Find the Maximum/Minimum Value -16x^2+64x+5

Solution:

Determine the Maximum/Minimum Value of $-16x^2+64x+5$

Step 1: Identify the Vertex

The vertex of a parabola described by $y=a{x}^2+bx+c$ 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 $y=f(-\frac{b}{2a})$.

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:

1. Quadratic Functions: A quadratic function is of the form $f(x)=a{x}^2+bx+c$, where $a$, $b$, and $c$ are constants, and $a\ne 0$. The graph of a quadratic function is a parabola.

2. Vertex of a Parabola: The vertex is the highest or lowest point on the graph of a parabola. For the function $f(x)=a{x}^2+bx+c$, the vertex's x-coordinate is given by $x=-\frac{b}{2a}$.

3. 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.

4. 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.

5. 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).