# How do you find the axis of symmetry, and the maximum or minimum value of the function y=x^2+5x-7?

Nov 22, 2017

Axis of symmetry is $x = - 2.5$

and

${y}_{\min} = - 13.5$

#### Explanation:

This is the equation of a parabola in standard form, i.e.:

$y = a {x}^{2} + b x + c$

Here, $a = 1 , b = 5 , c = - 7$

This parabola opens up as we can see:

graph{x^2+5x-7 [-30.32, 30.32, -15.16, 15.16]}

The axis of symmetry is a vertical line and goes through the vertex. The $x$ coordinate of the vertex can be found from:

$x = - \frac{b}{2 a} = \frac{- 5}{2 \left(1\right)} = - 2.5$

Therefore the axis of symmetry is $x = - 2.5$

This function does not have a maximum as it goes to infinity on both sides. Its minimum is at its vertex. We can find it by finding the value of $y$ at the vertex. We do this by plugging the value of $x$ of the vertex into the equation:

${y}_{\min} = {\left(- 2.5\right)}^{2} + 5 \left(- 2.5\right) - 7 = 6.25 - 12.5 - 7 = - 13.5$