Question 1: Given the function $f(x) = 3x^2 + 2x - 5$, determine: (a) The vertex of the parabola - HSC - SSCE Chemistry - Question 11 - 2017 - Paper 1

To find the vertex of the parabola defined by the function $f(x) = ax^2 + bx + c$, we use the formula for the x-coordinate of the vertex, given by:

$x_v = -\frac{b}{2a}$

For our function, $a = 3$ and $b = 2$. Thus:

$x_v = -\frac{2}{2 \cdot 3} = -\frac{1}{3}$

Now substitute $x_v$ back into the original function to find the y-coordinate:

$f(-\frac{1}{3}) = 3(-\frac{1}{3})^2 + 2(-\frac{1}{3}) - 5 = 3\cdot\frac{1}{9} - \frac{2}{3} - 5 = \frac{1}{3} - \frac{2}{3} - 5 = -5\frac{2}{3}$

Thus, the vertex is at $(-\frac{1}{3}, -5\frac{2}{3})$.