Question

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

Answer 1

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=ax^2+bx+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=-b/(2a)=(-5)/(2(1))=-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=(-2.5)^2+5(-2.5)-7=6.25-12.5-7=-13.5`