Answer
$(2, -1)$ is the minimum point.
Work Step by Step
First, we find the standard form but before, let's expand and simplify.
$g(x) = 2x(x-4)+7 = 2x^2-8x+7$
To find the standard form of some function $g(x) = ax^2+bx+c$, it would be $g(x) = a(x-h)^2+k$ where $h = -\frac{b}{2a}$ and $k = g(h)$. This is a standard result derived in the book simplifies the algebra and gives a closed form for the end result.
For this problem, $a= 2,b = -8, c = 7$. Plugging above, we get $h = 2, k = g(2) = -1$ and hence $g(x) = 2(x-2)^2-1.$
The vertex can be easily deduced from the standard form; it is the point $(h, k)$ so in this case $(2, -1)$.
The maximum or minimum of a quadratic function is attained at the vertex. To determine whether the vertex is a maximum or minimum, we look at $a$:
- if $a>0$, this tells us the function will grow towards positive infinity and hence, the vertex is a minimum.
- if $a<0$, this tells us that the function will grow towards negative infinity and hence, the vertex is a maximum.
Looking at $a$ in this case tells us that the vertex $(2, -1)$ is a minimum.
