Question 1044241
f(x) = 3x^2 +6x - 4
:
This is a parabola that curves upward, here is its graph
:
{{{ graph( 300, 200, -2.8, 0.8, -8, 4, 3x^2 +6x -4) }}}
:
The general form of a polynomial for the given function is
:
f(x) = ax^2 +bx +c, where a, b, c are real numbers
:
The x coordinate for the vertex is given by 
:
x = -b /2a, for our given function we have
:
x = -6 / (2 * 3) = -1
:
substitute for x in the given function for the y coordinate of the vertex
:
y = f(x) = 3(-1)^2 +6(-1) -4 = -7
:
***************************************
a) vertex is at (-1, -7)
***************************************
:
***************************************
b) axis of symmetry is x = -1
half of the parabola is to the left and half of the parabola is to the right 
***************************************
:
The y intercept is found by setting x = 0 and solve for y
y = 3(0)^2 +6(0) -4 = -4
:
x intercept/s are found by solving the given function for x, when f(x) = 0
:
3x^2 +6x -4 = 0
:
use quadratic formula to solve for x
:
x = (-6 + square root((-6)^2 -4*3*(-4))) / (2(3)) = (-3 + square root(21)) / 3
:
x = (-6 - square root((-6)^2 -4*3*(-4))) / (2(3)) = (-3 - square root(21)) / 3
:
*****************************************************************************
c) y intercept is -4
x intercepts are (-3 + square root(21)) /3, (-3 - square root(21)) / 3
note the intercepts are where the graph of f(x) crosses the y axis(y intercept) and where f(x) crosses the x axis (x intercepts
******************************************************************************
:
*************************************************
d) The domain and range of f(x) are the real numbers
*************************************************
:
***************************************************
e) The graph of f(x) shows us the f(x) is increasing on the interval [-1, +infinity) and f(x) is decreasing on the interval [-infinity, -1]
****************************************************
:
*****************************************************
f) see graph above
*****************************************************
: