Problem 63 - Algebra 2

Question

Find the value of the maximum or minimum of each quadratic function to the nearest hundredth. $$ f(x)=-4 x^{2}+5 x $$

Step-by-Step Solution

Answer

The maximum value is approximately 1.56.

Step 1: Identify the quadratic equation

The given quadratic function is \( f(x) = -4x^2 + 5x \). This is in the form \( ax^2 + bx + c \) where \( a = -4 \), \( b = 5 \), and \( c = 0 \).

Step 2: Determine if it is a maximum or minimum

Since \( a = -4 \) is less than zero, the parabola opens downwards, and the function has a maximum value.

Step 3: Find the vertex formula for x

The x-coordinate of the vertex for a quadratic equation \( ax^2 + bx + c \) is given by \( x = \frac{-b}{2a} \).

Step 4: Substitute values and calculate x

Substitute \( b = 5 \) and \( a = -4 \) into the vertex formula: \[ x = \frac{-5}{2(-4)} = \frac{-5}{-8} = \frac{5}{8} \approx 0.625 \]

Step 5: Find the maximum value by substituting x into f(x)

Substitute \( x = 0.625 \) back into the function to find the maximum value: \[ f(0.625) = -4(0.625)^2 + 5(0.625) \] Calculate each term: \[ (0.625)^2 = 0.390625 \] \[ -4 \times 0.390625 = -1.5625 \] \[ 5 \times 0.625 = 3.125 \] Thus, \[ f(0.625) = -1.5625 + 3.125 = 1.5625 \]

Step 6: Round to the nearest hundredth

Round 1.5625 to the nearest hundredth to obtain 1.56.

Key Concepts

Parabola

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.

Vertex Formula

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.

Maximum and Minimum Values

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.