SOLUTION: hi! please help me solve this one... In the xy-plane, the graph of y=x^2+bx+c is symmetric about the line x=3 (what is symmetric?) and passes through the point (5,2) . What is t

Question 643393: hi! please help me solve this one...
In the xy-plane, the graph of y=x^2+bx+c is symmetric about the line x=3 (what is symmetric?) and passes through the point (5,2) . What is the Value of c ?

Thank you in advance :)
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Symmetric means that one half of the graph reflects across this line to create the other half.

Notice how x = 3 is 2 units away from x = 5 (since 5-3 = 2). So if we have one point (5,2), then (1,2) is also on the parabola since this point has an x coordinate that is also 2 units away from x = 3 (and has the same y coordinate)

So again, we have 2 points (5,2) and (1,2) that lie on the parabola. We will use this information to find the values of b and c

------------------------------------------------------------------------------------

y=x^2+bx+c

2=(5)^2+b(5)+c ... Plug in (5,2), ie plug in x = 5 and y = 2

2=25+5b+c

Now solve for c

2=25+5b+c

2-25=5b+c

-23=5b+c

-23-5b=c

c = -23-5b

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Reset and plug in (1,2)

y=x^2+bx+c

2=(1)^2+b(1)+c ... Plug in (1,2), ie plug in x = 1 and y = 2

2=1+b+c

Now plug in c = -23-5b to get

2=1+b+(-23-5b)

2=1+b-23-5b

2=-22-4b

2+22 = -4b

24 = -4b

24/(-4) = b

-6 = b

b = -6

Finally, use this to find c

c = -23-5b

c = -23-5(-6)

c = -23+30

c = 7

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Answer:

So the values of b and c are: b = -6 and c = 7

which means that becomes

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