# What is the axis of symmetry and vertex for the graph y=x^2+5x-7?

Mar 17, 2017

Vertex $\Rightarrow \left(- \frac{5}{2} , - \frac{53}{4}\right)$

Axis of Symmetry$\Rightarrow x = - \frac{5}{2}$

#### Explanation:

• Method 1-
The graph of $y = {x}^{2} + 5 x - 7$ is -
graph{x^2+5x-7 [-26.02, 25.3, -14.33, 11.34]}
According to the above graph, We can find the vertex and axis of symmetry of the above graph.
Vertex $\Rightarrow \left(- \frac{5}{2} , - \frac{53}{4}\right)$
Axis of Symmetry$\Rightarrow x = - \frac{5}{2}$

• Method 2-

Check the derivative of the function.

$y = {x}^{2} + 5 x - 7$

$y ' = \frac{\mathrm{dy}}{\mathrm{dx}} = 2 x + 5$

The derivative of the function is zero at its vertex .

$y ' = 2 x + 5 = 0$

$x = - \frac{5}{2}$

Put the $x = - \frac{5}{2}$ in the function to get the value of the function at $x = - \frac{5}{2}$.

$y = \frac{25}{4} - \frac{25}{2} - 7$

$y = \frac{25 - 50 - 28}{4}$

$y = - \frac{53}{4}$

Vertex $\Rightarrow \left(- \frac{5}{2} , - \frac{53}{4}\right)$

Axis of Symmetry$\Rightarrow x = - \frac{5}{2}$

• Method 3-

The given function is a quadratic function.

$y = {x}^{2} + 5 x - 7$

The vertex of the parabola of the quadratic function $= \left(- \frac{b}{2 a} , - \frac{D}{4 a}\right)$

$= \left(- \frac{5}{2} , - \frac{53}{4}\right)$

Axis of Symmetry$\Rightarrow x = - \frac{5}{2}$