SOLUTION: I need help with how to solve this:write a quadratic equation in standard form that passes through these points: (-1, 5), (0,3), (3,9). I know it's a systems of equations, but the

Question 235135: I need help with how to solve this:write a quadratic equation in standard form that passes through these points: (-1, 5), (0,3), (3,9).
I know it's a systems of equations, but the variables are confusing me... I got
5 = -A^2 - B + C
3 = C, and
9 = 3A^2 + 3B + C
I tried substituting 3 for c, but I don't know what to do from there
show work please and explain

Answer by jim_thompson5910(35256) (Show Source): You can 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

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

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.

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

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)
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