Question 887945
A parabola has intercepts of x = -2, x = 3, and y = -4.
<pre>
Therefore it passes through the points (-2,0), (3,0), and (0,-4).

We plot those and sketch the graph approximately going through those
three points

{{{drawing(400,400,-5,5,-5,5,graph(400,400,-5,5,-5,5,(2/3)(x+2)(x-3)),
circle(-2,0,0.15),circle(-2,0,0.13),circle(-2,0,0.11),circle(-2,0,0.09),circle(-2,0,0.07),circle(-2,0,0.05),circle(-2,0,0.03),circle(-2,0,0.01),

circle(3,0,0.15),circle(3,0,0.13),circle(3,0,0.11),circle(3,0,0.09),circle(3,0,0.07),circle(3,0,0.05),circle(3,0,0.03),circle(3,0,0.01),

circle(0,-4,0.15),circle(0,-4,0.13),circle(0,-4,0.11),circle(0,-4,0.09),circle(0,-4,0.07),circle(0,-4,0.05),circle(0,-4,0.03),circle(0,-4,0.01)

)}}}
</pre>
a. Write the intercept form of the parabola.
<pre>
x=-2 becomes x+2=0 and x=3 becomes x-3=0

The intercept form is 

y = a(x+2)(x-3)

because when you set that = 0 you get the x-intercepts x=-2 and x=3
</pre>
b.  State the direction of the parabola. Explain.
<pre>
Looking at the graph above we can see that it can only open upward.           
</pre> 
c.  Write in ax<sup>2</sup> + bx + c form.
<pre>
We take the intercept form and substitute the y-intercept (0,-4)

 y = a(x+2)(x-3)
-4 = a(0+2)(0-3)
-4 = a(2)(-3)
-4 = -6a
{{{(-4)/(-6)}}} = a
{{{2/3}}} = a

[Note: We could have answered b above without looking at
the graph because if a is positive the graph opens
upward and if negative it opens downward]

Substitute for a:

 y = {{{2/3}}}{{{(x+2)(x-3)}}}
 y = {{{2/3}}}{{{(x^2-x-6)}}}
 y = {{{expr(2/3)x^2-expr(2/3)x-4}}}
</pre>
e.  What is the vertex?
<pre>
We use the vertex formula:

The vertex is the point with x-coordinate {{{-b/(2a)}}} = 
{{{-(-2/3)/(2(2/3))}}}{{{""=""}}}{{{(2/3)/(4/3)}}}{{{""=""}}}{{{(2/3)*(3/4)}}}{{{""=""}}}{{{1/2}}}

The y-coordinate of the vertex is found by substituting {{{x=1/2}}} into the
original equation:

 y = {{{expr(2/3)x^2-expr(2/3)x-4}}}
 y = {{{(2/3)(1/2)^2-(2/3)(1/2)-4}}}
 y = {{{(2/3)(1/4)-(1/3)-4}}}
 y = {{{1/6-1/3-4}}}
 y = {{{1/6-2/6-24/6}}}
 y = {{{-25/6}}}

So the vertex is {{{(matrix(1,3,1/2,",",-25/6))}}}, which is the
red point at the bottom of the graph below.
</pre>
d.	 What is the axis of symmetry? 
<pre>
The axis of symmetry is the vertical line whose equation is x=h, 
where h is the x-coordinate of the vertex. In this case it is {{{x=1/2}}}.
It is the vertical line passing through the vertex.

Axis of symmetry {{{x=1/2}}}, the green vertical line below:

{{{drawing(400,400,-5,5,-5,5,graph(400,400,-5,5,-5,5,(2/3)(x+2)(x-3)),
circle(-2,0,0.15),circle(-2,0,0.13),circle(-2,0,0.11),circle(-2,0,0.09),circle(-2,0,0.07),circle(-2,0,0.05),circle(-2,0,0.03),circle(-2,0,0.01),

circle(3,0,0.15),circle(3,0,0.13),circle(3,0,0.11),circle(3,0,0.09),circle(3,0,0.07),circle(3,0,0.05),circle(3,0,0.03),circle(3,0,0.01),
green(line(1/2,10,1/2,-10)),
circle(0,-4,0.15),circle(0,-4,0.13),circle(0,-4,0.11),circle(0,-4,0.09),circle(0,-4,0.07),circle(0,-4,0.05),circle(0,-4,0.03),circle(0,-4,0.01),

red(
circle(0.5,-4.16666667,0.15),circle(0.5,-4.16666667,0.13),circle(0.5,-4.16666667,0.11),circle(0.5,-4.16666667,0.09),circle(0.5,-4.16666667,0.07),circle(0.5,-4.16666667,0.05),circle(0.5,-4.16666667,0.03),circle(0.5,-4.16666667,0.01))


)}}}

Edwin</pre>