Find the Maximum/Minimum Value -16x^2+64x+5
The given mathematical problem is asking to determine the highest value (maximum) or the lowest value (minimum) that the quadratic function f(x) = -16x^2 + 64x + 5 can take. This function describes a parabola that opens downward because the coefficient of the x^2 term is negative. The task involves finding the vertex of the parabola, which will give the x-coordinate at which the maximum value of the function occurs since the parabola opens downwards. This can be done by completing the square or using the vertex formula for quadratic functions.
$- 16 x^{2} + 64 x + 5$
Answer
Solution:
Determine the Maximum/Minimum Value of $-16x^2+64x+5$
Step 1: Identify the Vertex
The vertex of a parabola described by $y = ax^2 + bx + c$ is found at $x = -\frac{b}{2a}$. For a downward-opening parabola ($a < 0$), this vertex represents the maximum point. The maximum value is then $y = f\left(-\frac{b}{2a}\right)$.
Step 2: Calculate the Vertex's x-coordinate
Find the x-coordinate of the vertex using $x = -\frac{b}{2a}$.
Step 2.1: Plug in the coefficients for $a$ and $b$
$x = -\frac{64}{2(-16)}$
Step 2.2: Simplify the expression
$x = -\frac{64}{-32}$
Step 2.3: Reduce the fraction
Step 2.3.1: Divide numerator and denominator by 2
$x = -\frac{32}{-16}$
Step 2.3.2: Simplify the negative signs
$x = \frac{32}{16}$
Step 2.3.3: Calculate the value of $x$
$x = 2$
Step 3: Evaluate the Function at the Vertex's x-coordinate
Compute $f(2)$ to find the maximum value.
Step 3.1: Substitute $x$ with 2 in the original equation
$f(2) = -16(2)^2 + 64(2) + 5$
Step 3.2: Perform the arithmetic operations
Step 3.2.1: Calculate each term separately
$f(2) = -16 \cdot 4 + 64 \cdot 2 + 5$
Step 3.2.2: Add the results together
$f(2) = -64 + 128 + 5$
Step 3.2.3: Find the sum
$f(2) = 69$
Step 4: Determine the Maximum Value and Its Location
The maximum value of the function is 69, occurring at $x = 2$.
Step 5: Conclusion
The maximum value of the quadratic function $-16x^2+64x+5$ is 69, at the point $(2, 69)$.
Knowledge Notes:
Quadratic Functions: A quadratic function is of the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $a \neq 0$. The graph of a quadratic function is a parabola.
Vertex of a Parabola: The vertex is the highest or lowest point on the graph of a parabola. For the function $f(x) = ax^2 + bx + c$, the vertex's x-coordinate is given by $x = -\frac{b}{2a}$.
Maximum/Minimum Value: For a parabola that opens upwards ($a > 0$), the vertex represents the minimum point. For a parabola that opens downwards ($a < 0$), the vertex represents the maximum point.
Completing the Square: This is a method used to find the vertex form of a quadratic function, which can also be used to determine the maximum or minimum value. However, in this solution, we use the formula for the vertex directly.
Arithmetic Operations: When evaluating the function at a specific point, it is important to follow the order of operations: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right).
