Problem 59 - Algebra 2

Question

Determine whether \(f(x)=3 x^{2}-12 x-7\) has a maximum or a minimum value. Then find the maximum or minimum value.

Step-by-Step Solution

Answer

The function has a minimum value of -19 at \(x = 2\).

Step 1: Identify the Form of the Quadratic Function

The given function is a quadratic equation of the form \(f(x) = ax^2 + bx + c\). In this case, \(a = 3\), \(b = -12\), and \(c = -7\). Since \(a > 0\), this parabola opens upwards and thus has a minimum value at its vertex.

Step 2: Calculate the Vertex of the Parabola

The vertex form of a parabola provides the point where the minimum (or maximum) value occurs. The x-coordinate of the vertex can be found using the formula \(x = -\frac{b}{2a}\). Substituting the given values, we have \(x = -\frac{-12}{2 \times 3} = 2\).

Step 3: Determine the Minimum Value of the Function

Substitute \(x = 2\) back into the function \(f(x)\) to find the minimum value. \[f(2) = 3(2)^2 - 12(2) - 7 = 3(4) - 24 - 7 = 12 - 24 - 7 = -19.\] Thus, the minimum value of the function is \(-19\).

Key Concepts

Parabola

A quadratic function forms a curve called a parabola. All quadratic functions are comparable to a bowl shape, either opening upwards or downwards. This configuration is determined by the value of the coefficient \(a\) in the general quadratic equation \(f(x) = ax^2 + bx + c\).
When \(a > 0\), the parabola opens upwards, resembling a U-shape, which means it will have a minimum point. Conversely, if \(a < 0\), the parabola opens downwards, like an upside-down U, indicating a maximum point.
Parabolas are symmetrical around a vertical line, called the axis of symmetry, which helps in easily locating the vertex of the parabola.

Vertex

The vertex of a parabola is the central point of the curve. It represents the minimum point for an upward-opening parabola and the maximum point for a downward-opening one.
The vertex can be easily calculated from the quadratic equation using the vertex formula for the x-coordinate: \(x = -\frac{b}{2a}\). Knowing this x-coordinate helps you determine the position of the vertex along the x-axis. In our exercise, this is how we determined that the x-coordinate is 2.
To find the complete vertex point, substitute this x-coordinate back into the quadratic function to calculate the corresponding y-coordinate. This provides the exact location and the value of the vertex, giving insight into the function's behavior at its extremities.

Minimum Value

For a quadratic function that forms an upward-opening parabola, such as in our exercise, the minimum value occurs at the vertex.
Finding the minimum value involves two steps:

• Calculate the x-coordinate of the vertex using \(x = -\frac{b}{2a}\).
• Substitute this x-value back into the function \(f(x)\) to determine the y-coordinate, which represents the minimum value.

This approach works efficiently, as demonstrated in the exercise where substituting \(x = 2\) led to finding the minimum value \(-19\). It's essential for accurately describing the function's lowest point and is especially useful in real-life applications where optimization is needed.