Find the Vertex y=-x^2+14x-49

$y = - x^{2} + 14 x - 49$

Rewrite the equation in vertex form.

Complete the square for $- x^{2} + 14 x - 49$ .

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 1$

$b = 14$

$c = - 49$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{14}{2 \cdot - 1}$

Simplify the right side.

Cancel the common factor of $14$ and $2$ .

Factor $2$ out of $14$ .

$d = \frac{2 \cdot 7}{2 \cdot - 1}$

Move the negative one from the denominator of $\frac{7}{- 1}$ .

$d = - 1 \cdot 7$

Multiply $- 1$ by $7$ .

$d = - 7$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = - 49 - \frac{14^{2}}{4 \cdot - 1}$

Simplify the right side.

Simplify each term.

Raise $14$ to the power of $2$ .

$e = - 49 - \frac{196}{4 \cdot - 1}$

Multiply $4$ by $- 1$ .

$e = - 49 - \frac{196}{- 4}$

Divide $196$ by $- 4$ .

$e = - 49 - - 49$

Multiply $- 1$ by $- 49$ .

$e = - 49 + 49$

Add $- 49$ and $49$ .

$e = 0$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x - 7)}^{2} + 0$ .

$- {(x - 7)}^{2} + 0$

Set $y$ equal to the new right side.

$y = - {(x - 7)}^{2} + 0$

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = - 1$

$h = 7$

$k = 0$

Find the vertex $(h, k)$ .

$(7, 0)$