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]#

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]}

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Answer 2

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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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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