# How do you find the axis of symmetry and vertex point of the function: y = 3x^2 - 7x - 8?

Oct 14, 2015

Vertex: $\left(\frac{7}{6} , - \frac{145}{12}\right)$.
Axis: $x = \frac{7}{6}$.

#### Explanation:

Since this is a parabola with axis of symmetry parallel to the $y$-axis (a "U"-shaped parabola, to be more clear), the vertex will be the minimum of the graph, and the axis will be the vertical line passing through that point.

To find the minimum, you could plug the formula, but I prefer to avoid them, if you can get them by reasoning :)

So, to find the minimum, simply differentiate the formula and find the root of the derivative:

$f ' \left(x\right) = 6 x - 7 = 0 \setminus \iff x = \frac{7}{6}$

So, at this point, we already know that the axis of symmetry is the line $x = \frac{7}{6}$.

Finding the vertex is even simplier: since it is a point on the parabola, and you know the $x$-coordinate, simply plug it into the expression:

$f \left(\frac{7}{6}\right) = 3 {\left(\frac{7}{6}\right)}^{2} + - 7 \left(\frac{7}{6}\right) - 8 = - \frac{145}{12}$