91Ó°ÊÓ

Parabolas are fascinating curves that appear in many mathematical contexts. The shape of a parabola is defined by a quadratic function, typically written in the standard form as \( ax^2 + bx + c \). Understanding this curve is vital, as it models various real-world phenomena such as the path of projectiles or structures like satellite dishes.
Key characteristics of a parabola:

• If the coefficient \( a \) is positive, the parabola opens upwards, resembling a U-shape.
• If \( a \) is negative, it opens downwards, like an upside-down U.

In the given exercise, the quadratic function \( f(x) = -4x^2 + 5x \) has a negative \( a \, (a = -4) \), indicating a downward opening parabola. This automatically suggests that the function has a maximum point since the parabola caps at its highest point before continuing downwards.

The vertex of a parabola is a significant point, representing the peak or the lowest point of the parabola. To find the vertex for a quadratic equation in the form \( ax^2 + bx + c \), we use the vertex formula for the x-coordinate:
\[ x = \frac{-b}{2a} \]
This formula is derived from completing the square or using calculus (derivative setting). It pinpoints where the curve changes direction or the highest/lowest point.
In the exercise, by substituting \( b = 5 \) and \( a = -4 \) into the formula, we calculate:
\[ x = \frac{-5}{2(-4)} = \frac{5}{8} = 0.625 \]
The x-coordinate of the vertex is therefore 0.625. This x-value, when inserted back into the original function, helps to find the y-coordinate, completing the vertex's location.

The concepts of maximum and minimum values in quadratic functions revolve around their parabolic nature.
When a parabola opens upwards (\( a > 0 \)), it has a minimum value at its vertex, forming a lowest point of the curve.
On the other hand, when a parabola opens downwards (\( a < 0 \)), it displays a maximum value at its vertex, being at the top of its arc.
In our exercise:

• The quadratic function \( f(x) = -4x^2 + 5x \) has a \( a < 0 \), meaning a downward-opening parabola with a maximum value.
• By substituting \( x = 0.625 \) into the function, the y-coordinate or the maximum value is found: \( f(0.625) = 1.5625 \).
• Rounding it off to the nearest hundredth gives the final answer: 1.56.

Recognizing these values is crucial for understanding the behavior of the quadratic function in real-world applications. It shows you the turning point from increase to decrease or vice versa, highlighting the extremities of the function, whether it's a maximum peak or a minimum trough.