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

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> 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       Log On


   

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) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form


y=-1+x%5E2%2B3+x-3 Start with the given equation



y%2B3=-1+x%5E2%2B3+x Add 3 to both sides



y%2B3=-1%28x%5E2-3x%29 Factor out the leading coefficient -1



Take half of the x coefficient -3 to get -3%2F2 (ie %281%2F2%29%28-3%29=-3%2F2).


Now square -3%2F2 to get 9%2F4 (ie %28-3%2F2%29%5E2=%28-3%2F2%29%28-3%2F2%29=9%2F4)





y%2B3=-1%28x%5E2-3x%2B9%2F4-9%2F4%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 9%2F4 does not change the equation




y%2B3=-1%28%28x-3%2F2%29%5E2-9%2F4%29 Now factor x%5E2-3x%2B9%2F4 to get %28x-3%2F2%29%5E2



y%2B3=-1%28x-3%2F2%29%5E2%2B1%289%2F4%29 Distribute



y%2B3=-1%28x-3%2F2%29%5E2%2B9%2F4 Multiply



y=-1%28x-3%2F2%29%5E2%2B9%2F4-3 Now add %2B3 to both sides to isolate y



y=-1%28x-3%2F2%29%5E2-3%2F4 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=-1, h=3%2F2, and k=-3%2F4. Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation y=-1x%5E2%2B3x-3 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-1x%5E2%2B3x-3%29 Graph of y=-1x%5E2%2B3x-3. Notice how the vertex is (3%2F2,-3%2F4).



Notice if we graph the final equation y=-1%28x-3%2F2%29%5E2-3%2F4 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-1%28x-3%2F2%29%5E2-3%2F4%29 Graph of y=-1%28x-3%2F2%29%5E2-3%2F4. Notice how the vertex is also (3%2F2,-3%2F4).



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 (3%2F2,-3%2F4) or (1.5,-0.75), this is one point on the graph.

Now lets pick any point after x=1.5. Lets evaluate x=2

f%28x%29=-x%5E2%2B3x-3 Start with the given polynomial


f%282%29=-%282%29%5E2%2B3%282%29-3 Plug in x=2


f%282%29=-%284%29%2B3%282%29-3 Raise 2 to the second power to get 4


f%282%29=-%284%29%2B6-3 Multiply 3 by 2 to get 6


f%282%29=-1 Now combine like terms

So we get the point (2,-1)

Lets pick another value x=3


f%28x%29=-x%5E2%2B3x-3 Start with the given polynomial


f%283%29=-%283%29%5E2%2B3%283%29-3 Plug in x=3


f%283%29=-%289%29%2B3%283%29-3 Raise 3 to the second power to get 9


f%283%29=-%289%29%2B9-3 Multiply 3 by 3 to get 9


f%283%29=-3 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 x=2 is x=1 and the counterpart to x=3 is x=0 (notice they are the same distance away from the vertex along the x-axis)

So here's the table of suitable values 





 xy


 0-3
 

 1-1


 1.5-0.75
 

 2-1
 

 3-3


Notice if we graph the equation y=-x%5E2%2B3x-3 and the table of points we get



Since the points lie on the curve, this verifies our answer.

Answer by stanbon(75887) About Me  (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)
---------------
graph%28400%2C300%2C-10%2C10%2C-10%2C10%2C-x%5E2%2B3x-3%29
==============
Cheers,
Stan H.