What is the vertexof y= -5x^2 − 3x ?

Mar 26, 2018

Vertex: $\left(\frac{- 3}{10} , \frac{9}{20}\right)$

Explanation:

First, use the axis of symmetry formula $\left(A o S : x = \frac{- b}{2 a}\right)$ to find the x-coordinate of the vertex $\left({x}_{v}\right)$ by substituting $- 5$ for $a$ and $- 3$ for $b$:

${x}_{v} = \frac{- b}{2 a}$

${x}_{v} = \frac{- \left(- 3\right)}{2 \left(- 5\right)}$

${x}_{v} = \frac{- 3}{10}$

Then find the y-coordinate of the vertex $\left({y}_{v}\right)$ by substituting $\frac{- 3}{10}$ for $x$ in the original equation:

${y}_{v} = - 5 {x}^{2} - 3 x$

${y}_{v} = - 5 {\left(\frac{- 3}{10}\right)}^{2} - 3 \left(\frac{- 3}{10}\right)$

${y}_{v} = - 5 \left(\frac{9}{100}\right) + \frac{9}{10}$

${y}_{v} = \frac{- 45}{100} + \frac{90}{100}$

${y}_{v} = \frac{45}{100}$

${y}_{v} = \frac{9}{20}$

Finally, express the vertex as an ordered pair:

Vertex: $\left({x}_{v} , {y}_{v}\right) = \left(\frac{- 3}{10} , \frac{9}{20}\right)$

Mar 26, 2018

The vertex is $\left(- \frac{3}{10} , \frac{9}{20}\right)$ or $\left(- 0.3 , 0.45\right)$.

Explanation:

Given:

$y = - 5 {x}^{2} - 3 x$ is a quadratic equation in standard form:

$a {x}^{2} + b x - 3 x$,

where:

$a = - 5$, $b = - 3$, $c = 0$

The vertex of a parabola is its maximum or minimum point. In this case, since $a < 0$, the vertex will be the maximum point and the parabola will open downward.

To find the $x$-value of the vertex, use the formula for the axis of symmetry:

$x = \frac{- b}{2 a}$

$x = \frac{- \left(- 3\right)}{2 \cdot \left(- 5\right)}$

$x = \frac{3}{- 10}$

$x = - \frac{3}{10}$

To find the $y$-value of the vertex, substitute $- \frac{3}{10}$ for $x$ and solve for $y$.

$y = - 5 {\left(- \frac{3}{10}\right)}^{2} - 3 \left(- \frac{3}{10}\right)$

Simplify.

$y = - {\textcolor{red}{\cancel{\textcolor{b l a c k}{5}}}}^{1} \left(\frac{9}{\textcolor{red}{\cancel{\textcolor{b l a c k}{100}}}} ^ 20\right) + \frac{9}{10}$

$y = - \frac{9}{20} + \frac{9}{10}$

Multiply $\frac{9}{10}$ by $\frac{2}{2}$ to get the common denominator $20$.

$y = - \frac{9}{20} + \frac{9}{10} \times \frac{2}{2}$

$y = - \frac{9}{20} + \frac{18}{20}$

$y = \frac{9}{20}$

The vertex is $\left(- \frac{3}{10} , \frac{9}{20}\right)$ or $\left(- 0.3 , 0.45\right)$.

graph{y=-5x^2-3x [-10, 10, -5, 5]}