Question

How do you find the axis of symmetry, and the maximum or minimum value of the function `y=(x+1)^2 - 4`?

Answer 1

Axis of Symmetry: `-1`.
Minimum: `x=-4`
Maximum: None
End Behavior: As `x` goes towards `-oo`, `y` heads towards `-oo`. However, as `x` goes towards `+oo`, `y` heads towards `+oo`.

Explanation:

graph{y=(x+1)^2-4 [-10, 10, -5, 5]}

As you can see on the graph of this parabola, there is no maximum value because the parabola opens up and will head towards infinity `(oo)`. However, there is a minimum value. The parabola will never go passed `x=-4`.

The axis of symmetry is found at `x=-1`.

However, without knowing this based on the graphed parabola, the equation gives us all the information we need.

`y=(x+1)^2-4`

The axis of symmetry is always the inverse of the number in the parenthesis. Therefore:

`x=-1`

The value outside of the parenthesis is always either the minimum or maximum value. How do you know whether the value is a minimum value or maximum value? Well, you graph the equation! Therefore:

`y=-4`

Therefore the `y` value is our minimum of this parabola.

Minimum`=-4`

The end behavior is what happens to a parabola as it approaches positive and negative infinity.

In this case, we can clearly see that as the graph opens towards negative infinity, the `x` and `y` values also become negative. So simply put, we write the first part of the end behavior as so:

As `x->-oo, x->-oo`

As `y->-oo, y->-oo`

(the second arrow represents the word "becomes").

The second part of the end behavior is what happens as it approaches positive infinity. Therefore the second part of this parabolas end behavior is as stated:

As `x->+oo, x->+oo`

As `y->+oo, y->+oo`