Hey guys! Ever stumbled upon an equation that looks like a jumbled mess of numbers and variables? Well, fear not! Today, we're going to dissect a fascinating equation: 0.4(x+3)=(y-6)^2. This isn't just any equation; it represents a parabola, a U-shaped curve with some seriously cool properties. Our mission? To unravel its secrets and pinpoint its key features: the vertex, directrix, focus, and focal width. Buckle up, because we're about to embark on a mathematical adventure!
Cracking the Parabola Code: Standard Form is Our Superpower
Before we dive headfirst into identifying the parabola's features, let's talk strategy. The key to unlocking this equation lies in transforming it into its standard form. Think of the standard form as a decoder ring; it reveals all the hidden information we need. For a parabola that opens horizontally (like this one), the standard form equation looks like this:
(y - k)^2 = 4p(x - h)
Where:
- (h, k) represents the vertex, the turning point of the parabola.
- p dictates the distance between the vertex and the focus (the magic spot where all reflected rays converge) and the distance between the vertex and the directrix (a line that helps define the parabola's shape).
- The focal width is the distance across the parabola at the focus. It's also known as the latus rectum.
Now, let's get our hands dirty and massage our given equation, 0. 4(x+3)=(y-6)^2, into this beautiful standard form. Notice anything? Our equation already looks pretty darn close! We just need to isolate the (x+3) term. To do that, we'll divide both sides of the equation by 0.4:
(y - 6)^2 = (1 / 0.4)(x + 3)
(y - 6)^2 = 2.5(x + 3)
Boom! We've done it! Our equation is now in standard form. This is where things get really exciting, guys. We can now directly compare this equation to the standard form and pluck out the values of h, k, and p. Remember, these values are the keys to unlocking the parabola's secrets. The power of standard form cannot be overstated. It transforms a seemingly complex equation into a treasure map, guiding us directly to the information we seek. So, let's put on our detective hats and start deciphering!
Pinpointing the Vertex: The Heart of the Parabola
The vertex is the cornerstone of a parabola. It's the point where the curve changes direction, the very heart of the shape. Think of it as the parabola's home base. In our standard form equation, (y - k)^2 = 4p(x - h), the vertex is represented by the coordinates (h, k). This is fantastic news for us because we've already transformed our equation into standard form! We can now directly read off the values of h and k.
Looking at our equation, (y - 6)^2 = 2.5(x + 3), we can see a clear correspondence:
- (y - k) corresponds to (y - 6), which means k = 6.
- (x - h) corresponds to (x + 3). Be careful here! Remember that the standard form has a minus sign. So, to make them match perfectly, we can rewrite (x + 3) as (x - (-3)). This tells us that h = -3.
Therefore, the vertex of our parabola is (-3, 6). We've successfully located the heart of our parabola! To truly understand the significance of the vertex, imagine it as the anchor point from which the entire parabola is constructed. It dictates the parabola's position on the coordinate plane and serves as the reference point for all other key features, such as the focus and directrix. Finding the vertex is often the first crucial step in analyzing any parabola, as it provides a fundamental understanding of its orientation and location.
Unveiling the Directrix: The Guiding Line
Now that we've found the vertex, let's hunt down the directrix. The directrix is a line that plays a crucial role in defining the parabola's shape. It's a line such that for any point on the parabola, the distance to the focus is equal to the distance to the directrix. This might sound a bit abstract, but it's a fundamental property of parabolas.
For a parabola that opens horizontally, the directrix is a vertical line. Its equation takes the form x = h - p, where (h, k) is the vertex and p is the distance between the vertex and the focus (and also the distance between the vertex and the directrix). We already know that the vertex is (-3, 6), so we have h = -3. But what about p? To find p, let's revisit our standard form equation:
(y - 6)^2 = 2.5(x + 3)
Remember that the standard form is (y - k)^2 = 4p(x - h). So, we can equate 4p to 2.5:
4p = 2.5
Now, we simply solve for p:
p = 2.5 / 4 = 0.625
Fantastic! We've found p. Now we can calculate the equation of the directrix:
x = h - p
x = -3 - 0.625
x = -3.625
Therefore, the directrix of our parabola is the vertical line x = -3.625. The directrix acts as a boundary line, influencing how the parabola curves. Points on the parabola are equidistant from the focus and the directrix, creating the characteristic U-shape. Visualizing the directrix helps us understand the parabola's orientation and how it opens.
Locating the Focus: The Reflection Point
The focus is another key feature of a parabola. It's a special point inside the curve with a fascinating property: any ray parallel to the axis of symmetry that hits the parabola will be reflected through the focus. This is the principle behind satellite dishes and parabolic mirrors. So, finding the focus is kind of a big deal!
For a horizontally opening parabola, the focus has coordinates (h + p, k), where (h, k) is the vertex and p is the same distance we used for the directrix. We already know that the vertex is (-3, 6), so h = -3 and k = 6. We also calculated p to be 0.625. Now, we just plug these values into the formula:
Focus = (h + p, k)
Focus = (-3 + 0.625, 6)
Focus = (-2.375, 6)
Therefore, the focus of our parabola is the point (-2.375, 6). The focus dictates the direction in which the parabola opens and the degree of curvature. The closer the focus is to the vertex, the tighter the curve. Conversely, the farther the focus is from the vertex, the wider the curve. Understanding the focus is crucial for applying parabolas in various real-world scenarios, such as designing antennas and solar collectors.
Measuring the Focal Width: The Latus Rectum
Our final task is to determine the focal width, also known as the latus rectum. The focal width is the length of the line segment that passes through the focus, is perpendicular to the axis of symmetry, and has endpoints on the parabola. It essentially tells us how