Quadratic Functions And Their Properties
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4Section 3.3 Notes Page 1
3.3 Quadratic Functions and Their Properties
This section is all about quadratic functions, which give U shaped graphs called parabolas. First we need to
define a couple of terms involving parabolas:
At the bottom of this graph we have the vertex. This is either the
lowest or highest point on a parabola. In the standard form
=
+
+
2
if a > 0 then the graph opens up and the vertex is
y
ax
bx
c
the lowest point on the graph. If a < 0 then the graph opens down
and the vertex is the highest point on the graph.
Another new term is the axis of symmetry. This is a vertical line
that always goes through the x-coordinate of the vertex. Since this
is a vertical line it will start with x = . This is considered a fold
line since the parabola can be folded on top of itself.
=
−
+
2
Vertex form:
y
a
(
x
h
)
k
. If an equation is written in this form, then the vertex is (h, k). If the a value is
negative, then the graph opens down. If the a value is positive, then the graph opens up.
=
−
+
+
2
EXAMPLE: Find the vertex, axis of symmetry, intercepts and graph of
y
( 2
x
) 3
8
.
=
−
−
−
+
2
In order to find h, we can rewrite the formula as:
y
( 2
x
(
3
))
8
.
Now we know h = -3. We also know that k = 8. So the vertex is (-3, 8).
The axis of symmetry goes through the x-coordinate of the vertex, so this
equation is x = -3. To find the y-intercept, put in a zero for x in our original
=
−
+
+
2
equation:
y
2
0 (
) 3
8
. Solving this we get y = -14. The y-intercept
is written as (0, -10). For the x-intercept, put in a zero for y:
=
−
+
+
−
=
−
+
2
2
0
( 2
x
) 3
8
. To solve this, subtract 4 from both sides:
8
( 2
x
) 3
.
= x
+
2
Now divide both sides by -2:
4
(
) 3
. Now take the square root of both
±
=
+
sides to get:
2
x
3
. This gives us two separate equations to solve:
x + 3 = 2 and x + 3 = -2. Solving each individually we get x = -1, -5.
We write the x-intercept as (-1, 0) and (-5, 0). Plot all these points to get the graph.
If you have a graph given in vertex form, you can also graph it using transformations like we learned before.
So now what do we do if the quadratic equation is not written in vertex form? The book shows that we can start
with the standard form of a quadratic and use the completing the square method to find a formula to find the
vertex if an equation is not written in vertex form. I will just give the formula here.
Vertex formula:
−
b
=
+
+
=
2
Given
y
ax
bx
c
, then the x-coordinate of the vertex is
x
. To find the y-coordinate put the x back
2
a
into the original equation.
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