.
If p > 0, the parabola opens upwards.
If p < 0, the parabola opens downwards.

Horizontal Parabolas

The standard form for a parabola that opens horizontally (left or right) is:

(y - k)^2 = 4p(x - h)

Where:

(h, k) is the vertex of the parabola.
p is the distance from the vertex to the focus and from the vertex to the directrix.
If p > 0, the parabola opens to the right.
If p < 0, the parabola opens to the left.

Identifying which form our given equation matches will be our first step in solving the problem. This will tell us the orientation of the parabola and guide us on how to find the vertex and focus. The vertex is a crucial point – it’s the turning point of the parabola. The focus is another key point, located inside the curve, which has a special property: any ray parallel to the axis of symmetry will reflect off the parabola and pass through the focus. Knowing these definitions will help us visualize and solve the problem more effectively.

Problem Statement: Unraveling the Parabola

Okay, guys, let's tackle the problem head-on! We are given the equation of a parabola:

y² - 4y + 4x + 12 = 0

Our mission is to find the vertex and focus of this parabola. To do this, we need to massage the given equation into one of the standard forms we discussed earlier. This involves a technique called completing the square, which is a powerful tool in algebra. Don't worry if it sounds intimidating; we'll break it down step by step. The vertex and focus are key characteristics of a parabola, and finding them allows us to fully understand the shape and position of the curve in the coordinate plane. This is essential not only in mathematics but also in various applications, such as designing antennas and reflectors. Let's get our hands dirty with some algebra and bring this parabola into focus!

Step-by-Step Solution: Completing the Square

The first step in solving this problem is to rewrite the given equation into the standard form of a parabola. To do this, we'll use the technique of completing the square. This method allows us to transform a quadratic expression into a perfect square trinomial, which is essential for identifying the vertex and focus.

1. Group the 'y' terms:

Let's group the terms involving 'y' together and move the other terms to the other side of the equation:

y² - 4y = -4x - 12

This sets us up nicely for the next step, where we'll complete the square on the left side of the equation. Remember, completing the square involves adding a constant to both sides of the equation to create a perfect square trinomial on one side. This constant is determined by taking half of the coefficient of the 'y' term and squaring it.

2. Complete the square for the 'y' terms:

To complete the square for the expression y² - 4y, we take half of the coefficient of the 'y' term (-4), which is -2, and square it: (-2)² = 4. Now, we add 4 to both sides of the equation:

y² - 4y + 4 = -4x - 12 + 4

Adding the same value to both sides ensures that the equation remains balanced. The left side is now a perfect square trinomial, which can be factored easily. This is a crucial step in getting our equation into standard form.

3. Factor the perfect square trinomial:

The left side of the equation, y² - 4y + 4, is a perfect square trinomial and can be factored as (y - 2)². On the right side, we simplify the expression:

(y - 2)² = -4x - 8

Factoring the perfect square trinomial is a key step because it brings us closer to the standard form of a parabola equation. This form will allow us to easily identify the vertex and the value of 'p', which is crucial for finding the focus.

4. Factor out the coefficient of 'x':

To get the equation into the standard form (y - k)² = 4p(x - h), we need to factor out the coefficient of 'x' on the right side of the equation:

(y - 2)² = -4(x + 2)

Now, our equation is in the standard form for a parabola that opens horizontally. This means we can directly read off the values of h, k, and p. Identifying these values is the final step in finding the vertex and focus of the parabola. Remember, the vertex is the turning point, and the focus is a key point inside the curve, which helps define its shape.

Identifying the Vertex and the Value of 'p'

Now that we've massaged our equation into the standard form, (y - 2)² = -4(x + 2), it's time to pinpoint the vertex and the value of 'p'. These are the golden nuggets that will lead us to the focus. Remember, the standard form for a horizontal parabola is (y - k)² = 4p(x - h), where (h, k) is the vertex and 'p' determines the distance from the vertex to the focus and the direction the parabola opens.

1. Finding the Vertex (h, k):

By comparing our equation with the standard form, we can directly identify the values of 'h' and 'k'. Notice that in our equation, we have (y - 2)² which corresponds to (y - k)², so k = 2. Similarly, we have -4(x + 2) which corresponds to 4p(x - h). This means that (x + 2) is equivalent to (x - h), so h = -2. Therefore, the vertex of the parabola is:

Vertex (h, k) = (-2, 2)

The vertex is a crucial point on the parabola, acting as its turning point. It's the point where the parabola changes direction, and it's essential for graphing the parabola accurately. We've successfully located the vertex, which is a significant step in understanding this curve.

2. Determining the Value of 'p':

Now, let's find the value of 'p'. In our equation, we have -4(x + 2), which corresponds to 4p(x - h). This means that 4p = -4. To find 'p', we simply divide both sides of the equation by 4:

p = -1

The value of 'p' is critical because it tells us the distance from the vertex to the focus and also the direction in which the parabola opens. Since p is negative (p = -1), this tells us that the parabola opens to the left. This is because for horizontal parabolas, a negative 'p' value indicates that the parabola opens towards the negative x-axis.

Locating the Focus: The Heart of the Parabola

With the vertex and the value of 'p' in hand, we're now ready to pinpoint the focus of the parabola. The focus is a special point inside the curve that has a unique property: any ray parallel to the axis of symmetry will reflect off the parabola and pass through the focus. This property makes parabolas incredibly useful in applications like satellite dishes and antennas.

Focus Formula for Horizontal Parabolas:

For a horizontal parabola in the standard form (y - k)² = 4p(x - h), the focus is located at the point (h + p, k).

Plugging in the Values:

We've already determined that the vertex (h, k) is (-2, 2) and the value of p is -1. Now, let's plug these values into the focus formula:

Focus = (h + p, k) = (-2 + (-1), 2) = (-3, 2)

Therefore, the focus of the parabola is:

Focus = (-3, 2)

We've successfully located the focus, which is a key characteristic of the parabola. Knowing the focus and the vertex allows us to fully describe the parabola's position and shape in the coordinate plane. This understanding is essential for both theoretical mathematics and practical applications. This point, along with the vertex, gives us a complete picture of the parabola's geometry. It's like finding the heart of the parabola, the point around which everything else is centered.

Final Answer and Summary

Alright, guys, we've done it! We've successfully navigated the world of parabolas and found both the vertex and the focus of the given equation. Let's recap our findings and make sure we've got everything crystal clear.

The Parabola:

We started with the equation:

y² - 4y + 4x + 12 = 0

Our Mission:

Our goal was to find the vertex and the focus of this parabola.

The Solution:

Vertex: After completing the square and transforming the equation into standard form, we identified the vertex as:

Vertex = (-2, 2)
Focus: Using the vertex and the value of 'p' (which we found to be -1), we calculated the focus using the formula for horizontal parabolas:

Focus = (-3, 2)

In Conclusion:

We successfully found the vertex and the focus of the given parabola by completing the square, converting the equation to standard form, and applying the relevant formulas. The vertex represents the turning point of the parabola, while the focus is a key point inside the curve that influences its shape and reflective properties. This exercise has not only given us the answer but also reinforced our understanding of parabolas, their equations, and their key characteristics. Understanding how to find the vertex and focus is super useful, not just in mathematics class, but also in real-world applications. Think about satellite dishes, solar ovens, and even the design of car headlights – parabolas are everywhere! So, mastering this concept is definitely a win. Keep practicing, and you'll become a parabola pro in no time!