Question

Explain the steps to finding the vertex of g(x) = 3x2 + 12x + 15

Answer 1

Answer:

The vertex of this parabola, $(-2, 3)$, can be found by completing the square.

Step-by-step explanation:

The goal is to express this parabola in its vertex form:

$g(x) = a\, (x - h)^2 + k$,

where $a$, $h$, and $k$ are constants. Once these three constants were found, it can be concluded that the vertex of this parabola is at $(h,\, k)$.

The vertex form can be expanded to obtain:

$\begin{aligned}g(x)&= a\, (x - h)^2 + k \\ &= a\, \left(x^2 - 2\, x\, h + h^2\right) + k = a\, x^2 - 2\, a\, h\, x + \left(a\,h^2 + k\right)\end{aligned}$.

Compare that expression with the given equation of this parabola. The constant term, the coefficient for $x$, and the coefficient for $x^2$ should all match accordingly. That is:

$\left\lbrace\begin{aligned}& a = 3 \\ & -2\,a\, h = 12 \\& a\, h^2 + k = 15\end{aligned}\right.$.

The first equation implies that $a$ is equal to $3$. Hence, replace the "$a\!$" in the second equation with $3\!$ to eliminate $\! a$:

$(-2\times 3)\, h = 12$.

$h = -2$.

Similarly, replace the "$a$" and the "$h$" in the third equation with $3$ and $(-2)$, respectively:

$3 \times (-2)^2 + k = 15$.

$k = 3$.

Therefore, $g(x) = 3\, x^2 + 12\, x + 15$ would be equivalent to $g(x) = 3\, (x - (-2))^2 + 3$. The vertex of this parabola would thus be:

$\begin{aligned}&(-2, \, 3)\\ &\phantom{(}\uparrow \phantom{,\,} \uparrow \phantom{)} \\ &\phantom{(}\; h \phantom{,\,} \;\;k\end{aligned}$.