4.2 – Quadratic Relations
MPM2D
The relation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 is called a _______________________________,
where 𝑎, 𝑏, and 𝑐 are real numbers and 𝑎 ≠ 0.
Investigation: How can you compare relations of the form 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐?
Make a table of values for each relation from −𝟑 to +𝟑.
a) 𝑦 = 𝑥 2
b) 𝑦 = 2𝑥 2
c) 𝑦 = 𝑥 2 + 2𝑥 + 3
𝒙
𝒙
𝒙
𝒚
d) 𝑦 = −𝑥 2
𝒙
𝒚
Graph a) b) and c) on the grid below.
𝒚
𝒚
e) 𝑦 = −0.5𝑥 2 + 3
𝒙
𝒚
Graph d) and e) on the grid below.
What do you notice about the 𝑎 value in 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 and how it affects the graphs?
Properties of Quadratic Relations
The graph (shape) of a quadratic relation is called a _______________________.
A ___________________ has a minimum or maximum point called the ______________ (ℎ, 𝑘)
The 𝑦-value of the vertex is called the _________________________________.
A parabola is symmetrical about a vertical line drawn through the vertex, called the
___________________________________________.
If the parabola crosses the x-axis, the x-coordinates of these points are called the ___________,
or ___________________________ of the relation.
Example 1 𝑦 = 𝑥 2 − 6𝑥 + 7
Label the vertex, zeros, and the axis of symmetry. What is the optimal value?
Is it a maximum or a minimum?
What do you notice about the 𝑐 value in 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐?
Parabola
Graph
Vertex
Optimal
Value
Axis of
Symmetry
Zeroes a.k.a.
x-intercepts
(if any)
Direction of
Opening
y–intercept
True or False?
______
The axis of symmetry and the y-intercept are the same.
______
The axis of symmetry is always located halfway between the zeroes (if there are zeros).
______
The y-coordinate of the vertex is always the same as the optimal value.
______
The x-coordinate of the vertex is always the same as the axis of symmetry.
______
All parabolas have 2 zeros (x-intercepts)
______
The y-intercept is always positive.
______
Some parabolas have no y-intercepts
Example 2: Draw parabolas that have the following characteristics.
a) a minimum value
b) a maximum value
c) two x-intercepts
d) no x-intercepts.
e) one x-intercept and an axis of symmetry of x = 2.
Example 3: First and Second Differences
𝑦 = 2𝑥 – 5
𝒙
−2
𝒚
First
Difference
Second
Difference
𝑦 = 𝑥2 – 4
𝒙
−2
−1
−1
0
0
1
1
2
2
𝒚
First
Difference
Second
Difference
How can you use first differences and second differences to determine whether a relation is
linear or quadratic?
Homework: Pg. 172 #5, 6, and 9