DETERMINE A QUADRATIC EQUATION GIVEN THE ROOTS AND A POINT

The quadratic function will be in the form

y = ax2 + bx + c

this will have two roots α and β.

Roots = zeroes = x-intercepts

To find quadratic function from the given roots, we follow the steps give below.

Step 1 :

The given roots can be converted into factored form

x = α ==> (x - α)

x = β ==> (x - β)

y = a(x - α)(x - β)

Step 2 :

To figure out the value of a, we apply the given point.

Problem 1 :

The roots of a quadratic equation are -2 and -6. The minimum point of the graph of its related function is at (-4, -2). Sketch the graph of the function.

Solution :

α = -2 and β = -6

α + β = -2 - 6 ==> -8

α β = -2 (-6) ==> 12

The quadratic function will be,

y = a(x2 - (-8)x + 12)

y = a(x2 + 8x + 12)

Since the quadratic function is having a minimum point (-4, -2), the will pass through this point.

-2 = a((-4)2 + 8(-4) + 12)

-2 = a(16 - 32 + 12)

-2 = a(-4)

a = 1/2

By applying the value of a, we get

y = (1/2)(x2 + 8x + 12)

So, the required quadratic function is

y = (1/2)(x2 + 8x + 12)

Problem 2 :

The roots of a quadratic equation are -6 and 0. The minimum point of the graph of its related function is at (-3, 4). Sketch the graph of the function.

Solution :

α = -6 and β = 0

α + β = -6 + 0 ==> -6

α β = -6 (0) ==> 0

The quadratic function will be,

y = a(x2 - (-6)x + 0)

y = a(x2 + 6x)

Since the quadratic function is having a minimum point (-3, 4), the will pass through this point.

4 = a((-3)2 + 6(-3))

4 = a(9 - 18)

4 = a(-9)

a = -4/9

By applying the value of a, we get

y = (-4/9)(x2 + 6x)

So, the required quadratic function is

y = (-4/9)(x2 + 6x)

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