How do you find the vertex and axis of symmetry, and then graph the parabola given by: #g(x)= 3x^2 + 12x + 15#?

1 Answer
Alan P.
Oct 9, 2015

Vertex: #(-2,3)#
Axis of symmetry: #x=-2#

Explanation:

Given
#color(white)("XXX")g(x)=3x^2+12x+15#

Part 1: The Vertex
Convert into vertex form (#g(x)=m(x-a)^2+b# with vertex at #(a,b)#)

#color(white)("XXX")#Extract the #m#
#color(white)("XXX")g(x)=3(x^2+4x) + 15#

#color(white)("XXX")#Complete the square
#color(white)("XXX")g(x)= 3(x^2+4x+2^2)+15 - 3(2^2)#

#color(white)("XXX")#Re-write as a squared binomial of form #(x-a)^2# and simply
#color(white)("XXX")g(x)=3(x-(-2))^2+3#

This is in vertex form with the vertex at #(-2,3)#

Part 2: The Axis of Symmetry
An parabolic equation in the form:
#color(white)("XXX")y=ax^2+bx+c#
has a vertical axis (i.e. #x=c# for some constant #c#) through the vertex.

Since the given #g(x)# is of this form and the vertex is at #(x,y)=(-2,3)#
the axis of symmetry is
#color(white)("XXX")x=(-2)#

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