91Ó°ÊÓ

The vertex of a parabola is a crucial concept in understanding quadratic functions. For a quadratic equation of the form \( f(x) = ax^2 + bx + c \), the vertex represents the highest or lowest point of the graph. This is because a parabola is symmetrical around its vertex.
To find the vertex, we use the formula for the vertex's x-coordinate, which is \( x = -\frac{b}{2a} \). For example, in the quadratic function \( f(s) = s^2 - 1.2s + 16 \), we identify \( a = 1 \) and \( b = -1.2 \). Plugging these values into the formula, we get
\[ s = -\frac{-1.2}{2 \times 1} = 0.6 \]
This means the vertex occurs at \( s = 0.6 \).

The vertex gives us a crucial point on the graph:

• It helps determine the direction of the parabola (whether it opens up or down).
• It provides the minimum or maximum value.
• It serves as a point of symmetry for the parabola.

Quadratic functions can have either a minimum or maximum value depending on the orientation of the parabola. If the parabola opens upwards, the vertex provides a minimum point. Conversely, if the parabola opens downwards, the vertex provides a maximum point.

In our function \( f(s) = s^2 - 1.2s + 16 \), since \( a = 1 \) is positive, the parabola opens upwards, indicating a minimum value at its vertex.

To find this minimum value, we substitute the s-coordinate of the vertex back into the function:
\[ f(0.6) = (0.6)^2 - 1.2 \times 0.6 + 16 = 0.36 - 0.72 + 16 \]
Which simplifies to 15.64.

Thus, the minimum value is:

• Determined by evaluating the function at the vertex.
• Provides the lowest point on the graph for a parabola that opens upward.

The quadratic formula is an essential tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). While it is primarily used for finding the roots of a quadratic equation, it indirectly relates to finding attributes such as the vertex and the minimum or maximum value.

The quadratic formula is expressed as:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
While solving for the vertex or the extremum of the quadratic function, we modify this formula to just \( x = -\frac{b}{2a} \), focusing only on the part without the square root.

Understanding how the quadratic formula connects different aspects of a quadratic equation can:

• Help in finding key points like roots and vertex.
• Offer insights into the nature and shape of the parabola.
• Provide a comprehensive toolkit for analyzing quadratic functions.

Using these interconnections, we can solve a variety of problems more effectively.