Question

What are the vertex, axis of symmetry, maximum or minimum value, domain, and range of the function `y = - x^2- 4x + 3`?

Answer 1

Vertex: `(-2,7)`
Axis of symmetry: `x=-2`
Maximum value : `7`
Domain: `(-oo,oo)`
Range: `(-oo,7]`

Explanation:

We are given a quadratic function `y=-x^2-4x+3`

On graphing it would graph a parabola.

Since the coefficient of `x^2` is negative the parabola would be open down.

The `x` coordinate of vertex would help in finding the axis of symmetry.

For the graph which opens down, there is only maximum and that can be found by the `y` coordinate of the vertex.

So first let us find the vertex. There are many different approaches to find the vertex.

Let us try one method.

To find the vertex `(h,k)` we can use the following.

`h=-b/(2a)` and `k= y(h)`

`h=-(-4)/(2(-1))`

`h=4/-2`

`h=-2`

`k=-(-2)^2-4(-2)+3`
`k=-4+8+3`

`k=7`

Vertex: `(-2,7)`
Axis of symmetry: `x=-2`
Maximum value : `7`
Domain: `(-oo,oo)`
Range: `(-oo,7]`

Check the graph to understand it.

graph{y=-x^2-4x+3 [-10, 10, -5, 8]}