How do you find the vertex of y=2x^2+3x-8?

Apr 7, 2018

The vertex is $\left(- \frac{3}{4} , - 9 \frac{1}{8}\right)$.

Here's how I did it:

Explanation:

$y = 2 {x}^{2} + 3 x - 8$

This equation is written in standard form, or $y = a {x}^{2} + b x + c$

To find the $x$-value of the vertex, or the axis of symmetry, we use the formula: $x = - \frac{b}{2 a}$.

We know that $a = 2$ and $b = 3$, so we can plug in these values into the formula and solve:
$x = - \frac{3}{2 \left(2\right)}$

$x = - \frac{3}{4}$

$- - - - - - - - - - - - - - - - - -$

Now, to find the $y$-value of the vertex, we just plug in the value of $x$ back into the original equation:
$y = 2 {x}^{2} + 3 x - 8$

$y = 2 {\left(- \frac{3}{4}\right)}^{2} + 3 \left(- \frac{3}{4}\right) - 8$

And now we simplify...
$y = 2 \left(\frac{9}{16}\right) - \frac{9}{4} - 8$

$y = \cancel{2} \textcolor{red}{1} \left(\frac{9}{\cancel{16} \textcolor{red}{8}}\right) - \frac{9}{4} - 8$

$y = \frac{9}{8} - \frac{9}{4} - 8$

Make both fractions have the same denominator so you can subtract them:
$y = \frac{9}{8} - \frac{18}{8} - 8$

$y = - \frac{9}{8} - 8$

Convert to mixed fraction form:
$y = - 1 \frac{1}{8} - 8$

$y = - 9 \frac{1}{8}$

$- - - - - - - - - - - - - - - - - -$

Finally, the vertex is $\left(- \frac{3}{4} , - 9 \frac{1}{8}\right)$.

Hope this helps!