Question

Determine the vertex of the quadratic relation y=2(x+2)(x-6)

Answer 1

(2, -32)

Explanation:

1) Expand them.

y = 2(x² - 6x + 2x - 12)

y = 2(x² - 4x - 12)

y = 2x² - 8x - 24.

2) Complete the square to find the vertex.

y = 2([x² - 4x)] - 24

y = 2[(x - 2)² - 4] - 24

y = 2(x - 2)² - 8 - 24

y = 2(x - 2)² - 32

The vertex: (2, -32)

or you can use this: x = -b/2a

x = -(-8)/2(2)

x = 8/4

x = 2

Substitute the value into the original equation for y.

y = 2(2)² - 8(2) - 24

y = 2(4) - 16 - 24

y = 8 - 16 - 24

y = -32

Vertex: (2, -32)

Answer 2

(2, -32)

Explanation:

The vertex of the graph of a quadratic equation is on the line of symmetry, halfway between the x-intercepts. It can be found by evaluating the equation at that point.

Line of symmetry

The given function is written in factored form, so the x-intercepts are easy to find. They are the values of x that make the factors zero:

(x +2) = 0 ⇒ x = -2

(x -6) = 0 ⇒ x = 6

The midpoint between these values of x is their average:

x = (-2 +6)/2 = 4/2

Then the x-coordinate of the vertex, and the equation of the line of symmetry is ...

x = 2

Vertex

Using this value of x in the quadratic relation, we find the y-value at the vertex to be ...

y = 2(2 +2)(2 -6) = 2(4)(-4)

y = -32

The coordinates of the vertex are (x, y) = (2, -32).