Question

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

Answer 1

`x=-1/2,"maximum at "(-1/2,-1/2)`

Explanation:

`"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`

`"here "a=-2<0" hence f(x) has a maximum"`

`"the maximum/minimum occurs at the vertex "`

`"the x-coordinate of the vertex which is also the axis"`
`"of symmetry is"`

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

`"here "a=-2" and "b=-2`

`rArrx_(color(red)"vertex ")=-(-2)/(-4)=-1/2`

`"substitute this value into f(x) for y-coordinate"`

`rArry_(color(red)"vertex")=-2(-1/2)^2-2(-1/2)-1=-1/2`

`"equation of axis of symmetry is "x=-1/2`

`"maximum value "=-1/2`
graph{(y+2x^2+2x+1)(y-1000x-500)=0 [-10, 10, -5, 5]}

Answer 2

Axis of symmetry is `x=-0.5` , maximum value is `-0.5`
and minimum value is `-oo`

Explanation:

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

This is a quadratic equanion of form `ax^2+bx+c` and

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

opens downward and minimum point is at `-oo`

Discriminant :`D= b^2-4ac = 4-8=-4`. The vertex is the

maximum point. Maximum vale is `-D/(4a)=-(-4)/(4*(-2))=-0.5`

Axis of symmetry is `x=-b/(2a)=2/-4=-0.5`

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