Quadratic functions
A quadratic function is a function of the form
f
(
x
) =
ax2
+
bx
+
c
where
a
= 0. Every quadratic function is a
transformation of the function
f
(
x
) =
x2
, so the graph of all quadratic functions are parabolas. If
a >
0 then the
function has a unique minimum value located at its vertex, or lowest point. Similarly, if
a <
0 then the function has a
unique maximum value located at its vertex, or highest point. To determine the vertex of a quadratic function, we
complete the square to write the function in vertex form,
f
(
x
) =
a
(
x−B
)
2
+
C
, where
B
=
b/
2
a
and
C
=
c−b2/
4
a
.
The vertex of the parabola is then located at the point (
B, C
). The parabola is symmetric about the line
x
=
B
, and
its domain is all real numbers, its range is [
C, ∞
) if
a >
0 and (
−∞, C
] if
a <
0, so that its minimum/maximum value is
C
occurring at
x
=
B
. The
y
intercept is
c
, and the
x
-intercepts or zeros of the function are
x
= (
−b±√b2−4ac
)
/
2
a
.
Example 1. Let f(x) = −2(x−1)2+ 8. Determine the following:
(1) Whether the parabola opens up or down.
Down, graph has been reflected over x-axis.
(2) The vertex.
Easy to read off from vertex form: (1,8)
(3) The x-intercepts and yintercept.
Set y= 0, x-intercepts are (3,0) and (−1,0). Set x= 0, y-intercept is (0,6).
(4) The axis of symmetry and the maximum/minimum value.
Symmetric about the line x= 1. Because the parabola opens down, the maximum value is 8.
(5) The domain and range.
The domain is all real numbers and the range is (−∞,8].
(6) Sketch the function.
−2−1 1 2 3 4
−3
−2
−1
1
2
3
4
5
6
7
8
9
x
y
Example 2. Let f(x) = 3x2+ 12x+ 5. Determine the following:
(1) Whether the parabola opens up or down.
Up, leading coefficient is positive.
(2) The vertex.
We need to write the function in vertex form. Complete the square:
3x2+ 12x+ 5 = 3(x2+ 4x) + 5 = 3(x2+ 4x+ 4 −4) + 5 = 3(x+ 2)2−12 + 5 = 3(x+ 2)2−7
The vertex is (−2,−7).
(3) The x-intercepts and yintercept.
Set y= 0, x-intercepts are ((−6 + √21)/3,0) and (−6−√21)/3,0). Set x= 0, y-intercept is (0,5).
(4) The axis of symmetry and the maximum/minimum value.
Symmetric about the line x=−2. Because the parabola opens up, the minimum value is -7.
(5) The domain and range.
The domain is all real numbers and the range is [−7,∞).
(6) Sketch the function.
1
