Question

What is the equation, in standard form of the parabola that contains the following points? (-2,18), (0,2), (4,42)

Answer 1

Answer: y = 3
`x^(2)` - 2x + 2

Explanation:

The equation in standard form of a parabola is given as :

y =
`ax^(2)` + bx + c

The points given are :

( -2 , 18 ) , ( 0,2) , ( 4 , 42)

This means that :

`x_(1)` = -2

`x_(2)` = 0

`x_(3)` = 4

`y_(1)` = 18

`y_(2)` = 2

`y_(3)` = 42

All we need do is to substitute each of this points into the equation , that is ,
`x_(1)` and
`y_(1)` will be substituted to get an equation ,
`x_(2)` and
`y_(2)` will be substituted to get an equation and
`x_(3)` ,
`y_(3)` will also be substituted to get an equation also.

Starting with the first one , we have :

y =
`ax^(2)` + bx + c

18 = a[
`(-2)^(2)`] + b (-2) + c

18 = 4a - 2b + c

Therefore :

4a - 2b + c = 18 ................ equation 1

substituting the second values , we have

2 = a (0) + b ( 0) + c

2 = c

Therefore c = 2 ............... equation 2

also substituting the third values , we have

42 = a[
`(4)^(2)`] + b (4) + c

42 = 16a + 4b + c

Therefore

16a + 4b + c = 42 ........... equation 3

Combining the three equations we have:

4a - 2b + c = 18 ................ equation 1

c = 2 ............... equation 2

16a + 4b + c = 42 ........... equation 3

Solving the resulting linear equations:

substitute equation 2 into equation 1 and equation 3 ,

substituting into equation 1 first we have

4a - 2b + 2 = 18

4a - 2b = 16

dividing through by 2 , we have

2a - b = 8 ............... equation 4

substituting c = 2 into equation 3 , we have

16a + 4b + c = 42

16a + 4b + 2 = 42

16a + 4b = 40

dividing through by 4 , we have

4a + b = 10 ................ equation 5

combining equation 4 and 5 , we have

2a - b = 8 ............... equation 4

4a + b = 10 ................ equation 5

Adding the two equations to eliminate b , we have

6a = 18

a = 18/6

a = 3

Substituting a = 3 into equation 4 to find the value of b , we have

2(3) - b = 8

6 - b = 8

b = 6 - 8

b = -2

Therefore :

a = 3 , b = -2 and c = 2

Substituting these values into the equation of parabola in standard form , we have

y = 3
`x^(2)` - 2x + 2