Find Points On The Parabola Y = -6x² + 3x + 8

by ADMIN 46 views

Hey math whizzes! Today we're diving into the awesome world of quadratic equations and how to figure out if a specific point actually sits on the curve of a parabola. Our mission, should we choose to accept it, is to determine which of the given points lies on the parabola defined by the equation y = -6x² + 3x + 8. This is a super common type of problem in algebra, and understanding it will give you a solid grasp on graphing and analyzing quadratic functions. So, grab your calculators, maybe a snack, and let's get this mathematical party started!

Understanding Quadratic Equations and Parabolas

Alright guys, let's first get our heads around what a quadratic equation is and what it looks like when we graph it. A quadratic equation is basically a polynomial equation of the second degree. That means the highest power of the variable (usually 'x' in our case) is 2. The general form you'll often see is ax² + bx + c = 0, but when we're talking about graphing, we use the form y = ax² + bx + c. The 'a', 'b', and 'c' are just numbers, and they tell us a ton about the shape and position of our graph. The most iconic feature of a graphed quadratic equation is its shape: a parabola. Depending on the sign of the 'a' coefficient, the parabola will either open upwards (if 'a' is positive, like a smiley face :)) or downwards (if 'a' is negative, like a frowny face :(). In our specific equation, y = -6x² + 3x + 8, the 'a' value is -6. Since it's negative, we know right away that our parabola is going to be pointing downwards. The 'b' term (which is 3 here) and the 'c' term (which is 8) help determine the parabola's position, including where its vertex (the highest or lowest point) is and where it crosses the y-axis. The y-intercept, which is the point where the parabola crosses the y-axis, is always at the 'c' value when x=0. So, for our equation, the y-intercept is at (0, 8). Pretty neat, huh?

Now, what does it mean for a point to be on the parabola? It means that if you plug the x-coordinate of that point into the equation, the resulting y-value you calculate will be exactly the same as the y-coordinate of the point. It's like a secret handshake between a point and an equation! If the numbers match up, that point is a loyal member of the parabola club. If they don't match, well, that point is just passing by and isn't part of the parabola's journey. This concept is fundamental because it allows us to verify our graphs, find specific points of interest, and solve various problems related to quadratic functions. We can use this principle to check if our plotted points are accurate or to find solutions to equations. So, whenever you're given a point and an equation, the first thing you should think is, "Can I plug this point into the equation and see if it holds true?"

Testing the Given Points

Alright team, it's time to put our detective hats on and test each of the given options to see which one is the true resident of our parabola y = -6x² + 3x + 8. Remember the golden rule: if a point (x, y) is on the parabola, then substituting the x-value into the equation should give us the y-value.

Option A: (-6, 206)

Let's start with option A, the point (-6, 206). Here, x = -6 and y = 206. We need to substitute x = -6 into our equation and see if we get y = 206.

  • Calculation: y = -6x² + 3x + 8 y = -6(-6)² + 3(-6) + 8 y = -6(36) - 18 + 8 y = -216 - 18 + 8 y = -234 + 8 y = -226

  • Comparison: We calculated y = -226, but the point's y-coordinate is 206. Since -226 is definitely not equal to 206, the point (-6, 206) is NOT on the parabola. So, we can strike this one off the list, guys.

Option B: (-3, 323)

Next up is option B, the point (-3, 323). Here, x = -3 and y = 323. Let's plug x = -3 into our equation.

  • Calculation: y = -6x² + 3x + 8 y = -6(-3)² + 3(-3) + 8 y = -6(9) - 9 + 8 y = -54 - 9 + 8 y = -63 + 8 y = -55

  • Comparison: Our calculation gives y = -55, but the point's y-coordinate is 323. Since -55 ≠ 323, the point (-3, 323) is NOT on the parabola. Another one bites the dust!

Option C: (-3, -55)

Now let's check out option C, the point (-3, -55). This one looks interesting, especially after our calculation for option B! Here, x = -3 and y = -55. We've already done most of the work for x = -3.

  • Calculation: From our test of option B, we already found that when x = -3, the equation yields: y = -6(-3)² + 3(-3) + 8 = -55.

  • Comparison: Our calculated y-value is -55, and the point's y-coordinate is also -55. Boom! Since -55 = -55, this means the point (-3, -55) IS on the parabola. We've found our winner, folks!

Option D: (-6, 8)

Just to be thorough, let's quickly check option D, the point (-6, 8). Here, x = -6 and y = 8.

  • Calculation: We already calculated the result for x = -6 when we tested option A. We found: y = -6(-6)² + 3(-6) + 8 = -226.

  • Comparison: Our calculated y-value is -226, but the point's y-coordinate is 8. Since -226 ≠ 8, the point (-6, 8) is NOT on the parabola.

Conclusion: The Winning Point!

So, after rigorously testing all four options, we found that only one point satisfies the equation y = -6x² + 3x + 8. That point is (-3, -55). This means that if you were to graph this parabola, the point with coordinates x = -3 and y = -55 would be located directly on the curve.

This whole process highlights a super important skill in mathematics: verification. Being able to plug in values and check if they work is crucial for confirming solutions, understanding functions, and building confidence in your mathematical abilities. Keep practicing these steps, and you'll be a parabola pro in no time! Remember, every point on the line or curve of an equation is a solution to that equation when its coordinates are substituted. It's all about finding those numbers that make the equation true. Keep up the great work, everyone!