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put this solution on YOUR website!Recall that each point is of the form (x,y). So for instance, the point (-1, 5) means x=-1 and y=5. This applies for each point given.
Also, remember that every quadratic can be represented as the equation:
where 'a', 'b' and 'c' are real numbers. These values are usually known (and we solve for 'x'), but in this case, we must set up a system of equations to find these values. Note: it turns out that there is only one unique solution to this problem.
So....
For the point (-1,5) we know that x=-1 and y=5. Since this point lies on the quadratic, we know that if we plug in x=-1 into the unknown quadratic, we know that we'll get y=5. So the idea is to plug in the given values to find the unknown values.
Plug these values into the general equation
to get
Now square -1 to get 1, which means it will absorb into 'a', and simplify:
So the first equation we get is
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Furthermore, since the parabola goes through (0,3) we can plug in x=0 and y=3 to get
which simplifies to
. Because we've already isolated 'c', we can use this and plug it into the first equation
to get
. Now solve for 'a' to get
. So whatever 'b' is, the value of 'a' will be 2 more than that.
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Finally, we see that the point (3,9) lies on the parabola. So just plug in x=3 and y=9 to get
and simplify:
From here, we'll use the previously solved for variables 'a' and 'c' to find 'b':
Start with the given equation.
Plug in
and
Distribute
Combine like terms.
Subtract 21 from both sides.
Combine like terms.
Divide both sides by 12 to isolate 'b'.
Reduce
So the value of 'b' is
. Remember that we found that
. So
which means
So after everything is said and done, we find that
,
and
giving us the quadratic
Notice how the parabola
goes through the points (-1, 5), (0,3), and (3,9). So this visually confirms our answer.
Graph of
through the points (-1, 5), (0,3), and (3,9)
(