Quadratic Function Analysis: Finding Maximum Value And Its Location For G(x) = -3x² + 30x - 72

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Hey guys! Today, we're diving deep into the fascinating world of quadratic functions, specifically focusing on the function g(x) = -3x² + 30x - 72. We'll be tackling some key questions about this function, like whether it has a minimum or maximum value, what that value is, and where it occurs. So, buckle up and get ready for a mathematical adventure!

Decoding the Quadratic Form: Maximum or Minimum?

The first question we need to answer is: Does the function g(x) = -3x² + 30x - 72 have a minimum or maximum value? To figure this out, we need to understand the fundamental shape of a quadratic function. Quadratic functions, like our g(x), are characterized by their x² term. When graphed, they form a U-shaped curve called a parabola. Now, this parabola can either open upwards or downwards, and that's the key to determining whether we have a minimum or a maximum.

Think of it like this: if the parabola opens upwards, it has a lowest point, a minimum. If it opens downwards, it has a highest point, a maximum. The direction the parabola opens is determined by the coefficient of the x² term. If this coefficient is positive, the parabola opens upwards; if it's negative, the parabola opens downwards. In our function, g(x) = -3x² + 30x - 72, the coefficient of x² is -3, which is negative. This tells us that the parabola opens downwards. So, drumroll please… g(x) has a maximum value! Understanding the sign of the leading coefficient is crucial in determining the concavity of the parabola and whether the quadratic function possesses a maximum or a minimum value. The negative coefficient in this case signifies a downward-opening parabola, indicating a maximum value. This concept is fundamental in various applications, including optimization problems in calculus and economics, where identifying maximum or minimum points is essential for decision-making and analysis. Furthermore, the coefficient's magnitude influences the parabola's 'steepness'; a larger magnitude results in a narrower parabola, while a smaller one produces a wider curve. Grasping these nuances provides a deeper understanding of quadratic functions and their behavior across diverse contexts. Moreover, the interplay between the quadratic, linear, and constant terms in the function dictates the parabola's positioning on the coordinate plane. The linear term, in particular, affects the horizontal placement of the vertex, while the constant term determines the y-intercept. A thorough comprehension of these elements enables a more complete and insightful analysis of quadratic functions.

Finding the Peak: What is the Maximum Value?

Now that we know g(x) has a maximum value, the next logical question is: What is the function's maximum value? This maximum value corresponds to the y-coordinate of the vertex of the parabola. There are a couple of ways we can find this vertex. One way is to complete the square, and the other is to use the vertex formula.

Let's use the vertex formula. For a quadratic function in the form f(x) = ax² + bx + c, the x-coordinate of the vertex is given by x = -b / 2a. In our case, a = -3 and b = 30. Plugging these values into the formula, we get:

x = -30 / (2 * -3) = -30 / -6 = 5

So, the x-coordinate of the vertex is 5. To find the y-coordinate (which is the maximum value of the function), we simply plug this value back into our original function:

g(5) = -3(5)² + 30(5) - 72 = -3(25) + 150 - 72 = -75 + 150 - 72 = 3

Therefore, the maximum value of the function g(x) is 3. Determining the maximum or minimum value of a quadratic function involves pinpointing the vertex of its parabolic graph. The vertex represents the point at which the function transitions from increasing to decreasing (for a maximum) or vice versa (for a minimum). As calculated using the vertex formula, the x-coordinate provides the input value that yields the extreme value, while the corresponding y-coordinate represents the actual maximum or minimum output. This vertex is not merely a mathematical point; it often holds significant practical meaning in real-world scenarios. For instance, in physics, it might represent the peak height of a projectile's trajectory, and in business, it could signify the production level that maximizes profit. Consequently, the ability to accurately calculate and interpret the vertex is an invaluable skill across various disciplines. Furthermore, the process of finding the vertex can be approached using alternative methods, such as completing the square, which transforms the quadratic function into vertex form, directly revealing the vertex coordinates. Choosing the most efficient method depends on the specific function and the context of the problem, but the end goal remains the same: to identify the turning point that signifies the function's extreme value.

Pinpointing the Location: Where Does the Maximum Occur?

We've found the maximum value, but now we need to know: Where does this maximum value occur? This is simply asking for the x-coordinate of the vertex, which we already calculated! Remember, we found that x = 5. This means that the maximum value of g(x) occurs when x = 5. Identifying the location of the maximum or minimum value is crucial for understanding the behavior of the quadratic function and its applications. As we determined earlier, the x-coordinate of the vertex reveals the input value at which the function attains its extreme value. In practical contexts, this location may have significant implications. For example, if the function models the profit of a business, the x-coordinate of the vertex would indicate the level of production that maximizes profit. This insight is invaluable for decision-making and optimization in various fields. Furthermore, the location of the vertex can be visualized on the graph of the parabola. It represents the axis of symmetry, a vertical line that divides the parabola into two mirror-image halves. This symmetry is a fundamental property of quadratic functions and can be used to simplify calculations and analysis. For instance, if we know the function's value at a certain distance from the axis of symmetry, we can infer its value at the same distance on the opposite side. Therefore, pinpointing the location of the maximum or minimum value not only provides information about the function's behavior but also unlocks additional insights into its symmetry and overall structure.

Wrapping Up: Mastering Quadratic Functions

So, to recap, for the quadratic function g(x) = -3x² + 30x - 72:

  • It has a maximum value.
  • The maximum value is 3.
  • This maximum value occurs at x = 5.

By understanding the shape of parabolas, the vertex formula, and the significance of the coefficients, we can confidently analyze and interpret quadratic functions. You guys have nailed it! Keep exploring the world of mathematics, and you'll uncover even more amazing concepts and applications.

  • Does the function g(x) = -3x² + 30x - 72 have a minimum or maximum value?
  • What is the function's minimum or maximum value for g(x) = -3x² + 30x - 72?
  • At what x-value does the minimum or maximum value occur for g(x) = -3x² + 30x - 72?