find the vertex and axis of symmetry

Question

y=2x²+2x+3

Answer 1

y = 2x² + 2x + 3

 

The above quadratic is in the "standard form".  There is also an equivalent "vertex form" for a quadratic equation:

 

y = a(x-h)² + k

 

Where (h,k) is the (as yet unknown) location of the vertex. Let's set the right hand sides of the two equations equal and solve for h and k, the location of the vertex.

 

2x² + 2x + 3 = a(x-h)² + k

2x² + 2x + 3 = a(x² - 2hx + h²) + k

2x² + 2x + 3 = ax² - 2hax + (h²a + k)

 

Equating terms:

• 2x² = ax², so a = 2

• 2x = -2hax, so -2ha = 2

Since a = 2:

-2h*2 = 2

h = -1/2

• 3 = h²a+k

Since a = 2 and h = 1/2:

3 = (1/2)²(2) + k

3 = 1/2 + k

5/2 = k

 

The vertex is located at the point (-1/2, 5/2).

 

The axis of symmetry is a vertical line through the vertex, x = -1/2.