Question

The equation of a parabola is y = a(x - 1)^2 + k and the points (2, 6.5) and (-1, -1) lie on the parabola. Determine the maximum value of the parabola.

Answer 1

( 2 ,6.5 ) lied on the parabola which mean when we put the x=2 in the equation the y value would be 6.5 :

`6.5 = a * ({2 - 1})^(2) + k`

`6.5 = a + k \: \: \: \: \: \: \: ( \alpha )`

In the otherhand ( -1 , - 1 ) has the same condition:

`- 1 = a * ({ - 1 - 1})^(2) + k`

`- 1 = 4a + k \: \: \: \: \: \: \: \: ( \beta )`

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`4a + k = - 1 \: \: \: \: \: \: \: \: \: \: ( \alpha )`

`a + k = 6.5 \: \: \: \: \: \: \: \: ( \beta )`

Multiply Beta by (-1) :

`4a + k = - 1`

`- a - k = - 6.5`

then we have to find the submission of alpha and beta :

`\alpha + \beta`

`4a - a + k - k = - 1 - 6.5`

`3a = - 7.5`

Divide both sides by 3

`(3a)/(3) = ( - 7.5)/(3) \\`

`a = - 2.5`

Put the value of a in one of the above equations to find the value of k .I'm gonna use the second equation ( beta ) :

`a + k = 6.5`

`- 2.5 + k = 6.5`

Add both sides by 2.5

`- 2.5 + 2.5 + k = 6.5 + 2.5`

`k = 9`

Thus the equation of the parabola is :

`y = - 2.5 ({x - 1})^(2) + 9`

The maximum value of the parabola is for the vertex which is x = 1 , so we have to put it in the equation and find it's y coordinate :

`y = - 2.5 ({1 - 1})^(2) + 9`

`y = - 2.5 * {0}^(2) + 9`

`y = 0 + 9`

`y = 9`

Adn we're done ...