Give the equation of the parabola that has a vertex at (2,-3) and a point at (1,-10/3).

Answer 1

y = C(x-h)^2 + k

Vertex at (2,-3)

Then y = C(x-2)^2 - 3

It eats (1, -10/3)

So -10/3 = C(1-2)^2 - 3

-10/3 = C(-1)^2 - 3

-10/3 = C(1) - 3

-10/3 = C - 3

3 - 10/3 = C

9/3 - 10/3 = C

C = -1/3

The parabola is (-1/3)(X-2)^2 - 3

= (-1/3)(x^2 - 4x + 4) - 3

= (-1/3)x^2 + (4/3)x + (-4/3 - 3)

= (-1/3)x^2 + (4/3)x + (-4/3 - 9/3)

= (-1/3)x^2 + (4/3)x - 13/3

= (-1/3)(x^2 - 4x + 13)

Note that in either equation, y=-3 when x=2 and y = -10/3 when x = 1

Answer 2

Hi Parisa D.,

The equations for a parabola are:

(x - h)² = ± 4p(y - k), with +4p if the parabola opens up and -4p if the parabola opens down.

(y - k)² = ± 4p(x - h), with +4p if the parabola opens to the right and -4p if the parabola open to the left.

A quick plot of the points V(2, -3) and point (1, -10/3) shows two possible parabolas. One parabola that opens down, and another that opens to the left. Lets first look at the parabola that opens down.

(x - h)² = ± 4p(y - k)

(x - 2)² = - 4p(y + 3), V(h, k) and -4p because it opens down

(1 - 2)² = - 4p(-10/3 + 3), point (1, -10/3)

1 = -4p(-1/3)

3 = 4p

(x - 2)² = -3(y + 3), the equation for the parabola that opens down.

Now lets look at the parabola that opens to the left:

(y - k)² = ± 4p(x - h)

(y + 3)² = - 4p(x - 2), - 4p because the parabola opens to the left

(-10/3 + 3)² = - 4p(1 - 2)

1/9 = 4p

(y + 3)² = -1/9(x - 2), the equation for the parabola the open to the left.

I hope this helps, Joe.

Answer 3

The equation of a parabola in general form is y=ax²+bx+c.

The expression -b/(2a) is x-coordinate of the vertex, so you get -b/(2a)=2 which this gives you

b= - 4a. Hence the parabola now is y=ax²-4ax+c.

As two points (2,3) and (1, -10/3) are located on the parabola, you can form the following equations

-3 = a(2)² -4a(2) + c ⇒ 4a - 8a + c=-3 ⇒ -4a + c = -3

-10/3 = a(1)² -4a(1) + c . ⇒ a - 4a+c = -10/3 ⇒ -3a +c = -10/3

By solving the system of the equations we get a=-1/3, b=4/3 and c = -13/3. Hence the parabola is

y=-1/3 x² +4/3 x - 13/3.