Problem 40 - Algebra and Trigonometry

Question

Find the maximum or minimum value of the function. $$ f(x)=-\frac{x^{2}}{3}+2 x+7 $$

Step-by-Step Solution

Answer

The maximum value of the function is 10.

Step 1: Identify the Type of Function

The function given is quadratic because it includes a term with \(x^2\). The standard form of a quadratic function is \(f(x) = ax^2 + bx + c\). In this case, \(a = -\frac{1}{3}\), \(b = 2\), and \(c = 7\).

Step 2: Determine the Parabola Direction

The parabola opens downward because \(a = -\frac{1}{3}\) is negative. Therefore, the function has a maximum point.

Step 3: Calculate the Vertex

The vertex of a quadratic function \(ax^2 + bx + c\) can be found using the vertex formula \(x = -\frac{b}{2a}\). Substitute \(b = 2\) and \(a = -\frac{1}{3}\), we get:\[x = -\frac{2}{2(-\frac{1}{3})} = -\frac{2}{-\frac{2}{3}} = 3\]

Step 4: Find the Function Value at the Vertex

Substitute \(x = 3\) into the original function to find the maximum value:\[f(3) = -\frac{(3)^2}{3} + 2(3) + 7 = -3 + 6 + 7 = 10\]Thus, the maximum value of the function is 10.

Key Concepts

Vertex Formula

The vertex formula is a key player when working with quadratic functions. It helps us find the vertex of a parabola, which is the point where the parabola changes direction. This is crucial in determining either the maximum or the minimum value of the function, depending on the curvature of the parabola. The generic formula to find the x-coordinate of the vertex is given by:\[ x = -\frac{b}{2a} \]where

• \(a\) is the coefficient of \(x^2\)
• \(b\) is the coefficient of \(x\)
• \(c\) is the constant term

Substituting the given values from the quadratic formula allows us to know exactly where the vertex lies along the x-axis. Once the x-coordinate is determined, it can be plugged back into the function to find the corresponding y-coordinate. Together, these point coordinates (x, f(x)) represent the vertex, which can be the highest or lowest point of the parabola.

Parabola Direction

Understanding the direction of a parabola is essential in analyzing quadratic functions. The direction in which the parabola opens is determined by the sign of the coefficient \(a\). If \(a\) is positive, the parabola opens upwards, resembling a U-shape, indicating that the function has a minimum point at its vertex. Conversely, if \(a\) is negative, the parabola opens downwards, resembling an upside-down U-shape, indicating a maximum point.

For the given quadratic function \(f(x) = -\frac{x^{2}}{3} + 2x + 7\), \(a = -\frac{1}{3}\), which is negative, signaling that the parabola opens downwards. Therefore, this quadratic function has a maximum value at its vertex. Recognizing the direction of the parabola early in problem-solving helps predict whether you're solving for a maximum or minimum, guiding the subsequent steps.

Maximum Value Determination

Determining the maximum value of a quadratic function is a step-by-step process that involves calculating coordinates of the vertex, particularly when the parabola opens downward. In our exercise, after finding the x-coordinate of the vertex using the vertex formula \(x = -\frac{b}{2a}\), we substitute this value back into the original quadratic function to find the function's value at that specific point.

In the function given, we calculate the vertex's x-coordinate as 3. Substitute \(x = 3\) back into the equation:

\[ f(3) = -\frac{(3)^2}{3} + 2(3) + 7 \]
Simplifying this step-by-step gives us the maximum value is 10.

Therefore, the highest point on the graph occurs at \( (3, 10) \). This comprehensive approach provides clarity to assess whether the vertex represents the maximum or minimum, based on the parabola direction. Being methodical with these calculations ensures accuracy in finding extreme values.