How do i sketch the graph of this function??? PLEASE HELP!!!?

Question

y= 4x squared+ 6x+ 3

the direction also states that i need to label the vertex but i don't even konw what it is so please tell me...

thanks in advance!

Answer 1

Complete the square on the right side:

4x² + 6x + 3 = 4(x² + (3/2)x ) + 3

=  4(x² + (3/2)x + [(3/2)/2]² - [(3/2)/2]²) + 3

= 4(x² + (3/2)x + (3/4)² - (3/4)²) + 3

= 4((x + 3/4)²) - 4((3/4)²) + 3

= 4((x + 3/4)²) - 9/4 + 3

= 4((x + 3/4)²) + 3/4

So y = 4((x + 3/4)²) + 3/4

This is the so-called vertex form of the equation. It is a parabola, opening upward, with axis x = -3/4 and vertex at (-3/4, 3/4)