Parabola Point Test: Find The Coordinate!
Let's figure out which of the given coordinate pairs actually sits on the parabola described by the equation y = -4x² - 53x - 56. We need to test each point to see if it satisfies the equation. Basically, we'll plug in the x-value of each coordinate pair into the equation and check if we get the corresponding y-value. It's like a high-stakes game of coordinate matching! So, let's dive in and test each option one by one.
Testing the Coordinate Pairs
We're going to plug in the x value from each coordinate pair into our parabola equation: y = -4x² - 53x - 56. If the resulting y value matches the y value in the coordinate pair, then that point lies on the parabola. Let's get started!
Option A: (-1, 13)
Okay, let's test the first coordinate pair, (-1, 13). We'll substitute x = -1 into the equation:
y = -4(-1)² - 53(-1) - 56
First, calculate (-1)² which equals 1:
y = -4(1) + 53 - 56
Now, simplify:
y = -4 + 53 - 56
y = 49 - 56
y = -7
So, when x = -1, we get y = -7. But the coordinate pair given is (-1, 13). Since -7 ≠13, the point (-1, 13) is not on the parabola. Option A is incorrect, guys. On to the next one!
Option B: (-1, -7)
Let's test the coordinate pair (-1, -7). Again, we'll substitute x = -1 into the equation:
y = -4(-1)² - 53(-1) - 56
We already calculated this in the previous step, but let's quickly recap:
y = -4(1) + 53 - 56
y = -4 + 53 - 56
y = 49 - 56
y = -7
In this case, when x = -1, we get y = -7. The coordinate pair given is (-1, -7). Since -7 = -7, the point (-1, -7) is on the parabola! Woo-hoo! We found a match. But let's check the other options just to be absolutely sure.
Option C: (-4, 220)
Now, let's test the coordinate pair (-4, 220). Substitute x = -4 into the equation:
y = -4(-4)² - 53(-4) - 56
First, calculate (-4)² which equals 16:
y = -4(16) + 212 - 56
Now, simplify:
y = -64 + 212 - 56
y = 148 - 56
y = 92
So, when x = -4, we get y = 92. But the coordinate pair given is (-4, 220). Since 92 ≠220, the point (-4, 220) is not on the parabola. Option C is incorrect.
Option D: (-4, -56)
Finally, let's test the coordinate pair (-4, -56). Substitute x = -4 into the equation:
y = -4(-4)² - 53(-4) - 56
Again, we've already done most of this calculation. From the previous step, we know that when x = -4:
y = -64 + 212 - 56
y = 92
So, when x = -4, we get y = 92. But the coordinate pair given is (-4, -56). Since 92 ≠-56, the point (-4, -56) is not on the parabola. Option D is incorrect.
Conclusion
After meticulously testing each coordinate pair, we found that only one of them satisfies the equation y = -4x² - 53x - 56. The coordinate pair (-1, -7) is the only point that lies on the parabola. Therefore, the correct answer is B. (-1, -7).
In summary, to determine if a point lies on a parabola, simply substitute the x-coordinate into the equation and check if the resulting y-coordinate matches the y-coordinate of the given point. It's a straightforward process, but it's crucial to perform the calculations accurately to avoid any errors.
Understanding Parabolas
Before we wrap up, let's touch upon the basics of parabolas to provide a bit more context. A parabola is a U-shaped curve that can open upwards, downwards, leftwards, or rightwards. The equation y = ax² + bx + c represents a parabola that opens upwards (if a > 0) or downwards (if a < 0). The vertex of the parabola is the point where the curve changes direction, and it's a crucial point to understand the behavior of the parabola.
In our case, the equation y = -4x² - 53x - 56 represents a parabola that opens downwards because the coefficient of the x² term (a) is negative (-4). This means that the vertex of the parabola will be the highest point on the curve. Finding the vertex involves using the formula x = -b / 2a, which gives the x-coordinate of the vertex. Once you have the x-coordinate, you can plug it back into the equation to find the y-coordinate of the vertex.
Understanding the properties of parabolas can help you visualize the curve and make predictions about its behavior. For instance, you can determine whether a point lies on the parabola by simply plugging in the x-coordinate and checking if the resulting y-coordinate matches the y-coordinate of the point. This is precisely what we did in our problem, and it's a fundamental technique for working with parabolas.
Common Mistakes to Avoid
When working with parabolas, there are a few common mistakes that you should be aware of to ensure accuracy. One of the most frequent errors is making mistakes in the arithmetic, especially when dealing with negative numbers and exponents. Always double-check your calculations to avoid these types of errors. Another common mistake is confusing the x and y coordinates when substituting them into the equation. Make sure you plug in the x-coordinate for x and the y-coordinate for y. Finally, be careful with the order of operations (PEMDAS/BODMAS) to ensure you perform the calculations in the correct sequence.
By avoiding these common mistakes, you can increase your chances of solving parabola-related problems accurately and efficiently. Remember, practice makes perfect, so keep working on these types of problems to improve your skills and build confidence.
Real-World Applications of Parabolas
You might be wondering,