Vertex Of G(x) = 8x² - 48x + 65 Finding The Vertex Of A Quadratic Function

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Finding the vertex of a quadratic function is a fundamental concept in algebra, with applications ranging from optimization problems to graphing parabolas. In this comprehensive guide, we will delve deep into the process of determining the vertex of the quadratic function g(x) = 8x² - 48x + 65. We will explore multiple methods, including completing the square and using the vertex formula, to ensure a thorough understanding. By the end of this article, you will not only be able to solve this specific problem but also have the tools to tackle similar challenges with confidence.

Understanding Quadratic Functions and the Vertex

Before diving into the solution, let's first establish a solid understanding of quadratic functions and the significance of the vertex. A quadratic function is a polynomial function of degree two, generally expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is a parabola, a U-shaped curve that opens either upwards (if a > 0) or downwards (if a < 0). The vertex is the point where the parabola changes direction, representing either the minimum (if the parabola opens upwards) or the maximum (if the parabola opens downwards) value of the function.

Why is the Vertex Important?

The vertex holds significant importance in various applications. In optimization problems, it helps identify the maximum or minimum value of a quantity represented by a quadratic function. For example, if a quadratic function models the profit of a business, the vertex would indicate the production level that maximizes profit. Similarly, in physics, the vertex can represent the highest point reached by a projectile or the minimum potential energy of a system. Understanding the vertex is also crucial for accurately graphing parabolas and analyzing their behavior.

Method 1: Completing the Square

One of the most powerful methods for finding the vertex of a quadratic function is completing the square. This technique involves rewriting the quadratic expression in vertex form, which directly reveals the coordinates of the vertex. The vertex form of a quadratic function is given by f(x) = a(x - h)² + k, where (h, k) represents the vertex of the parabola. Let's apply this method to our function, g(x) = 8x² - 48x + 65.

Step-by-Step Guide to Completing the Square

  1. Factor out the leading coefficient from the x² and x terms: In our case, the leading coefficient is 8. Factoring it out from the first two terms gives us: g(x) = 8(x² - 6x) + 65
  2. Complete the square inside the parentheses: To complete the square, we need to add and subtract the square of half the coefficient of the x term. The coefficient of the x term is -6, so half of it is -3, and the square of -3 is 9. Therefore, we add and subtract 9 inside the parentheses: g(x) = 8(x² - 6x + 9 - 9) + 65
  3. Rewrite the expression inside the parentheses as a perfect square: The expression x² - 6x + 9 is a perfect square trinomial, which can be factored as (x - 3)²: g(x) = 8((x - 3)² - 9) + 65
  4. Distribute the leading coefficient and simplify: Distribute the 8 to both terms inside the parentheses: g(x) = 8(x - 3)² - 72 + 65 g(x) = 8(x - 3)² - 7

Now, the function is in vertex form, g(x) = 8(x - 3)² - 7. Comparing this to the general vertex form f(x) = a(x - h)² + k, we can identify the vertex as (h, k) = (3, -7). Therefore, the vertex of the quadratic function g(x) = 8x² - 48x + 65 is (3, -7).

Method 2: Using the Vertex Formula

Another efficient method for finding the vertex is by using the vertex formula. For a quadratic function in the standard form f(x) = ax² + bx + c, the x-coordinate of the vertex (h) is given by h = -b / 2a. Once we have the x-coordinate, we can find the y-coordinate (k) by substituting h back into the original function, k = f(h). Let's apply this formula to our function, g(x) = 8x² - 48x + 65.

Step-by-Step Guide to Using the Vertex Formula

  1. Identify the coefficients a, b, and c: In our function, g(x) = 8x² - 48x + 65, we have a = 8, b = -48, and c = 65.
  2. Calculate the x-coordinate of the vertex (h): Using the formula h = -b / 2a, we get: h = -(-48) / (2 * 8) = 48 / 16 = 3
  3. Calculate the y-coordinate of the vertex (k): Substitute h = 3 back into the original function: k = g(3) = 8(3)² - 48(3) + 65 k = 8(9) - 144 + 65 k = 72 - 144 + 65 k = -7

Thus, the vertex of the quadratic function g(x) = 8x² - 48x + 65 is (3, -7). This confirms the result we obtained using the completing the square method.

Comparing the Methods

Both completing the square and the vertex formula are effective methods for finding the vertex of a quadratic function. However, they have their own advantages and disadvantages.

  • Completing the square: This method provides a deeper understanding of the structure of the quadratic function and how it relates to the vertex form. It is particularly useful when you need to rewrite the function in vertex form for other purposes, such as graphing. However, it can be more algebraically intensive and time-consuming, especially for functions with complex coefficients.
  • Vertex formula: This method is more direct and efficient for finding the vertex coordinates, especially when you only need the vertex and not the vertex form of the function. It involves simple arithmetic calculations and is less prone to algebraic errors. However, it does not provide as much insight into the function's structure as completing the square.

Ultimately, the choice of method depends on your preference and the specific requirements of the problem. It's beneficial to be proficient in both methods to tackle a wide range of quadratic function problems.

Graphical Interpretation

Visualizing the graph of the quadratic function can provide further insight into the significance of the vertex. The graph of g(x) = 8x² - 48x + 65 is a parabola that opens upwards (since the coefficient of x² is positive). The vertex (3, -7) represents the minimum point on the parabola. This means that the function reaches its lowest value, -7, when x = 3. The parabola is symmetric about the vertical line that passes through the vertex, which is the line x = 3. Understanding the graphical representation can help you verify your algebraic calculations and interpret the results in a real-world context.

Conclusion

In this comprehensive guide, we have thoroughly explored the process of finding the vertex of the quadratic function g(x) = 8x² - 48x + 65. We have demonstrated two powerful methods: completing the square and using the vertex formula. Both methods lead to the same conclusion: the vertex of the function is (3, -7). We also discussed the importance of the vertex in various applications and provided a graphical interpretation to enhance understanding. By mastering these techniques, you will be well-equipped to analyze quadratic functions and solve related problems with confidence. Remember to practice these methods with different quadratic functions to solidify your understanding and develop your problem-solving skills.

Therefore, the correct answer to the question “What is the vertex of g(x) = 8x² - 48x + 65?” is C. (3, -7).