MHB Finding equation of parabola with 3 points given

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To find the equation of a parabola given the points (8,10), (11,10), and (10,20/3), a system of equations is established using the standard form y=ax^2+bx+c. By rewriting the equations derived from the points, a 2x2 system can be formed to eliminate c and solve for a and b. The final equation obtained is y=5/3x^2 -95/3x +470/3, which passes through the given points. The user initially confused the y-intercept with the vertex's y-value but clarified their understanding through the discussion. The method outlined is effective for similar problems in the future.
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Hello, I am supposed to find the equation of a parabola with the points (8,10)(11,10)(10,20/3). I have tried putting these values into y=ax^2+bx+c, but get different answers each time, like c=-160, which is not right! A step by step explanation would be greatly appreciated, as I am very unsure what I'm doing wrong :cool:
 
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Your technique is good and will work:

$$a(8)^2+b(8)+c=10$$

$$a(11)^2+b(11)+c=10$$

$$a(10)^2+b(10)+c=\frac{20}{3}$$

We can rewrite this system as:

$$64a+8b+c=10$$

$$121a+11b+c=10$$

$$300a+30b+3c=20$$

Now, here is what I suggest...Subtract the first equation from the second and subtract the third equation from 3 times the second to eliminate $c$ and obtain a 2X2 system in $a$ and $b$. The using this system, subtract the first from the second to eliminate $b$ and solve for $a$. Use this value of $a$ in either of these two equations to find $b$ and then use the values of $a$ and $b$ in the third equation above to find $c$.

What do you find?
 
I end up with y=5/3x^2 -95/3x +470/3.. I think it should work, but I was also given a not-to-scale diagram with the question, and it shows the TP to have a smaller y value, the TP is above the x-axis and below 10, it's confusing, hmm
 
Those are the correct values, and I have verified the parabola passes through the given points. Here is a plot:

View attachment 2244
 

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Haha, accidentally thought for a second(more than a few seconds) finding the Y intercept was finding the y value of the tp (Giggle) thanks for the help, will try applying this to future situations!
 
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