How do I find an equation in standard form of the parabola through the points (1,-2),(2,-4),(3,-4)?

Answer 1

Many authorities refer to y = ax² + bx + c as standard form for a vertical parabola. In this case, you would need to solve for a, b, and c. Plugging in gives 3 simultaneous linear equations:

(1) -2 = a + b + c

(2) -4 = 4a + 2b + c

(3) -4 = 9a + 3b + c

Doubling (1), and subtracting from (2), gives 0 = 2a - c, and therefore c = 2a.

Subtracting (3) - (2) gives 5a + b = 0, and therefore b = -5a.

Plugging these results into (1), everything in terms of a, gives -2 = a - 5a + 2a, so a = 1.

It follows from c = 2a that c = 2, and b = -5a = -1.

The equation is then y = x² - 5x + 2.

The alternative "standard form" can be written as 4p(y - k) = (x - h)²

The value of h will be half of b in order to use the perfect square [(x - 2.5)² = x² - 5x + 6.25]

In order to keep the 2 forms of the equation equal, 4p must be = 1, and 4.25 must be added to 2 in order to give 6.25; therefore we will add 4.25 to the left side of the equation and obtain

y + 4.25 = (x - 2.5)²

Answer 2

The idea is to write the general quadratic form y=a x² + bx + c. Then subs the three points in this equation to find the a, b, and c. If you write down these equations and solve, you should get a=1, b=-5, c=2.

So, y= x² -5x +2.

Then, h=-b/2a=5/2 and k=f(h)= (5/2)² -5(5/2) +2= -17/4.

Then, the standard form is , y= ( x - 5/2)² - 17/4.

Answer 3

First would be to realize that this parabolas are reflective about some vertical line. Since our second two points share a y value of -4. This means that the local minimum (or maximum) is between the x values of 2 and 3, and that we can use the Vertical Axis of Symmetry standard form.

(x-h)² = 4p(y-k)

Since the parabola is reflective about the vertical line at x=2.5, then h=2.5

(x-2.5)² = 4p(y-k)

Plugging in our points:

(1-h)² = 4p(-2-k)

(2-h)² = 4p(-4-k)

(3-h)² = 4p(-4-k)

1+h² -2h = 4p(-2-k)

4+h² -4h =4p(-4-k)

h² = 4p(-2-k) -1+2h

h² =4p(-4-k) -4+4h

4p(-2-k) -1+2h = 4p(-4-k) -4+4h

4p(-2-k) +3 = 4p(-4-k) +2h

-8p+4pk +3 = -16p+4pk +2h

8p+3 = 2h

Since h = 2.5

8p+3 = 5

8p =2

p = 1/4

Substituting 2.5 for h and 1/4 for p

(1-h)² = 4p(-2-k)

(2-h)² = 4p(-4-k)

(3-h)² = 4p(-4-k)

Becomes:

(1-2.5)² = (-2-k)

(2-2.5)² = (-4-k)

(3-2.5)² = (-4-k)

(-1.5)² = (-2-k)

(-0.5)² = (-4-k)

(0.5)² = (-4-k)

2.25 = -2-k

0.25 = -4-k

0.25 = -4-k

k = -4.25

Thus the standard formula is :

(x-2.5)² = 4(1/4)(y-(-4.25))

or

(x-2.5)² = y+4.25