THE GRAPH OF A QUADRATIC FUNCTION

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SECTION 2.4

• THE GRAPH OF A QUADRATIC FUNCTION

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GRAPHS OF QUADRATIC FUNCTIONS
As weve 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.
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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.
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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.
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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.
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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.
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GRAPHS OF QUADRATIC FUNCTIONS
Compare f(x) x2 to the following f(x) 2x2
f(x) .5x2 f(x) -.5x2 If a gt 0, then
the parabola opens up If a lt 0, then the parabola
opens down
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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
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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
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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.
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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.
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GRAPHS OF QUADRATIC FUNCTIONS
a(x - h) 2 k Vertex (highest or lowest
point) (h,k) If a gt 0, then the parabola opens
up If a lt 0, then the parabola opens down
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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.
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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
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GRAPHS OF QUADRATIC FUNCTIONS
This will require to process of completing the
square.
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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
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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
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CHARACTERISTICS OF THE GRAPH OF A QUADRATIC
FUNCTION
f(x) ax2 bx c
Parabola opens up if a gt 0. Parabola opens down
if a lt 0.
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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
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THE X-INTERCEPTS OF A QUADRATIC FUNCTION

• If the discriminant b2 4ac gt 0, the graph of
  f(x) ax2 bx c has two distinct x-intercepts
  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 lt 0, the graph of
  f(x) ax2 bx c has no x-intercept and will
  not cross or touch the x-axis.

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FINDING A QUADRATIC FUNCTION
Determine the quadratic function whose vertex is
(1,- 5) and whose y-intercept is -3.
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• CONCLUSION OF SECTION 2.4