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

2 Answers
Jun 16, 2015

Answer:

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

Explanation:

x of vertex = x of axis of symmetry: #x = (-b/2a) = -3/-6 = 1/2#

y of vertex: #y = f(1/2) = - 3/4 + 3/2 - 2 = - 5/4#

y intercept: y = -2

x-intercepts -> y = 0

D = b^2 - 4ac = 9 - 24 = - 15 < 0. There are no real roots (no x-intercepts) because D < 0.
Since a < 0, the parabola opens downward. The parabola is completely below the x-axis.

Jun 16, 2015

Answer:

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#.

Explanation:

#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]}