The parabola of (0,6),(2,16),(4,10)

Answer 1

Hi Alyssa R

For your (0,6), (2,16), (4,10) parabola

f(x) = y = -2x² + 9x + 6

I assume you want the equation of the parabola which passes through the three points given

You will need to determine a, b, and c based on the Standard Form equation for a parabola

You will have 3 equations in 3 unknowns

Standard Form

f(x) = y = ax² + bx + c

Using the first set of coordinates (0,6) will can easily determine c

6 = a(0)² + b(0) + c

6 = c

We have one of the unknowns, c and we can substitute this as needed in the other two equations and find a and b

y = ax² + bx + c

16 = a(2)² + b(2) + 6

10 = a(4)² + b(4) + 6

Above if we complete the multiplication, and combine like terms we will have two equations in two unknowns

10 = 4a + 2b

4 = 16a + 4b

I will eliminate b by multiplying (10 = 4a + 2b) by negative 2 then combine the equations

-20 = -8a - 4b

4 = 16a + 4b

-16 = 8a

Divide both sides by 8

-2 = a

I can substitute the values for a and c into one of the original equations above to find b

16 = a(2)² + b(2) + 6

16 = -2(2)² + b(2) + 6

16 = -8 + 2b + 6

16 = -2 + 2b

16 + 2 = 2b

18 = 2b

18/2 = b

9 = b

Your Standard Form Equation for your (0,6), (2,16), (4,10)

f(x) = y = -2x² + 9x + 6

You can graph this at Desmos.com

You can also check this equation against all the a coordinates, we know (0, 6) works

You can try (2,16)

16 = -2(2)² + 9(2) + 6

16 = -8 + 18 + 6

16 = 10 + 6

16 = 16

I hope this helps

Answer 2

y-y₁ = (x-x₁)²

y-6 = x² --> y = x +6

y-16 = (x-2)² --> y = x²-4x+20

y-10 = (x-4)² --> y = x²-8x+26