How To Find The Vertex Of The Parabola Y=2x^2-16x+34

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Finding the vertex of a parabola is a fundamental concept in algebra and is crucial for understanding the behavior and characteristics of quadratic functions. If you're grappling with the equation y=2x^2-16x+34 and need to pinpoint its vertex, you've come to the right place! In this comprehensive guide, we'll break down the process step-by-step, ensuring you grasp the underlying principles and confidently solve similar problems. Let's dive in, guys!

Understanding Parabolas and the Vertex

Before we jump into the calculations, let's establish a solid understanding of parabolas and their key features. A parabola is a U-shaped curve that represents the graph of a quadratic function. Quadratic functions are expressed in the general form of y = ax^2 + bx + c, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The vertex is the most crucial point on the parabola – it's either the minimum point (if the parabola opens upwards) or the maximum point (if the parabola opens downwards). Essentially, it’s the turning point of the curve.

The coefficient 'a' plays a significant role in determining the parabola's orientation. If 'a' is positive, the parabola opens upwards, resembling a smile. If 'a' is negative, the parabola opens downwards, like a frown. The vertex's coordinates, represented as (h, k), provide valuable information about the parabola's position on the coordinate plane. The 'h' value represents the x-coordinate of the vertex, and the 'k' value represents the y-coordinate. Knowing the vertex allows us to easily sketch the parabola and understand its range, axis of symmetry, and other essential properties. So, mastering the vertex calculation is a key step in understanding quadratic functions and their graphical representations. In the equation y=2x^2-16x+34, we can see that a=2, b=-16, and c=34. Since 'a' is positive, we know this parabola opens upwards, and the vertex will be the minimum point. This gives us a visual idea of what we are trying to find, ensuring our answer makes sense within the context of the problem. Understanding these fundamentals will help us approach the calculation more intuitively and confidently.

Methods to Find the Vertex

There are two primary methods for finding the vertex of a parabola: completing the square and using the vertex formula. Both methods are effective, but they approach the problem from slightly different angles. We'll explore both methods in detail, giving you the flexibility to choose the one that resonates best with your understanding and problem-solving style.

1. Completing the Square

Completing the square is a powerful algebraic technique that transforms a quadratic expression into a perfect square trinomial, plus a constant term. This form directly reveals the vertex coordinates. Let's walk through the steps for our equation, y = 2x^2 - 16x + 34.

  1. Factor out the coefficient of the x^2 term (if it's not 1): In our case, the coefficient of x^2 is 2, so we factor it out from the first two terms: y = 2(x^2 - 8x) + 34. It's crucial to only factor out from the terms containing 'x'. The constant term remains untouched for now.
  2. Complete the square inside the parentheses: To complete the square for the expression inside the parentheses (x^2 - 8x), we take half of the coefficient of the x term (-8), square it ((-8/2)^2 = 16), and add it inside the parentheses. However, since we're adding it inside the parentheses that are multiplied by 2, we also need to subtract 2 * 16 = 32 outside the parentheses to maintain the equation's balance. This gives us: y = 2(x^2 - 8x + 16) + 34 - 32. The critical point here is to remember to adjust for the factored-out coefficient when subtracting the term outside the parentheses.
  3. Rewrite the expression as a perfect square: The expression inside the parentheses is now a perfect square trinomial, which can be factored as (x - 4)^2. So, our equation becomes: y = 2(x - 4)^2 + 2. This form is called the vertex form of a quadratic equation, which is incredibly useful for identifying the vertex.
  4. Identify the vertex: The vertex form of a parabola is y = a(x - h)^2 + k, where (h, k) is the vertex. Comparing this with our equation y = 2(x - 4)^2 + 2, we can see that h = 4 and k = 2. Therefore, the vertex of the parabola is (4, 2). Completing the square not only gives us the vertex but also transforms the equation into a more insightful form, highlighting the parabola's symmetry and vertical shift. It’s a valuable technique to have in your mathematical toolkit.

2. Using the Vertex Formula

The vertex formula provides a direct way to calculate the coordinates of the vertex without going through the process of completing the square. The formula is derived from the process of completing the square but offers a shortcut for those who prefer a more formulaic approach. The x-coordinate of the vertex (h) is given by h = -b / 2a, and the y-coordinate (k) is found by substituting the value of h back into the original equation, k = f(h). Let's apply this formula to our equation, y = 2x^2 - 16x + 34.

