Question

How do you find the axis of symmetry, and the maximum or minimum value of the function `y=x^2+3x-5`?

Answer 1

The axis of symmetry is `x=-3/2` or `-1.5`.

The vertex is `(-3/2,-29/4)` or `(-1.5,-7.25)`.

Explanation:

Given:

`y=x^2+3x-5` is a quadratic equation in standard form:

`y=ax^2+bx+c`,

where:

`a=1`, `b=3`, `c=-5`

Axis of symmetry: vertical line that divides the parabola into two equal halves. It is also the `x`-coordinate of the vertex.

The formula to find the axis of symmetry:

`x=(-b)/(2a)`

Plug in the known values.

`x=(-3)/(2*1)`

`x=-3/2` or `-1.5`

The axis of symmetry is `x=-3/2` or `-1.5`.

Vertex: the minimum or maximum point on the parabola. The axis of symmetry is the `x`-coordinate. To find the `y`-coordinate, substitute `-3/2` into the equation and solve for `y`.

`y=(-3/2)^2+3(-3/2)-5`

`y=9/4-9/2-5`

Multiply `9/2` by `2/2`, and `5` by `4/4` so each term has `4` as its denominator.

`y=9/4-9/2xxcolor(red)2/color(red)2-5xxcolor(green)4/color(green)4`

`y=9/4-18/4-20/4`

Simplify.

`y=((9-18-20))/4`

`y=-29/4`

The vertex is `(-3/2,-29/4)` or `(-1.5,-7.25)`.

graph{y=x^2+3x-5 [-16.02, 16.01, -8.01, 8.01]}