SOLUTION: Identify the axis of symmetry, create a suitable table of values, then sketch the graph (including the axis of symmetry). y = �x^2 + 3x � 3

Question 88331: Identify the axis of symmetry, create a suitable table of values, then sketch the graph (including the axis of symmetry).
y = �x^2 + 3x � 3

Found 2 solutions by jim_thompson5910, stanbon:
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form

Start with the given equation

Add to both sides

Factor out the leading coefficient

Take half of the x coefficient to get (ie ).

Now square to get (ie )

Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of does not change the equation

Now factor to get

Distribute

Multiply

Now add to both sides to isolate y

Combine like terms

Now the quadratic is in vertex form where , , and . Remember (h,k) is the vertex and "a" is the stretch/compression factor.

Check:

Notice if we graph the original equation we get:

Graph of . Notice how the vertex is (,).

Notice if we graph the final equation we get:

Graph of . Notice how the vertex is also (,).

So if these two equations were graphed on the same coordinate plane, one would overlap another perfectly. So this visually verifies our answer.

Since we know the vertex is (,) or (1.5,-0.75), this is one point on the graph.

Now lets pick any point after . Lets evaluate

Start with the given polynomial

Plug in

Raise 2 to the second power to get 4

Multiply 3 by 2 to get 6

Now combine like terms

So we get the point (2,-1)

Lets pick another value

Start with the given polynomial

Plug in

Raise 3 to the second power to get 9

Multiply 3 by 3 to get 9

Now combine like terms

So another point is (3,-3)

Now since the graph is symmetrical with respect to the axis of symmetry, this means x-values on the other side of the vertex will have the same y-values as their respective counterparts. For instance, the counterpart to is and the counterpart to is (notice they are the same distance away from the vertex along the x-axis)

So here's the table of suitable values

x y

0 -3

1 -1

1.5 -0.75

2 -1

3 -3

Notice if we graph the equation and the table of points we get

Since the points lie on the curve, this verifies our answer.
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
then sketch the graph (including the axis of symmetry).
y = �x^2 + 3x � 3
----------------------------
y+3 + (3/2)^2 = -(x^2-3x+(3/2)^2)
y + (21/4) = -(x-(3/2))^2
-------------------------
Vertex at ((3/2),(-21/4))
Axis of symmetry at x = (3/2)
---------------

==============
Cheers,
Stan H.
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