Problem 59 - Algebra 2

Question

Find the value of the maximum or minimum of each quadratic function to the nearest hundredth. $$ f(x)=-5 x^{2}+8 x $$

Step-by-Step Solution

Answer

The maximum value of the quadratic function is 3.20.

Step 1: Identify the Form of the Quadratic Function

The given quadratic function is \( f(x) = -5x^2 + 8x \). This is in the standard form \( ax^2 + bx + c \), where \( a = -5 \), \( b = 8 \), and \( c = 0 \). Recognize that it's a downward-opening parabola since \( a = -5 < 0 \). Thus, it has a maximum point.

Step 2: Find the Vertex x-coordinate

The x-coordinate of the vertex of a quadratic function \( ax^2+bx+c \) is given by \( x = \frac{-b}{2a} \). Substitute the values of \( a \) and \( b \) into the vertex formula: \( x = \frac{-8}{2(-5)} = \frac{-8}{-10} = 0.8 \).

Step 3: Calculate the Vertex y-coordinate (Function value at Maximum)

Substitute the x-coordinate of the vertex back into the function to find the maximum value: \( f(0.8) = -5(0.8)^2 + 8(0.8) \). First, calculate \( (0.8)^2 = 0.64 \). Then compute \( -5 imes 0.64 + 8 imes 0.8 = -3.2 + 6.4 = 3.2 \).

Step 4: Round the Value to the Nearest Hundredth

The maximum value is calculated as 3.2. Since it already has two decimal places, it is already rounded to the nearest hundredth.

Key Concepts

Vertex of a Parabola

In a quadratic function, the vertex is a critical point that can represent either the highest or the lowest point of a parabola. This point is central when analyzing quadratic functions because it helps to understand the function's overall shape and direction.

• The quadratic function given is in the standard form, \( ax^2 + bx + c \). For the function \( f(x) = -5x^2 + 8x \), the vertex provides us with valuable information about where the parabola will achieve its peak because the coefficient \( a \) is negative.
• The x-coordinate of the vertex is determined by the formula \( x = \frac{-b}{2a} \). This formula is derived from the completion of the square process and it gives us the horizontal point where the vertex of the parabola is located.

By calculating \( x = \frac{-8}{2(-5)} = 0.8 \), we find the specific x-point at which the function reaches its maximum value. This calculation is essential for determining the full coordinates of the vertex and understanding the behavior of the function.

Maximum and Minimum Values of Quadratics

Quadratic functions can either reach maximum or minimum values, depending on the orientation of the parabola.

• If the parabola opens upwards (\( a > 0 \)), it has a minimum vertex.
• If it opens downwards (\( a < 0 \)), like our function \( f(x) = -5x^2 + 8x \), it reaches a maximum.

The maximum or minimum value refers to the y-coordinate of the vertex. For our function, after finding the vertex x-coordinate (0.8), we substitute it back into the function to find the corresponding y-value: \[ f(0.8) = -5(0.8)^2 + 8(0.8) \]Calculating these values yields a maximum value of 3.2. This y-coordinate indicates the peak of the parabola and tells us the highest value the function can achieve, based on its quadratic equation properties.

Parabola Properties

Parabolas have several defining properties that make them unique among curves. Understanding these properties helps us to interpret quadratic functions more efficiently.

• A parabola is symmetric about its vertex. This means that for every point on the parabola, there is a corresponding point on the opposite side of the axis of symmetry.
• The direction in which a parabola opens is determined by the sign of the coefficient \( a \). If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards.

Parabolas can model many real-world situations, like projectile motion, where the shape represents the trajectory of an object. The vertex becomes significantly useful in such cases as it indicates the point of highest elevation (maximum).
A deeper understanding of these properties helps in solving complex problems involving quadratics, allowing for predictions and analysis of the conic sections they produce.