A parabola has intercepts of x = -2, x = 3, and y = -4.
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
a. Write the intercept form of the parabola.
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
b. State the direction of the parabola. Explain.
Looking at the graph above we can see that it can only open upward.
c. Write in ax2 + bx + c form.
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
= a
= 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 =
y =
y =
e. What is the vertex?
We use the vertex formula:
The vertex is the point with x-coordinate =
The y-coordinate of the vertex is found by substituting into the
original equation:
y =
y =
y =
y =
y =
y =
So the vertex is , which is the
red point at the bottom of the graph below.
d. What is the axis of symmetry?
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 .
It is the vertical line passing through the vertex.
Axis of symmetry , the green vertical line below:
Edwin