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

Vertex:

Axis of symmetry:

Maximum value :

Domain:

Range:

On graphing it would graph a parabola.

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

Let us try one method.

Check the graph to understand it.

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

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The vertex of the function ( y = -x^2 - 4x + 3 ) can be found using the formula ( x = \frac{-b}{2a} ) and then substituting this value of ( x ) into the equation to find the corresponding ( y ) value.

The axis of symmetry is the vertical line passing through the vertex, which is ( x = \frac{-b}{2a} ).

To determine whether the vertex is a maximum or minimum value, you can observe the coefficient of ( x^2 ). Since it is negative, the parabola opens downwards, indicating that the vertex represents a maximum value.

The domain of the function is all real numbers because there are no restrictions on the possible values of ( x ) in the given function.

To find the range, you consider the behavior of the parabola. Since it opens downwards, the maximum value of ( y ) occurs at the vertex. Therefore, the range is from negative infinity up to the maximum value of ( y ) at the vertex.

By completing the square or using the vertex formula, the vertex of the function ( y = -x^2 - 4x + 3 ) can be found to be ( (-2, 7) ). The axis of symmetry is ( x = -2 ). The maximum value of the function is ( y = 7 ). The domain is all real numbers, and the range is ( y \leq 7 ).

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