Graphs of Quadratic
Functions
Focus 10 Learning Goal –
(HS.A-REI.B.4, HS.F-IF.B.4, HS.F-IF.C.7, HS.F-IF.C.8)
=
Students will sketch graphs of quadratics using key features and solve quadratics using the
quadratic formula.
4
3
2
1
In addition to
level 3, students
make
connections to
other content
areas and/or
contextual
situations
outside of math.
Students will
sketch graphs of
quadratics using
key features and
solve quadratics
using the
quadratic
formula.
- Students will be
able to write,
interpret and
graph quadratics
in vertex form.
Students will be
able to use the
quadratic
formula to solve
quadratics and
are able to
identify some key
features of a
graph of a
quadratic.
Students
will have
partial
success at
a 2 or 3,
with help.
0
Even with
help, the
student is
not
successful
at the
learning
goal.
Quadratic Graphs Vocabulary
 Axis of Symmetry: Given a
quadratic function in
standard form,
f(x) = ax2 + bx + c, the vertical
line given by the graph of the
equation x = -b/2a, is called
the axis of symmetry.
 Vertex: The point where the
graph of the quadratic
function and its axis of
symmetry intersect.
Quadratic Graphs Vocabulary
 End Behavior of a Graph: Given a
quadratic function in the form f(x) =
ax2 + bx + c or f(x) = a(x-h)2 + k, the
quadratic function is said to open up
if a > 0 and open down if a < 0.
 If a > 0, then f has a minimum at the
x-coordinate of the vertex (f is
decreasing for x-values less than, or
to the left of, the vertex, and f is
increasing for x-values greater than,
or to the right of, the vertex.)
 If a < 0, then f has a maximum at xcoordinate of the vertex. (f is
increasing for x-values less than, or to
the left of, the vertex, and f is
decreasing for x-values greater than,
or to the right of, the vertex.)
Architecture Around
the World
 The photographs of
architectural features might
be closely represented by
graphs of quadratic
functions.
 How would you describe the
overall shape of a graph of a
quadratic function?
 What is similar or different
about the overall shape of
the curves to the left?
Graphs of Quadratic Functions – Graph A
1. Fill in the table of values based off of the
graph.
2. State the x-intercepts:
x
f(x)
-1
8
3. State the vertex:
0
3
4. If we wrote the
equation for this graph,
what would be the sign
of the leading
coefficient?
1
0
2
-1
3
4
0
3
5
8
5. Does the vertex represents a minimum
or a maximum?
Graphs of Quadratic Functions – Graph A
 Look at the table and
the graph, state points
of symmetry.
 (-1, 8) and (5, 8)
 (0, 3) and (4, 3)
 What interval is the
graph increasing?
 [2, ∞]
 What interval is the graph
decreasing?
 [-∞, 2]
x
f(x)
-1
8
0
3
1
0
2
-1
3
0
4
3
5
8
Graphs of Quadratic Functions – Graph B
1. Fill in the table of values based off of the
graph.
2. State the x-intercepts:
x
f(x)
-5
-5
3. State the vertex:
-4
0
4. If we wrote the
equation for this graph,
what would be the sign
of the leading
coefficient?
-3
3
-2
4
-1
3
0
0
1
-5
5. Does the vertex represents a minimum
or a maximum?
Graphs of Quadratic Functions – Graph B
 Look at the table and
the graph, state points
of symmetry.
 (-5, -5) and (1, -5)
 (-4, 0) and (0, 0)
 What interval is the
graph increasing?
 [-∞, -2]
 What interval is the graph
decreasing?
 [-2 , ∞]
x
f(x)
-5
-5
-4
0
-3
3
-2
4
-1
3
0
0
1
-5
Patterns in the tables of values…
Table A
Table B
x
f(x)
x
f(x)
-1
8
-5
-5
0
3
-4
0
1
0
-3
3
2
-1
-2
4
3
0
-1
3
4
3
0
0
5
8
1
-5
What patterns do you see in
the tables of values?
How can we know the xcoordinate of the vertex by
looking at two symmetric
points?
What happens to the yvalues of the functions as the
x-values increase to very
large numbers?
Quadratic Challenge Time!
The graph at the left is half of a quadratic
function.
With a partner, complete the graph by
plotting 3 additional points.
Be prepared to explain how you found
these points.
Quadratic Challenge Time!
x
f(x)
 Fill in the table of values:
-3
-6
 What are the coordinates of the
x-intercepts?
-2
-1
-1
2
0
 What are the coordinates of the
y-intercept?
3
1
2
2
-1
 What are the coordinates of the vertex?
Is it a minimum or a maximum?
3
-6
 If we knew the equation for this curve, what
would the sign of the leading coefficient be and
why?
Find the equation for the axis of symmetry for the
graph of a quadratic function with the given pair
of coordinates. If not possible, explain why.
1. (3, 10) and (15, 10)
1. Since these ordered pairs are points of symmetry (same yvalue), you can find the axis of symmetry.
2. The middle of 3 and 15 is 9.
3. The axis of symmetry would be x = 9.
2. (-2, 6) and (6, 4)
1. Since these ordered pairs are not points of symmetry
(different y-values), you cannot find the axis of symmetry.
Conclusion:
Graphs of quadratic functions have a unique
symmetrical nature with a maximum or minimum
function value corresponding to the vertex.
When the leading coefficient of the quadratic
expression representing the function is negative
the graph opens down and when positive it
opens up.