Finding The Value Of 'a' In A Quadratic Function Vertex Form
Hey guys! Today, we're diving into the fascinating world of quadratic functions, specifically focusing on how to determine the value of 'a' in the vertex form of a quadratic equation. You know, that format that sometimes looks intimidating but is actually super helpful! We've got a juicy problem on our hands, so let's break it down step by step and make sure we nail this concept. Get ready to unleash your inner math whiz!
Setting the Stage: Understanding Vertex Form and Key Information
So, the problem throws us some key pieces of information. First, it tells us that the vertex of our quadratic function is (6, 2). Remember, the vertex is that turning point of the parabola, the highest or lowest point on the graph. In the vertex form equation, , the vertex is represented by the coordinates (h, k). This is crucial information because it immediately gives us the values of h and k! In our case, h = 6 and k = 2. We're already making progress, guys! It's like unlocking a secret level in a video game.
Next, we're given that the y-intercept of the function is (0, -70). The y-intercept is the point where the parabola crosses the y-axis. This means that when x = 0, the function's value, f(x), is -70. This is another vital clue in our quest to find 'a'. Think of it as a hidden treasure chest filled with mathematical gold! The problem even gives us a starting point by showing the equation with the y-intercept plugged in: $-70=a(0-6)^2+2$. This equation is our roadmap to solving for 'a'. It’s like having the GPS coordinates to our final destination. Now, let's talk about why understanding the vertex form is so darn important. The vertex form is incredibly useful because it directly reveals the vertex of the parabola. This makes it easy to visualize the graph and understand the function's behavior. The 'a' value, which we're trying to find, plays a significant role in determining the parabola's shape. If 'a' is positive, the parabola opens upwards, like a smiley face. If 'a' is negative, it opens downwards, like a frown. The magnitude of 'a' also affects how wide or narrow the parabola is. A larger absolute value of 'a' means a narrower parabola, while a smaller absolute value means a wider parabola. This connection between 'a' and the parabola's shape is a fundamental concept in understanding quadratic functions. So, by finding 'a', we're not just solving for a variable; we're gaining deeper insight into the function's characteristics. It's like becoming fluent in the language of parabolas!
Decoding the Equation: Solving for 'a'
Alright, let's get down to business and solve for 'a'! We've got the equation: $-70 = a(0 - 6)^2 + 2$. Our mission is to isolate 'a' on one side of the equation. It's like a mathematical puzzle, and we're the master puzzle solvers! First, let's simplify the equation. Inside the parentheses, we have (0 - 6), which is simply -6. So, the equation becomes: $-70 = a(-6)^2 + 2$. Now, let's deal with the exponent. (-6)^2 means -6 multiplied by -6, which equals 36. Remember, a negative number multiplied by a negative number gives a positive result! So, our equation now looks like this: $-70 = 36a + 2$. We're getting closer, guys! It's like we're closing in on the treasure. Next, we need to get rid of that pesky '+ 2' on the right side of the equation. To do that, we subtract 2 from both sides. This is a crucial step in maintaining the equation's balance. What we do to one side, we must do to the other! Subtracting 2 from both sides gives us: $-70 - 2 = 36a + 2 - 2$ which simplifies to: $-72 = 36a$. Almost there! We're in the home stretch. Now, to isolate 'a', we need to divide both sides of the equation by 36. This will undo the multiplication of 'a' by 36. So, we have: $ rac{-72}{36} = rac{36a}{36}$. Dividing -72 by 36 gives us -2, and dividing 36a by 36 gives us 'a'. Therefore, we have: $a = -2$. We did it! We found the value of 'a'! It's like cracking the code and unlocking the secret message. This process of isolating a variable is a fundamental skill in algebra, and mastering it will empower you to solve a wide range of mathematical problems. So, congratulations on conquering this equation!
Confirming the Answer: Ensuring Accuracy
But wait, we're not quite done yet! It's always a good idea to double-check our work to make sure we haven't made any silly mistakes. Think of it as our quality control check, ensuring that our answer is accurate and reliable. To verify our solution, we can plug the value of a = -2 back into the original equation and see if it holds true. Our original equation was: $-70 = a(0 - 6)^2 + 2$. Now, let's substitute a = -2: $-70 = -2(0 - 6)^2 + 2$. We'll simplify this equation step by step, just like we did before. First, simplify the parentheses: $-70 = -2(-6)^2 + 2$. Next, square -6: $-70 = -2(36) + 2$. Now, multiply -2 by 36: $-70 = -72 + 2$. Finally, add -72 and 2: $-70 = -70$. Voila! The equation holds true. This confirms that our value of a = -2 is indeed correct. It's like getting a gold star on our math assignment! This step of verification is crucial in any mathematical problem-solving process. It not only ensures the accuracy of our answer but also reinforces our understanding of the concepts involved. By plugging our solution back into the original equation, we're solidifying our grasp of the relationship between the variables and the equation itself. So, never skip this important step, guys! It's the key to becoming a confident and accurate problem solver.
The Final Verdict: Selecting the Correct Option
Okay, we've done the hard work, and now it's time to choose the correct answer from the given options. We found that the value of 'a' is -2. Looking at the options provided:
A. -6 B. -2 C. 2 D. 6
The correct answer is B. -2. We nailed it! It's like reaching the top of the mountain after a challenging climb. We can now confidently say that we understand how to determine the value of 'a' in the vertex form of a quadratic equation when given the vertex and a point on the parabola. This is a valuable skill that will serve you well in your mathematical journey. Remember, guys, math isn't about memorizing formulas; it's about understanding the concepts and applying them to solve problems. By breaking down the problem into smaller steps, simplifying equations, and verifying our answers, we can conquer any mathematical challenge that comes our way. So, keep practicing, keep exploring, and keep having fun with math! And remember, the world of quadratic functions is just one small part of the vast and fascinating world of mathematics. There's always more to learn and discover!
Key Takeaways: Mastering Quadratic Functions
Before we wrap things up, let's quickly recap the key takeaways from this problem-solving adventure. These are the nuggets of wisdom that you can carry with you as you continue your exploration of quadratic functions. Remember these points, and you'll be well-equipped to tackle similar problems in the future.
- Vertex Form: The vertex form of a quadratic equation, , is your best friend when you need to identify the vertex of the parabola. The vertex is simply (h, k). Knowing the vertex is like having a map to the parabola's most important landmark.
- Y-intercept: The y-intercept is the point where the parabola crosses the y-axis. It's given by the coordinates (0, y), where y is the value of the function when x = 0. The y-intercept provides another crucial piece of information about the parabola's position on the coordinate plane.
- The Role of 'a': The value of 'a' in the vertex form determines the parabola's shape and direction. A positive 'a' means the parabola opens upwards, a negative 'a' means it opens downwards, and the magnitude of 'a' affects the parabola's width. Understanding 'a' is like understanding the personality of the parabola.
- Solving for 'a': To find the value of 'a', you'll typically be given the vertex (h, k) and another point on the parabola (like the y-intercept). Plug these values into the vertex form equation and solve for 'a' using algebraic manipulation. It's like solving a puzzle, where each step brings you closer to the solution.
- Verification is Key: Always, always, always verify your solution by plugging the value of 'a' back into the original equation. This ensures that your answer is correct and reinforces your understanding of the concepts. It's like having a safety net, ensuring that you land on your feet.
By mastering these key takeaways, you'll be well on your way to becoming a quadratic function pro! So, keep practicing, keep exploring, and keep having fun with math. And remember, guys, there's a whole universe of mathematical concepts waiting to be discovered. So, let's continue our journey of learning and exploration!
