Question

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

Answer 1

Answer

  $y = -3x^2-4x+3$

Explanation

The standard form of a parabola is

  $y = ax^2 + bx + c,$

where $a \ne 0$, and $a,b,c$ are real numbers.

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

  $3 = a(0)^2 + b(0) + c \implies c = 3$

so $y = ax^2 + bx + 3$.

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

  $\begin{aligned}-4 &= a(1)^2 + b(1) + 3 \\a+b+3 &= -4 \\a+b &= -7 && \text{(I).}\end{aligned}$

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

  $\begin{aligned}4 &= a(-1)^2 + b(-1) + 3 \\a-b+3 &= 4 \\a-b &= 1 && \text{(II).}\end{aligned}$

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

  $\begin{cases}a+b &= -7\qquad\text{(I)}\\a-b &= 1\qquad\text{(II)}\end{cases}$

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

  $\begin{aligned}(a+a) + (b-b) &= -7 + 1 \\ 2a&= -6 \\a &= -3.\end{aligned}$

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

  $\begin{aligned}a+b &= -7\\-3 + b &= -7 \\ b &= -4\end{aligned}$

Thus the parabola in standard form is

  $y = -3x^2-4x+3.$

Answer 2

Steps

• Standard Form of a Parabola: y = ax² + bx + c (a ≠ 0)

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:

$3=a*0^2+b*0+c\\3=c$

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):

$-4=a*1^2+1*b+3\\-4=a+b+3\\-7=a+b$

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

$4=a*(-1)^2+b*(-1)+3\\4=a-b+3\\1=a-b$

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

$-7=a+b\\1=a-b$

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

$-6=2a$

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

$-3=a$

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

$-7=-3+b\\-4=b\\\\1=-3-b\\4=-b\\-4=b$

Answer

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

$y=-3x^2-4x+3$