  1. Identify a, b, and c: In our equation, a = 2, b = -16, and c = 34. These coefficients are the key ingredients for our formula.
  2. Calculate the x-coordinate (h): Using the formula h = -b / 2a, we substitute the values of a and b: h = -(-16) / (2 * 2) = 16 / 4 = 4. So, the x-coordinate of the vertex is 4. This tells us the horizontal position of the vertex on the graph.
  3. Calculate the y-coordinate (k): To find the y-coordinate, we substitute h = 4 back into the original equation: k = 2(4)^2 - 16(4) + 34 = 2(16) - 64 + 34 = 32 - 64 + 34 = 2. Therefore, the y-coordinate of the vertex is 2. This gives us the vertical position of the vertex.
  4. Write the vertex coordinates: The vertex of the parabola is (h, k) = (4, 2). The vertex formula is a quick and efficient method, particularly useful when you primarily need the vertex coordinates without necessarily needing the vertex form of the equation. It’s a straightforward application of a formula that bypasses the algebraic manipulation of completing the square. Both methods, completing the square and using the vertex formula, lead us to the same answer, providing flexibility in how we approach the problem.

Applying the Methods to y=2x^2-16x+34

Now, let's consolidate our understanding by applying both methods to our example equation, y = 2x^2 - 16x + 34, and verify that we arrive at the same answer. This reinforces the concepts and demonstrates the versatility of each method.

Method 1: Completing the Square

As we walked through earlier, completing the square involves transforming the quadratic expression into vertex form. Let’s recap the steps:

  1. Factor out the coefficient of x^2: y = 2(x^2 - 8x) + 34
  2. Complete the square: y = 2(x^2 - 8x + 16) + 34 - 32
  3. Rewrite as a perfect square: y = 2(x - 4)^2 + 2
  4. Identify the vertex: (4, 2)

Method 2: Using the Vertex Formula

The vertex formula provides a more direct route to the vertex coordinates. Let's apply the formula:

  1. Identify a, b, and c: a = 2, b = -16, c = 34
  2. Calculate h: h = -b / 2a = -(-16) / (2 * 2) = 4
  3. Calculate k: k = 2(4)^2 - 16(4) + 34 = 2
  4. Write the vertex coordinates: (4, 2)

As you can see, both methods confidently point to the same vertex: (4, 2). This not only confirms our calculations but also emphasizes the reliability and consistency of these techniques. Whether you prefer the algebraic manipulation of completing the square or the direct application of the vertex formula, you now have two powerful tools at your disposal for finding the vertex of a parabola. The key is to practice and become comfortable with both methods, allowing you to choose the one that best suits the problem and your personal preferences.

Common Mistakes to Avoid

While finding the vertex of a parabola is a straightforward process, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can help you avoid them and ensure accurate results. Let's highlight some of the most frequent errors:

  1. Incorrectly factoring out the coefficient of x^2: When completing the square, it's crucial to factor out the coefficient of the x^2 term only from the terms containing 'x'. A common mistake is factoring it out from the constant term as well, which changes the equation. Remember to leave the constant term untouched until after completing the square inside the parentheses.
  2. Forgetting to adjust for the factored coefficient: After adding a term inside the parentheses to complete the square, you must subtract a corresponding term outside the parentheses to maintain the equation's balance. The subtracted term should account for the factored-out coefficient. For example, if you add 16 inside parentheses that are multiplied by 2, you need to subtract 2 * 16 = 32 outside the parentheses. Neglecting this adjustment will lead to an incorrect vertex.
  3. Mixing up the signs in the vertex formula: The vertex formula is h = -b / 2a. A common mistake is forgetting the negative sign in front of 'b' or miscalculating the denominator. Double-check your signs and calculations to avoid this error. Also, remember that the y-coordinate (k) is found by substituting 'h' back into the original equation, not by using a separate formula. Getting these steps mixed up can lead to incorrect results.
  4. Misinterpreting the vertex form: The vertex form of a parabola is y = a(x - h)^2 + k, where (h, k) is the vertex. Pay close attention to the signs. The x-coordinate of the vertex is the value being subtracted from 'x' inside the parentheses, and the y-coordinate is the constant term added outside the parentheses. For example, in the equation y = 2(x - 4)^2 + 2, the vertex is (4, 2), not (-4, 2) or (4, -2). Careful interpretation of the vertex form is essential for correctly identifying the vertex coordinates.
  5. Arithmetic errors: Simple arithmetic mistakes can derail the entire process. Be meticulous with your calculations, especially when dealing with fractions, negative numbers, and exponents. It’s always a good idea to double-check your arithmetic to catch any errors before they propagate through the problem. Using a calculator can help, but make sure you understand the steps involved and are not just relying on the calculator as a black box. By being mindful of these common mistakes and practicing careful calculation, you can significantly improve your accuracy and confidence in finding the vertex of a parabola.

Conclusion

In conclusion, finding the vertex of the parabola y = 2x^2 - 16x + 34 is a clear-cut process using either completing the square or the vertex formula. We've shown that the vertex is (4, 2) using both methods. Understanding these techniques will empower you to tackle similar problems with confidence. So, go ahead and practice, and you'll be a parabola pro in no time! Remember, guys, math isn't about memorizing; it's about understanding and applying the concepts. Keep practicing, and you'll ace it!