Problem 38 - College Algebra

Question

Determine, without graphing, whether the given quadratic function has a minimum value or \(a\) maximum value. Then find the coordinates of the minimum or the maximum point. \(f(x)=-2 x^{2}-12 x+3\)

Step-by-Step Solution

Answer

The given quadratic function has a maximum point. The coordinates of the maximum point are (1.5, -3.0).

Step 1: Determine the nature of the vertex point

Observe the coefficient of \(x^{2}\) in \(f(x)=-2 x^{2}-12 x+3\). In this case, it is -2 which is less than zero. This indicates that the given quadratic function has a maximum value.

Step 2: Find the x-coordinate of the maximum point

The x-coordinate of the vertex point is given by \(-\frac{b}{2a}\). Substituting \(a=-2\) and \(b =-12\) from the function \(f(x)=-2 x^{2}-12 x+3\) into this formula gives \(-\frac{-12}{2(-2)}= 1.5\). Therefore, the x-coordinate of the maximum point is 1.5.

Step 3: Find the y-coordinate of the maximum point

Substitute \(x = 1.5\) into the function to get the y-coordinate which is \(f(1.5)=-2 (1.5)^{2}-12 (1.5)+3\). Simplifying this gives \(f(1.5)=-3.0\). Therefore, the y-coordinate of the maximum point is -3.0.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a crucial concept in understanding quadratic functions. It represents the highest or lowest point on the graph of a quadratic equation, which is a curve called a parabola. In the case of the quadratic function from our exercise, where the general form is given by \(f(x) = ax^2 + bx + c\), the vertex can be found using the formula \(x = -\frac{b}{2a}\). The significance lies in determining the direction of the parabola as well. If the coefficient \(a\) is positive, the parabola opens upwards, and we look for the minimum value. Conversely, if \(a\) is negative, the parabola opens downwards, yielding a maximum value at the vertex.

Understanding the vertex helps in drawing the parabola and identifying properties such as symmetry, as the vertex is also the axis of symmetry for the graph. If the equation of the parabola is rewritten in vertex form, \(f(x) = a(x - h)^2 + k\), where \((h, k)\) are the coordinates of the vertex, one can easily sketch the graph and determine the vertex's location with respect to the x and y-axes.

Coordinates of the Maximum Point

The coordinates of the maximum point of a quadratic function are especially important when dealing with optimization problems. The maximum point, in this context, refers to the vertex of the parabola since the coefficient of \(x^2\) is negative, and thus the parabola opens downwards. To find the vertex or the maximum point, we first calculate the x-coordinate using the formula \(x = -\frac{b}{2a}\).

Once we have found the x-coordinate of the vertex, finding the y-coordinate is simply a matter of substituting this value back into the original quadratic equation. The resulting value of \(f(x)\) gives us the y-coordinate. Together, \((x, f(x))\) forms the coordinates of the maximum point on the graph. This technique is essential for determining the peak value that the function can attain without visually graphing the function.

Quadratic Function Analysis

When analyzing quadratic functions, it's key to consider both the algebraic expression and the graphic representation. Our exercise focuses on an algebraic approach to determine the maximum value of a quadratic function without graphing. We use the sign of the coefficient \(a\) to decide whether the function has a maximum or minimum value. Since in our case, \(a < 0\), we know the parabola opens downwards and has a maximum value.

It's also important to note that a quadratic function is symmetrical around the vertical axis passing through its vertex. The maximum point, therefore, not only gives us a peak value but also divides the parabola into two mirror images. In the broader context of mathematical analysis, the vertex and maximum or minimum values assist in solving real-world problems involving projectile motion, cost optimization, or design of structures, making it a fundamental skill for students to master.