SECTION 2.4
THE GRAPH OF A QUADRATIC
FUNCTION
GRAPHS OF QUADRATIC
FUNCTIONS
As we’ve already seen, f(x) = x2 graphs
into a PARABOLA.
This is the simplest quadratic function
we can think of. We will use this one as
a model by which to compare all other
quadratic functions we will examine.
VERTEX OF A PARABOLA
All parabolas have a VERTEX, the lowest
or highest point on the graph
(depending upon whether it opens up or
down.
AXIS OF SYMMETRY
All parabolas have an AXIS OF
SYMMETRY, an imaginary line which
goes through the vertex and about
which the parabola is symmetric.
HOW PARABOLAS DIFFER
Some parabolas open up and some
open down.
Parabolas will all have a different vertex
and a different axis of symmetry.
Some parabolas will be wide and some
will be narrow.
GRAPHS OF QUADRATIC
FUNCTIONS
The standard form of a quadratic
function is:
f(x) = ax2 + bx + c
The position, width, and orientation of a
particular parabola will depend upon
the values of a, b, and c.
GRAPHS OF QUADRATIC
FUNCTIONS
Compare f(x) = x2 to the following:
f(x) = 2x2 f(x) = .5x2
f(x) = -.5x2
If a > 0, then the parabola opens up
If a < 0, then the parabola opens down
GRAPHS OF QUADRATIC
FUNCTIONS
Now compare f(x) = x2 to the following:
f(x) = x 2 + 3
f(x) = x 2 - 2
Vertical shift up
Vertical shift down
GRAPHS OF QUADRATIC
FUNCTIONS
Now compare f(x) = x2 to the following:
f(x) = (x + 2)2
f(x) = (x – 3)2
Horizontal shift to
the left
Horizontal shift to
the right
GRAPHS OF QUADRATIC
FUNCTIONS
When the standard form of a quadratic
function f(x) = ax2 + bx + c is written in the
form:
a(x - h) 2 + k
We can tell by horizontal and vertical shifting
of the parabola where the vertex will be.
The parabola will be shifted h units
horizontally and k units vertically.
GRAPHS OF QUADRATIC
FUNCTIONS
Thus, a quadratic function written in the form
a(x - h) 2 + k
will have a vertex at the point (h,k).
The value of “a” will determine whether the
parabola opens up or down (positive or
negative) and whether the parabola is narrow
or wide.
GRAPHS OF QUADRATIC
FUNCTIONS
a(x - h) 2 + k
Vertex (highest or lowest point): (h,k)
If a > 0, then the parabola opens up
If a < 0, then the parabola opens down
GRAPHS OF QUADRATIC
FUNCTIONS
Axis of Symmetry
The vertical line about which the graph
of a quadratic function is symmetric.
x=h
where h is the x-coordinate of the
vertex.
GRAPHS OF QUADRATIC
FUNCTIONS
So, if we want to examine the
characteristics of the graph of a
quadratic function, our job is to
transform the standard form
f(x) = ax2 + bx + c
into the form
f(x) = a(x – h)2 + k
GRAPHS OF QUADRATIC
FUNCTIONS
This will require to process of
completing the square.
GRAPHING QUADRATIC
FUNCTIONS
Graph the functions below by hand by
determining whether its graph opens up
or down and by finding its vertex, axis of
symmetry, y-intercept, and x-intercepts,
if any. Verify your results using a
graphing calculator.
f(x) = 2x2 - 3
g(x) = x2 - 6x - 1
h(x) = 3x2 + 6x
k(x) = -2x2 + 6x + 2
DERIVING THE FORMULA
FOR THE VERTEX
A formula for the x-coordinate of the
vertex can be found by completing the
square on the standard form of a
quadratic function.
f(x) = ax2 + bx + c
CHARACTERISTICS OF THE
GRAPH OF A QUADRATIC
FUNCTION
f(x) = ax2 + bx + c
 -b
 - b 
VERTEX 
, f
 
 2a  
 2a
-b
AXISOF SYMMETRY: x 
2a
Parabola opens up if a > 0.
Parabola opens down if a < 0.
EXAMPLE
Determine without graphing whether the
given quadratic function has a maximum
or minimum value and then find the
value. Verify by graphing.
f(x) = 4x2 - 8x + 3
g(x) = -2x2 + 8x + 3
THE X-INTERCEPTS OF A
QUADRATIC FUNCTION
1. If the discriminant b2 – 4ac > 0, the graph
of f(x) = ax2 + bx + c has two distinct xintercepts and will cross the x-axis twice.
2. If the discriminant b2 – 4ac = 0, the graph
of f(x) = ax2 + bx + c has one x-intercept
and touches the x-axis at its vertex.
3. If the discriminant b2 – 4ac < 0, the graph
of f(x) = ax2 + bx + c has no x-intercept and
will not cross or touch the x-axis.
FINDING A QUADRATIC
FUNCTION
Determine the quadratic function whose
vertex is (1,- 5) and whose y-intercept is -3.
CONCLUSION OF SECTION 2.4