Problem 35 - 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. \(f(x)=3 x^{2}-12 x-1\)

Step-by-Step Solution

Answer

The quadratic function \(f(x) = 3x^2 -12x -1\) has a minimum value. The coordinates of this minimum point are (2, -7).

Step 1: Identify a, b and c

From the quadratic function \(f(x) = 3x^2 -12x -1\), identify: \(a = 3\), \(b = -12\), and \(c = -1\).

Step 2: Compute the x-coordinate of the vertex

Use the formula \(-\frac{b}{2a}\) to find the x-coordinate of the vertex. Substitute \(a = 3\) and \(b = -12\) into the formula to get \(h = -\frac{-12}{2*3} = 2\). So, the x-coordinate of the vertex is 2.

Step 3: Compute the y-coordinate of the vertex

To find the y-coordinate of the vertex, substitute \(x = 2\) into the function: \(f(2) = 3(2)^2 -12(2) -1 = -7\). So, the y-coordinate of the vertex is -7.

Step 4: Determine the function's minimum or maximum

A quadratic function takes a minimum value if \(a > 0\) and a maximum value if \(a < 0\). Here \(a = 3 > 0\), hence the function has a minimum value at the point (2, -7).

Key Concepts

Vertex Formula

The vertex formula is a powerful tool when dealing with quadratic functions. It helps to find the vertex of a parabola, which is either the minimum or maximum point, depending on the orientation of the parabola. This vertex can be identified using the formula for the x-coordinate:
\(-\frac{b}{2a}\).
In this expression, \(b\) and \(a\) are the coefficients from the quadratic equation in the standard form \(ax^2 + bx + c\). By substituting the values you find from the equation into this formula, you can quickly determine the x-coordinate of the vertex.
In our case, for the function \(f(x) = 3x^2 - 12x - 1\):

• \(a = 3\)
• \(b = -12\)

Substitute in the vertex formula:
\(-\frac{-12}{2 \times 3} = 2\).
Thus, the x-coordinate of the vertex is 2, which is a part of the coordinate for our parabola's vertex.

Minimum and Maximum Values

Understanding whether a quadratic function has a minimum or maximum value is essential to solving many problems. This depends on the sign of the coefficient \(a\) in the quadratic function.
A positive \(a\) (\(a > 0\)) indicates the parabola opens upwards, meaning it has a minimum point.
Conversely, a negative \(a\) (\(a < 0\)) shows that the parabola opens downwards, indicating a maximum point.
For the function \(f(x) = 3x^2 - 12x - 1\), we have \(a = 3\), which is positive. Hence, the function has a minimum value at its vertex.
Once we know the x-coordinate of the vertex (which is 2), we can calculate the y-coordinate by substituting \(x = 2\) back into the original equation:

• \(f(2) = 3(2)^2 - 12(2) - 1 = -7\)

This confirms that the function has a minimum point at the coordinates (2, -7).

Parabola Properties

Parabolas are beautiful curves, and every quadratic function represents a parabola. They have several defining properties that can help us understand their behavior.
Key properties include:

• The vertex, which is the highest or lowest point on the parabola, depending on its orientation.
• The axis of symmetry, a vertical line that passes through the vertex and divides the parabola into two mirror-image halves. For any quadratic equation \(f(x) = ax^2 + bx + c\), the line \(x = -\frac{b}{2a}\) is the axis of symmetry.
• Focus and directrix, which are not typically needed for basic problems but are part of the geometrical definition of a parabola.

For the function \(f(x) = 3x^2 -12x -1\):

• The vertex is at \((2, -7)\).
• The axis of symmetry is the line \(x = 2\).

This parabola opens upwards because \(a = 3 > 0\), giving it a minimum at the vertex. Understanding these properties helps break down the complexities of quadratic functions.