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Quadratic Graphs and Their
Properties
Section 9-1
Goals
Goal
• To graph quadratic
functions of the form y = ax2
and y = ax2 + c.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to solve
simple problems.
Level 4 – Use the goals to solve
more advanced problems.
Level 5 – Adapts and applies
the goals to different and more
complex problems.
Vocabulary
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Quadratic Function
Standard Form of a Quadratic Function
Quadratic parent function
Parabola
Axis of symmetry
Vertex
Minimum
Maximum
Quadratic Equation
• Solutions of the equation y = x2 are shown
in the graph.
• Notice that the graph is not linear.
• The equation y = x2 is a quadratic equation.
• A quadratic equation in two variables can
be written in the form y = ax2 + bx + c,
where a, b, and c are real numbers and
a ≠ 0.
• The equation y = x2 can be written as
y = 1x2 + 0x + 0, where a = 1, b = 0, and
c = 0.
Quadratic Equations and Their Graphs
For any quadratic equation in two variables
• all points on its graph are solutions to the equation.
• all solutions to the equation appear on its graph.
Quadratic Function
• Notice that the graph of y = x2
represents a function because each
domain value is paired with exactly
one range value.
• A function represented by a
quadratic equation is a quadratic
function.
• the simplest quadratic function f(x) =
x2 or y = x2 is the parent function.
Standard Form
Parabola
• The graph of a quadratic function is a
curve called a parabola.
• A parabola is a U-shaped curve as
shown at the right.
• To graph a quadratic function, generate
enough ordered pairs to see the shape of
the parabola. Then connect the points
with a smooth curve.
Vertex
• The highest or lowest point on a parabola is the vertex.
• If a parabola opens upward, the vertex is the lowest point.
• If a parabola opens downward, the vertex is the highest
point.
Vertex is the
highest point
Vertex is the
lowest point
Quadratic Functions of the
Form f(x) = ax2
Graph the functions on one coordinate plane.
f(x) = x2
y
g(x) = –x2
8
x
f(x)
x
g(x)
6
2
4
2
4
4
1
1
1
1
2
0
0
0
0
1
1
1
1
2
4
2
4
f(x) = x2
x
8
6
4
2
4
6
Notice that the graph of g(x)
is a reflection of the graph of
f(x) over the x-axis.
8
2
4
6
8
g(x) = – x2
Quadratic Functions
If a > 0 in y = ax2 + bx + c, the
parabola opens upward.
If a < 0 in y = ax2 + bx + c, the
parabola opens upward.
The vertex is a minimum point.
The vertex is a maximum point.
Vertex is the
maximum
Opens up
a>0
Vertex is the
minimum
Opens down
a<0
Minimums and Maximums
Example:
Identify the vertex of each parabola. Then give the
minimum or maximum value of the function.
A.
B.
The vertex is (–3, 2), and
the minimum is 2.
The vertex is (2, 5), and the
maximum is 5.
Your Turn:
Identify the vertex of each parabola. Then give the
minimum or maximum value of the function.
a.
b.
The vertex is (–2, 5) and the
maximum is 5.
The vertex is (3, –1), and
the minimum is –1.
Axis of Symmetry
• The vertical line that divides a parabola into two symmetrical
halves is the axis of symmetry.
• The axis of symmetry always passes through the vertex of the
parabola.
Axis of
symmetry
Vertex
Vertex
Axis of
symmetry
Axis of Symmetry & Zeros
Example:
Find the axis of symmetry of each parabola.
A.
Identify the x-coordinate of
(–1, 0)
the vertex.
The axis of symmetry is x = –1.
B.
Find the average of
the zeros.
The axis of symmetry is x = 2.5.
Your Turn:
Find the axis of symmetry of each parabola.
a.
(–3, 0)
Identify the x-coordinate of
the vertex.
The axis of symmetry is x = –3.
b.
Find the average of
the zeros.
The axis of symmetry is x = 1.
Graphing y =
2
ax
• You can use a table of values to graph the
quadratic.
• Also, use the fact that a parabola is symmetric.
– First, find the coordinates of the vertex and several
points on one side of the vertex.
– Then reflect the points across the axis of symmetry.
• For graphs of functions of the form y = ax2, the
vertex is at the origin.
• The axis of symmetry is the y-axis, or x = 0.
Example:
Use a table of values to graph the quadratic
function.
x
–2
–1
0
1
2
y
4
3
1
3
0
1
3
4
3
Make a table of values.
Choose values of x and
use them to find values
of y.
Graph the points. Then
connect the points with a
smooth curve.
Your Turn:
Use a table of values to graph the quadratic
function.
y = –4x2
x
y
–2
–16
–1
–4
0
0
1
–4
2
–16
Make a table of values.
Choose values of x and
use them to find values
of y.
Graph the points. Then connect
the points with a smooth curve.
Domain and Range
Unless a specific domain is given, you may
assume that the domain of a quadratic function
is all real numbers. You can find the range of a
quadratic function by looking at its graph.
For the graph of y = x2 – 4x + 5, the range
begins at the minimum value of the function,
where y = 1. All the y-values of the function
are greater than or equal to 1. So the range is
y  1.
Example:
Find the domain and range.
Step 1 The graph opens downward,
so identify the maximum.
The vertex is (–5, –3), so the
maximum is –3.
Step 2 Find the domain and range.
