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

Jun 16, 2015

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

#### Explanation:

x of vertex = x of axis of symmetry: $x = \left(- \frac{b}{2} a\right) = - \frac{3}{-} 6 = \frac{1}{2}$

y of vertex: $y = f \left(\frac{1}{2}\right) = - \frac{3}{4} + \frac{3}{2} - 2 = - \frac{5}{4}$

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

Find the axis of symmetry using the equation $x = \frac{- b}{2 a}$.

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:

$f \left(x\right) = - 3 {x}^{2} + 3 x - 2$

The general formula for a quadratic equation is $a {x}^{2} + b x + c$.

$a = - 3$

$b = 3$

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 $x = \frac{- b}{2 a}$. The vertex is the point where the parabola crosses its axis of symmetry, and is defined as a point $\left(x , y\right)$.

Axis of Symmetry

$x = \frac{- b}{2 a} = \frac{- 3}{2 \left(- 3\right)} = - \frac{3}{-} 6 = \frac{1}{2}$

The axis of symmetry is the line $x = \frac{1}{2}$

Vertex

Determine the value for $y$ by substituting $y$ for $f \left(x\right)$ and by substituting $\frac{1}{2}$ for $x$ in the equation,

$y = - 3 {x}^{2} + 3 x - 2$

$y = - 3 {\left(\frac{1}{2}\right)}^{2} + 3 \left(\frac{1}{2}\right) - 2$

$y = - 3 \left(\frac{1}{4}\right) + \frac{3}{2} - 2$

$y = - \frac{3}{4} + \frac{3}{2} - 2$

The common denominator is $8$.

$y = - \frac{3}{4} \cdot \frac{2}{2} + \frac{3}{2} \cdot \frac{4}{4} - 2 \cdot \frac{8}{8}$ =

$y = - \frac{6}{8} + \frac{12}{8} - \frac{16}{8}$ =

$y = - \frac{10}{8}$

$y = - \frac{5}{4}$

The vertex is $\left(x , y\right) = \left(\frac{1}{2} , - \frac{5}{4}\right)$

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 $y$.

$y = - 3 {\left(0\right)}^{2} + 3 \left(0\right) - 2$ =

$y = - 2$

The y-intercept is $- 2$.

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