# What is the vertex of  y=x^2/7-7x+1 ?

Jun 11, 2018

$\left(24.5 , - 84.75\right)$

#### Explanation:

$y = \implies a = \frac{1}{7} , b = - 7 , c = 1$
for co-ordinate of vertex $\left(h , k\right)$
$h = - \frac{b}{2 a} = \frac{7}{2. \left(\frac{1}{7}\right)} = \frac{49}{2}$
put $x = \frac{49}{2}$ to find $y$ and corresponding point $k$
$k = - 84.75$
co-ordinate is $\left(24.5 , - 84.75\right)$

best method : by calculus
vertex is the lowermost(or uppermost) point $i . e$ minimum or maximum of the function
we have
$y = {x}^{2} / 7 - 7 x + 1$
$\implies \frac{\mathrm{dy}}{\mathrm{dx}} = 2 \frac{x}{7} - 7$
at minimum or maximum slope of curve is 0 or $\frac{\mathrm{dy}}{\mathrm{dx}} = 0$
$\implies 2 \frac{x}{7} - 7 = 0 \implies x = \frac{49}{2}$

check if this point is of maximum or minimum by second derivative test(thisstep is not necessarily needed)
if second derivative is -ve it corresponds to point of maximum
if second derivative is +ve it corresponds to point of minimum

$\frac{{d}^{2} y}{{\mathrm{dx}}^{2}} = \frac{2}{7} = + v e \implies x = \frac{49}{2}$ corresponds to point of minimum
now put $x = \frac{49}{2}$ to find $y$
and you will find coordinates as
$\left(24.5 , - 84.75\right)$
and it's evident from the graph

graph{x^2/7-7x+1 [-10, 10, -5, 5]}