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

Feb 14, 2018

Axis of symmetry: $x = - 2$
Vertex: $\left(- 2 , - 14\right)$

#### Explanation:

This equation $y = 3 {x}^{2} + 12 x - 2$ is in standard form, or $a {x}^{2} + b x + c$.

To find the axis of symmetry, we do $x = - \frac{b}{2 a}$.

We know that $a = 3$ and $b = 12$, so we plug them into the equation.
$x = - \frac{12}{2 \left(3\right)}$

$x = - \frac{12}{6}$

$x = - 2$

So the axis of symmetry is $x = - 2$.

Now we want to find the vertex. The $x$-coordinate of the vertex is the same as the axis of symmetry. So the $x$-coordinate of the vertex is $- 2$.
To find the $y$-coordinate of the vertex, we just plug in the $x$ value into the original equation:
$y = 3 {\left(- 2\right)}^{2} + 12 \left(- 2\right) - 2$

$y = 3 \left(4\right) - 24 - 2$

$y = 12 - 26$

$y = - 14$

So the vertex is $\left(- 2 , - 14\right)$.

To visualize this, here is a graph of this equation:

Hope this helps!

Feb 14, 2018

Axis of Symmetry is the line color(blue)(x=-2

Vertex is at: color(blue)((-2, -14).It is a minimum.

#### Explanation:

Given:

color(red)(y=f(x) = 3x^2+12x-2

We use the Quadratic formula to find the Solutions:

color(blue)(x_1, x_2=(-b+-sqrt(b^2-4ac))/(2a)

Let us look at color(red)(f(x)

We observe that color(blue)(a=3; b=12; and c=(-2)

Substitute these values in our Quadratic formula:

We know that our discriminant ${b}^{2} - 4 a c$ is greater than zero.

color(blue)(x_1, x_2=[-12+-sqrt[12^2-4(3)(-2)])/(2(3))

Hence, we have two real roots.

${x}_{1} , {x}_{2} = \frac{- 12 \pm \sqrt{\left[144 + 24\right]}}{6}$

${x}_{1} , {x}_{2} = \frac{- 12 \pm \sqrt{168}}{6}$

${x}_{1} , {x}_{2} = \frac{- 12 \pm \sqrt{4 \cdot 42}}{6}$

${x}_{1} , {x}_{2} = \frac{- 12 \pm \sqrt{4} \cdot \sqrt{42}}{6}$

${x}_{1} , {x}_{2} = \frac{- 12 \pm 2 \cdot \sqrt{42}}{6}$

${x}_{1} , {x}_{2} = \frac{- 12}{6} \pm \frac{2 \cdot \sqrt{42}}{6}$

${x}_{1} , {x}_{2} = - 2 \pm \frac{\cancel{2} \cdot \sqrt{42}}{\cancel{6} \textcolor{red}{3}}$

${x}_{1} , {x}_{2} = - 2 + \frac{\sqrt{42}}{3} , - 2 - \frac{\sqrt{42}}{3}$

Using a calculator, we can simplify and get the values:

color(blue)(x_1=0.160247, x_2=-4.16025

Hence, our x-intercepts are: color(green)((0.16,0),(-4.16,0)

To find the Vertex,

we can use the formula: color(blue)((-b))/color(blue)((2a)

Vertex: -12/(2(3)

$\Rightarrow - \frac{12}{6} = - 2$

This is our x-coordinate value of our Vertex.

To find the y-coordinate value of our Vertex:

Substitute the value of color(blue)(x=-2 in

color(red)(y = 3x^2+12x-2

$y = 3 {\left(- 2\right)}^{2} + 12 \left(- 2\right) - 2$

$y = 3 \left(4\right) - 24 - 2$

$y = 12 - 24 - 2 = 14$

Vertex is at: color(blue)((-2, -14)

The coefficient of the color(green)(x^2 term is Positive and hence, our Parabola Opens Upward, and it has a minimum. Please refer to the image of the graph below to verify our solutions:

The Axis of symmetry of a parabola is a vertical line that divides the parabola into two congruent halves.

The Axis of Symmetry always passes through the Vertex of the Parabola. The $x$ coordinate of the vertex is the equation of the Axis of Symmetry of the Parabola.

Axis of Symmetry is the line color(blue)(x=-2