Write the equation of the parabola in standard form that passes through the points (0, 3), (1, -4), and (-1, 4).

Answer

Explanation

The standard form of a parabola is

where , and are real numbers.

If it passes through (0,3) then when x = 0, y = 3 so this means that

so .

If it passes through (1,-4), then when x = 1, y = -4 so

If it passes through (-1,4) then when x = -1, y = 4 so

Because both (I) and (II) need to be satisfied, we have the system of equations,

which we can easily solve by adding the two equations up to get

Then we take any of the previous equations to solve for b:

Thus the parabola in standard form is

So with the three points that are given to us, we will plug them into the standard form formula that I had mentioned earlier. Firstly, plug (0,3) into the equation since the 0 will cancel out the a and b variable:

Now we know that the value of c is 3.

Next, plug (1,-4) into the standard form formula and simplify (Remember to plug 3 into the c variable):

Next, plug (-1,4) into the standard form formula and simplify:

With the last two simplified equations, we will create a system of equations:

With this, I will be using the elimination method. Add the two equations together, and the following equation is the result:

From here we can solve for a. For this, just divide both sides by 2:

Now that we have the value of a, plug it into either equation to solve for b:

Now, plug the obtained values in our standard form equation and your final answer will be:

the answer is 5636.7