How do you find the axis of symmetry, graph and find the maximum or minimum value of the function #f(x)=x^2+6x+6#?
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See explanation
As this is in calculus we do the following:
Shortcut approach:
Given:
Set
By substitution set
Vertex
So the graph is of form
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This is at
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
As the vertex is
Using Completing the square:
Set
Set
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Just for reference: The first calculus part only using first principles.
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To find the axis of symmetry, graph, and locate the maximum or minimum value of the function ( f(x) = x^2 + 6x + 6 ), follow these steps:

Axis of Symmetry: The axis of symmetry for a parabola given by the equation ( y = ax^2 + bx + c ) is given by the formula ( x = \frac{b}{2a} ). For the given function ( f(x) = x^2 + 6x + 6 ): [ a = 1, \quad b = 6 ] Use the formula ( x = \frac{b}{2a} ) to find the axis of symmetry.

Graphing: Plot the function ( f(x) = x^2 + 6x + 6 ) on a coordinate plane. You can use points, the axis of symmetry, and any additional information to sketch the graph accurately.

Maximum or Minimum Value: To find the maximum or minimum value of the function, you need to determine whether the parabola opens upwards (in which case, it has a minimum value) or downwards (in which case, it has a maximum value). For the function ( f(x) = x^2 + 6x + 6 ), since the coefficient of ( x^2 ) is negative (1), the parabola opens downwards, indicating a maximum value. To find the maximum value, evaluate the function at the vertex (which corresponds to the axis of symmetry). Plug the value of ( x ) obtained from the axis of symmetry into the function ( f(x) ).

Summary:
 Axis of Symmetry: Use the formula ( x = \frac{b}{2a} ).
 Graph: Plot the function on a coordinate plane.
 Maximum or Minimum Value: Determine the direction of the parabola and evaluate the function at the vertex.
By following these steps, you can find the axis of symmetry, graph the function, and locate the maximum or minimum value of ( f(x) = x^2 + 6x + 6 ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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