Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of parabolas. We've got a quadratic equation, y=a(x+2)(x-4), and a mission: to uncover the value of 'a' and pinpoint the coordinates of the turning point. This isn't just about crunching numbers; it's about understanding the elegant dance of curves and equations. So, grab your thinking caps, and let's get started!
Cracking the Code: Determining the Value of 'a'
First things first, we need to find the value of 'a'. We know that the graph of the equation y=a(x+2)(x-4) passes through the point (2,16). This is our golden ticket! It means that when x is 2, y is 16. We can plug these values into our equation and solve for 'a'. Think of it as a detective story, where the point (2,16) is our clue, and 'a' is the hidden treasure.
So, let's substitute x=2 and y=16 into the equation:
16 = a(2+2)(2-4)
Now, let's simplify:
16 = a(4)(-2)
16 = -8a
To isolate 'a', we divide both sides of the equation by -8:
a = 16 / -8
a = -2
Eureka! We've found our first piece of the puzzle. The value of 'a' is -2. This tells us something important about our parabola: since 'a' is negative, the parabola opens downwards, forming a graceful arch. It's like a mountain, not a valley. This is a crucial detail to keep in mind as we move forward.
Knowing 'a' is negative also gives us a visual sense of the parabola. Imagine a downward-facing curve, a smile turned upside down. This helps us anticipate the location of the turning point, which we know will be the highest point on the curve, also known as the vertex. Remember, in the world of parabolas, the sign of 'a' is a compass, guiding us towards understanding the curve's shape and orientation. So far, so good! We've conquered the first challenge. Now, let's move on to the next exciting part: finding the turning point.
Locating the Vertex: Unveiling the Turning Point Coordinates
Now that we've successfully determined the value of 'a', which is -2, our equation looks like this: y = -2(x+2)(x-4). The next step in our mathematical journey is to find the coordinates of the turning point, also known as the vertex of the parabola. The turning point is where the parabola changes direction – it's the peak of our mountain, the point where the curve transitions from going up to going down. To find this crucial point, we have a couple of awesome methods at our disposal.
Method 1: The Axis of Symmetry Approach
One of the most elegant ways to find the turning point is by using the axis of symmetry. The axis of symmetry is a vertical line that cuts the parabola perfectly in half, and the turning point always lies smack-dab on this line. For a quadratic equation in the form y = a(x-r)(x-s), the x-coordinate of the axis of symmetry is simply the average of the roots, 'r' and 's'. In our case, the roots are -2 and 4 (the values of x that make y=0). Think of it like finding the midpoint between two friends standing on a line – the axis of symmetry is the midpoint between our parabola's roots.
So, let's calculate the x-coordinate of the axis of symmetry:
x = (-2 + 4) / 2
x = 2 / 2
x = 1
This means the axis of symmetry is the vertical line x = 1. The turning point's x-coordinate is 1. Now, to find the y-coordinate, we simply plug x = 1 back into our equation:
y = -2(1+2)(1-4)
y = -2(3)(-3)
y = 18
Ta-da! The turning point is (1, 18). This is the peak of our parabolic mountain, the highest point on the curve.
Method 2: Expanding and Completing the Square
Another fantastic method involves expanding the equation and then completing the square. This approach transforms our equation into vertex form, which directly reveals the coordinates of the turning point. It's like using a secret decoder ring to unlock the vertex's location. Let's walk through it step by step:
First, expand the equation y = -2(x+2)(x-4):
y = -2(x² - 4x + 2x - 8)
y = -2(x² - 2x - 8)
y = -2x² + 4x + 16
Now, complete the square. This involves manipulating the equation to create a perfect square trinomial. It might sound intimidating, but it's a powerful technique for revealing the vertex.
y = -2(x² - 2x) + 16
To complete the square, we need to add and subtract (b/2)² inside the parentheses, where 'b' is the coefficient of the x term (-2 in this case). So, (b/2)² = (-2/2)² = 1.
y = -2(x² - 2x + 1 - 1) + 16
y = -2((x - 1)² - 1) + 16
y = -2(x - 1)² + 2 + 16
y = -2(x - 1)² + 18
Now our equation is in vertex form: y = a(x - h)² + k, where (h, k) is the turning point. In our case, h = 1 and k = 18. So, the turning point is (1, 18). We arrived at the same answer using a different route! This is the beauty of mathematics – multiple paths can lead to the same destination.
Putting It All Together: The Grand Finale
We've successfully navigated the world of parabolas, found the value of 'a', and pinpointed the coordinates of the turning point. Let's recap our journey:
- The value of 'a' is -2. This tells us the parabola opens downwards.
- The turning point (vertex) is (1, 18). This is the highest point on the parabola.
By combining these pieces of information, we have a complete picture of the parabola described by the equation y = -2(x+2)(x-4). We know its shape, its orientation, and its highest point. This is the power of mathematical problem-solving – taking a set of clues and weaving them into a coherent understanding.
So, guys, whether you prefer the axis of symmetry approach or the completing the square method, you now have the tools to conquer similar parabola challenges. Keep exploring, keep questioning, and keep unlocking the secrets of mathematics! The world of curves and equations is vast and fascinating, and there's always more to discover. Until next time, happy calculating!