# What is the vertex form of y=-13x^2-x-19?

May 27, 2017

$y = - 13 {\left(x + \frac{1}{26}\right)}^{2} - \frac{987}{52}$ is vertex form

This gives the vertex as $\left(- \frac{1}{26} , - 18 \frac{51}{52}\right)$

#### Explanation:

The equation of a parabola can be written as:

$y = a {x}^{2} + b x + c \text{ }$ or in vertex form: $\text{ } y = a {\left(x + b\right)}^{2} + c$

We have $\text{ } y = - 13 {x}^{2} - x - 19$

To change an equation into vertex form:

Step 1. Make $1 {x}^{2}$. Divide out $a$, the coefficient of ${x}^{2}$

$y = - 13 \left({x}^{2} + \frac{x}{13} + \frac{19}{13}\right)$

Step 2. Add and subtract $\textcolor{b l u e}{{\left(\frac{b}{2}\right)}^{2}}$ (same as +0)

$y = - 13 \left({x}^{2} + \frac{x}{13} \textcolor{b l u e}{+ {\left(\frac{1}{26}\right)}^{2} - {\left(\frac{1}{26}\right)}^{2}} + \frac{19}{13}\right)$

Step 3. Write 3 terms as a perfect square and simplify the others

$y = - 13 \left(\left(\textcolor{red}{{x}^{2} + \frac{x}{13} + {\left(\frac{1}{26}\right)}^{2}}\right) - \textcolor{g r e e n}{\left(\frac{1}{676}\right) + \frac{19}{13}}\right)$

$y = - 13 \left(\textcolor{red}{{\left(x + \frac{1}{26}\right)}^{2}} \textcolor{g r e e n}{+ \frac{987}{676}}\right)$

Step 4: Multiply by $a$ outside the bracket

$y = - 13 \textcolor{red}{{\left(x + \frac{1}{26}\right)}^{2}} \textcolor{g r e e n}{- \frac{987}{52}} \text{ } \leftarrow$ vertex form

This gives the vertex as $\left(- \frac{1}{26} , - 18 \frac{51}{52}\right)$

graph{y = -13x^2-x-19 [-0.2135, 0.4115, -19.2013, -18.8888]}