Question

What is the axis of symmetry and vertex for the graph `y = 3x^2 - 9x + 12`?

Answer 1

`x=3/2," vertex "=(3/2,21/4)`

Explanation:

`"given a quadratic in "color(blue)"standard form"`

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

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

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

`y=3x^2-9x+12" is in standard form"`

`"with "a=3,b=-9" and "c=12`

`x_("vertex")=-(-9)/6=3/2`

`"substitute this value into the equation for y-coordinate"`

`y_("vertex")=3(3/2)^2-9(3/2)+12=21/4`

`color(magenta)"vertex "=(3/2,21/4)`

`"equation of axis of symmetry is "x=3/2`
graph{(y-3x^2+9x-12)((x-3/2)^2+(y-21/4)^2-0.04)=0 [-14.24, 14.24, -7.12, 7.12]}

Answer 2

`x=3/2` & `(3/2, 21/4)`

Explanation:

Given equation:

`y=3x^2-9x+12`

`y=3(x^2-3x)+12`

`y=3(x^2-3x+9/4)-27/4+12`

`y=3(x-3/2)^2+21/4`

`(x-3/2)^2=1/3(y-21/4)`

The above equation shows an upward parabola: `X^2=4AY` which has

Axis of symmetry : `X=0\implies x-3/2=0`

`x=3/2`

Vertex: `(X=0, Y=0)\equiv (x-3/2=0, y-21/4=0)`

`(3/2, 21/4)`