Find the Maximum/Minimum Value -x^2-6x-8
The problem presented is a mathematical optimization problem where you are asked to find the maximum or minimum value of a quadratic function. Specifically, you need to determine the point on the graph of the given quadratic equation, $-x^2 - 6x - 8$, where the function reaches its highest or lowest value. Quadratic functions have parabolic graphs, and their maxima or minima can be found by analyzing the coefficients or by completing the square, or by using calculus methods like finding the derivative and setting it equal to zero to find critical points. The sign of the coefficient of the $x^2$term typically indicates whether the parabola opens upwards (minimum) or downwards (maximum).
$- x^{2} - 6 x - 8$
Answer
Solution:
Step:1
To find the maximum value of a quadratic function $f(x) = ax^2 + bx + c$, we use the vertex formula $x = -\frac{b}{2a}$. For a negative $a$, this gives us the maximum value at $f\left(-\frac{b}{2a}\right)$.
Step:2
Determine the $x$-coordinate of the vertex using $x = -\frac{b}{2a}$.
Step:2.1
Insert the coefficients $a$ and $b$ into the formula: $x = -\frac{-6}{2(-1)}$.
Step:2.2
Eliminate the parentheses: $x = -\frac{-6}{-2}$.
Step:2.3
Simplify the expression $-\frac{-6}{-2}$.
Step:2.3.1
Reduce the fraction by cancelling out common factors.
Step:2.3.1.1
Factor out a $2$ from $-6$: $x = -\frac{2 \cdot (-3)}{2 \cdot (-1)}$.
Step:2.3.1.2
Apply the negative sign from the denominator: $x = -((-1) \cdot (-3))$.
Step:2.3.2
Rewrite $(-1) \cdot (-3)$ as $3$: $x = -(-3)$.
Step:2.3.3
Perform the multiplication.
Step:2.3.3.1
Multiply $-1$ by $-3$: $x = 3$.
Step:2.3.3.2
The result is $x = 3$.
Step:3
Calculate $f(3)$.
Step:3.1
Substitute $x$ with $3$ in the function: $f(3) = -(3)^2 - 6(3) - 8$.
Step:3.2
Simplify the expression.
Step:3.2.1
Simplify each term individually.
Step:3.2.1.1
Square $3$: $f(3) = -9 - 6(3) - 8$.
Step:3.2.1.2
Multiply $-6$ by $3$: $f(3) = -9 + 18 - 8$.
Step:3.2.1.3
Combine the terms: $f(3) = 9 - 8$.
Step:3.2.2
Add and subtract the terms: $f(3) = 1$.
Step:3.2.3
The maximum value of the function is $1$.
Step:4
The maximum occurs at the point $(-3, 1)$.
Step:5
There is no further step required as the maximum value has been found.
Solution: "The maximum value of the quadratic function $-x^2 - 6x - 8$ is $1$, which occurs at the point $(-3, 1)$."
Knowledge Notes:
The vertex form of a quadratic function is useful in determining the maximum or minimum value of the function.
For a quadratic function $f(x) = ax^2 + bx + c$, the vertex is located at $x = -\frac{b}{2a}$.
If the coefficient $a$ is negative, the function opens downwards, and the vertex represents the maximum point.
Simplification of algebraic expressions involves combining like terms and reducing fractions.
Substituting the vertex $x$-coordinate back into the function gives the maximum or minimum $y$-value, completing the vertex as an ordered pair $(x, f(x))$.
