Find the Maximum/Minimum Value -x^2-6x-8

Solution:

Step:1

To find the maximum value of a quadratic function $f(x)=a{x}^2+bx+c$, we use the vertex formula $x=-\frac{b}{2a}$. For a negative $a$, this gives us the maximum value at $f(-\frac{b}{2a})$.

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:

1. The vertex form of a quadratic function is useful in determining the maximum or minimum value of the function.

2. For a quadratic function $f(x)=a{x}^2+bx+c$, the vertex is located at $x=-\frac{b}{2a}$.

3. If the coefficient $a$ is negative, the function opens downwards, and the vertex represents the maximum point.

4. Simplification of algebraic expressions involves combining like terms and reducing fractions.

5. Substituting the vertex $x$-coordinate back into the function gives the maximum or minimum $y$-value, completing the vertex as an ordered pair $(x,f(x))$.