How do you find its vertex, axis of symmetry, y-intercept and x-intercept for ##f(x) = -3x^2 + 3x - 2##?

Find the axis of symmetry using the equation ##x=(-b)/(2a)##.

Find the vertex by substituting the value for ##x## into the equation and solving for ##y##.

There are no x-intercepts.

To get the y-intercept, substitute 0 for ##x## in the equation and solve for ##y##.

##f(x)=-3x^2+3x-2##

The general formula for a quadratic equation is ##ax^2+bx+c##.

##a=-3##

##b=3##

The graph of a quadratic equation is a parabola. A parabola has an axis of symmetry and a vertex. The axis of symmetry is a vertical line the divides the parabola into to equal halves. The line of symmetry is determined by the equation ##x=(-b)/(2a)##. The vertex is the point where the parabola crosses its axis of symmetry, and is defined as a point ##(x,y)##.

Axis of Symmetry

##x=(-b)/(2a)=(-3)/(2(-3))=-3/-6=1/2##

The axis of symmetry is the line ##x=1/2##

Vertex

Determine the value for ##y## by substituting ##y## for ##f(x)## and by substituting ##1/2## for ##x## in the equation,

##y=-3x^2+3x-2##

##y=-3(1/2)^2+3(1/2)-2##

##y=-3(1/4)+3/2-2##

##y=-3/4+3/2-2##

The common denominator is ##8##.

##y=-3/4*2/2+3/2*4/4-2*8/8## =

##y=-6/8+12/8-16/8## =

##y=-10/8##

##y=-5/4##

The vertex is ##(x,y)=(1/2,-5/4)##

X-Intercept

The x-intercepts are where the parabola crosses the x-axis.There are no x-intercepts for this equation because the vertex is below the x-axis and the parabola is facing downward.

Y-Intercept

The y-intercept is where the parabola crosses the y-axis. To find the y-intercept, make ##x=0##, and solve the equation for ##y##.

##y=-3(0)^2+3(0)-2## =

##y=-2##

The y-intercept is ##-2##.

graph{y=-3x^2+3x-2 [-14, 14.47, -13.1, 1.14]}