Question

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

Answer 1

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.

Answer 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`.

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]}