D: all real numbers
R: y ≤ –3
Your Turn:
Find the domain and range.
Step 1 The graph opens upward, so
identify the minimum.
The vertex is (–2, –4), so the
minimum is –4.
Step 2 Find the domain and range.
D: all real numbers
R: y ≥ –4
Your Turn:
Find the domain and range.
Step 1 The graph opens downward,
so identify the maximum.
The vertex is (2, 3), so the
maximum is 3.
Step 2 Find the domain and range.
D: all real numbers
R: y ≤ 3
Width of the Parabola
• The coefficient of the x2 term in a quadratic
function affects the width of a parabola.
• When |m| < |n|, the graph of y = mx2 is wider than
the graph of y = nx2.
• As we learned earlier the sign of the coefficient of
the x2 term tells the direction it opens.
• Positive it opens up and negative it opens down.
Example:
Graph the functions on one coordinate plane.
y
f(x) = x2
g(x) = 2x2
f(x)
x
g(x)
2
4
2
8
1
1
1
2
0
0
0
0
1
1
1
2
2
4
2
8
x
f(x) = x2
8
6
4
g(x) = 2x2
2
x
8
6
4
2
4
6
8
2
4
6
8
Notice that the graph
of g(x) is narrower
than the graph of f(x).
Example:
Graph the functions on one coordinate plane.
f(x) =
x2
y
1
g(x) = x2
2
x
f(x)
x
g(x)
2
4
2
2
1
1
1
1
2
0
0
0
0
1
1
1
1
2
2
4
2
2
f(x) = x2
8
6
1
g(x) = x2
2
4
2
x
8
6
4
2
4
6
8
2
4
6
8
Notice that the graph
of g(x) is wider than
the graph of f(x).
Your Turn:
Graph f(x) = x2
Graph g(x) = 3x2 and h(x) = (1/3)x2
• How do the shapes of the graphs compare?
Solution:
y
f(x) = x2
g(x) = 3x2
x
h(x) = (1/3)x2
Solution:
Graph f(x) = x2
Graph g(x) = 3x2 and h(x) = (1/3)x2
• How do the shapes of the graphs compare?
The graph of g(x) is narrower than f(x) and
the graph of h(x) is wider than f(x).
Properties of the Form
f(x) = ax2
• If a > 0, the graph of f(x) = ax2 will open upward. In addition,
if 0 < a < 1, the opening in the graph will be “wider” than that
of y = x2. If a > 1, the opening in the graph will be “narrower”
then that of y = x2.
• If a < 0, the graph of f(x) = ax2 will open downward. In
addition, if 0 < |a| < 1, the opening in the graph will be “wider”
than that of y = x2. If |a| > 1, the opening in the graph will be
“narrower” then that of y = x2.
• When |a| > 1, we say that the graph is vertically stretched by a
factor of |a|. When 0 < |a| < 1, we say that the graph is
vertically compressed by a factor of |a|.
Graphing y =
2
ax
+c
• The y-axis is the axis of symmetry for graphs of
functions of the form y = ax2 + c.
• The value of c translates the graph up or down.
• When c is positive the curve shifts up.
• When c is negative the curve shifts down.
Quadratic Functions of the Form
f(x) = x2 + c
y
Graph the functions on one coordinate
plane.
8
g(x) = x2 + 3
6
f(x) = x2
g(x) = x2 + 3
4
f(x) = x2
2
x
f(x)
x
g(x)
2
4
2
7
1
1
1
4
0
0
0
3
1
1
1
4
6
2
4
2
7
8
x
8
6
4
2
4
2
4
6
8
Notice that the graph
of g(x) is the graph of
f(x) shifted 3 units
upward.
Quadratic Functions of the Form
f(x) = x2 – c
Graph the functions on one coordinate
plane.
y
f(x) = x2
g(x) = x2 – 2
x
f(x)
x
g(x)
2
4
2
2
1
1
1
1
0
0
0
2
1
1
1
1
2
4
2
2
f(x) = x2
8
6
4
g(x) = x2 – 2
2
x
8
6
4
2
4
6
8
2
4
6
8
Notice that the graph
of g(x) is the graph of
f(x) shifted 2 units
downward.
Your Turn:
Graph f(x) = x2
Note that a = 1 in standard form.
Which way does it open?
What is the vertex?
What is the axis of symmetry?
Graph g(x) = x2 + 3 and h(x) = x2 – 3
What is the vertex of each function?
What is the axis of symmetry of each function?
Solution:
y
f(x) = x2
g(x) = x2 + 3
x
h(x) = x2 – 3
Solution:
Graph f(x) = x2
Note that a = 1 in standard form.
Which way does it open? Up
What is the vertex? (0, 0)
What is the axis of symmetry? x = 0
Graph g(x) = x2 + 3 and h(x) = x2 – 3
What is the vertex of each function? (0, 3) & (0, -3)
What is the axis of symmetry of each function?
Both x = 0
Properties of the Form
f(x) = ax2 + c
Graphing the Parabola Defined by f(x) = ax2 + c
If c is positive, the graph of f(x) = x2 + c is the graph
of y = x2 shifted upward k units.
If c is negative, the graph of f(x) = x2 + c is the graph
of y = x2 shifted downward |k| units.
The vertex is (0, c), and the axis of symmetry is the
y-axis.
Assignment
• 9-1 Exercises Pg. 556 - 558: #8 – 46 even