UNIT 5 – QUADRATIC RELATIONS
LESSON 1 – QUADRATIC RELATIONS
A relation whose equation is in the form 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, where
𝑎, 𝑏, and 𝑐 are real numbers and 𝑎 ≠ 0.
The shape of the graph of a quadratic relation, which is U-shaped and
symmetrical.
QUADRATIC RELATION:
PARABOLA:
BASIC GRAPH OF 𝒚 = 𝒙𝟐
𝒙
−3
−2
−1
0
1
2
3
𝒚 = 𝒙𝟐
PROPERTIES OF 𝒚 = 𝒙𝟐 :
 It is a Quadratic Function
 Its graph is called a ____________
 Vertex is ________
 The axis of symmetry is ________
 To graph (from vertex):
over 1, up 1
over 2, up 4
over 3, up 9, etc.
VERTEX: The point on a parabola where the curve
changes direction (maximum/minimum).
 The maximum point if the parabola opens down
 The minimum point if the parabola opens up
AXIS OF SYMMETRY: is the line that divides a
figure into two congruent parts
FINITE DIFFERENCES are the differences found
from the y-values in tables with evenly spaced
x-values
FINITE DIFFERENCES
EXAMPLE ① Complete the table of values for the following linear functions and determine the first and
second differences.
𝑦 = 3𝑥 − 2
a)
x
-2
-1
0
1
2
3
y
1st
Difference
𝑦 = 2𝑥 + 3
b)
2nd
Difference
x
-2
-1
0
1
2
3
y
1st
Difference
The first differences for a linear function are ______________________.
2nd
Difference
EXAMPLE ②
Complete the table of values for the following quadratic functions and determine the first
and second differences.
𝑦 = 𝑥2 + 𝑥 + 1
a)
𝑥
−2
−1
0
1
2
3
𝑦
1st
Difference
b)
2nd
Difference
𝑥
−2
−1
0
1
2
3
𝑦 = 2𝑥 2 – 𝑥 + 3
𝑦
1st
Difference
2nd
Difference
For a quadratic function, the first difference increases by a constant amount and the
second difference is a ______________________.
EXAMPLE ③ The entrance to a garden is an arch that can be approximated by the relation
𝑦 = −0.2𝑥 2 + 3.2, where y is the height, in metres, above the ground and x is the width, in
metres, from the centre of the bridge.
a) Graph the quadratic relation
𝒙
−4
−3
−2
−1
0
1
2
3
4
𝒚
b) Describe the shape of the arch:
The shape of the arch is __________________ . The parabola is ___________________
about a vertical line → the _________ . The graph has a __________________ point.
c) How tall and how wide is the arch?
Since the maximum value of 𝑦 is _______, the height of the arch is _______ . The x-axis represents the
ground of the garden. The width of the arch is the difference between the x intercepts.
The x intercepts are ______ and ______, therefore the arch is ___________ wide