# How do you find the axis of symmetry, and the maximum or minimum value of the function  f(x) = -2x^2-2x -1 ?

Dec 16, 2017

$x = - \frac{1}{2} , \text{maximum at } \left(- \frac{1}{2} , - \frac{1}{2}\right)$

#### Explanation:

$\text{given the equation of a parabola in "color(blue)"standard form}$

•color(white)(x)f(x)=ax^2+bx+c color(white)(x);a!=0

• " if "a>0" then parabola has a minimum value "uuu

• " if "a<0" then parabola has a maximum value "nnn

$\text{here "a=-2<0" hence f(x) has a maximum}$

$\text{the maximum/minimum occurs at the vertex }$

$\text{the x-coordinate of the vertex which is also the axis}$
$\text{of symmetry is}$

•color(white)(x)x_(color(red)"vertex ")=-b/(2a)

$\text{here "a=-2" and } b = - 2$

$\Rightarrow {x}_{\textcolor{red}{\text{vertex }}} = - \frac{- 2}{- 4} = - \frac{1}{2}$

$\text{substitute this value into f(x) for y-coordinate}$

$\Rightarrow {y}_{\textcolor{red}{\text{vertex}}} = - 2 {\left(- \frac{1}{2}\right)}^{2} - 2 \left(- \frac{1}{2}\right) - 1 = - \frac{1}{2}$

$\text{equation of axis of symmetry is } x = - \frac{1}{2}$

$\text{maximum value } = - \frac{1}{2}$
graph{(y+2x^2+2x+1)(y-1000x-500)=0 [-10, 10, -5, 5]}

Dec 16, 2017

Axis of symmetry is $x = - 0.5$ , maximum value is $- 0.5$
and minimum value is $- \infty$

#### Explanation:

fx)=-2x^2-2x-1; a=-2 ,b =-2 ,c=-1.

This is a quadratic equanion of form $a {x}^{2} + b x + c$ and

equation of parabola. Since $a$ is negative the parabola

opens downward and minimum point is at $- \infty$

Discriminant :$D = {b}^{2} - 4 a c = 4 - 8 = - 4$. The vertex is the

maximum point. Maximum vale is $- \frac{D}{4 a} = - \frac{- 4}{4 \cdot \left(- 2\right)} = - 0.5$

Axis of symmetry is $x = - \frac{b}{2 a} = \frac{2}{-} 4 = - 0.5$

graph{-2x^2-2x-1 [-10, 10, -5, 5]} [Ans]