THE GRAPH OF A QUADRATIC FUNCTION

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• THE GRAPH OF A QUADRATIC FUNCTION

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QUADRATIC FUNCTIONS
A quadratic function is a function of the
form f(x) ax 2 bx c
where a, b c are real numbers and a ? 0 The
domain of a quadratic function is all real
numbers.
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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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y (x - 3)2 - 4
Example
Y-intercept
x
y
Axis of Symmetry
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5
5
0
Roots or x intercepts
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-3
3
-4
Vertex
2
-3
1
0
0
5
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GRAPHS OF QUADRATIC FUNCTIONS
The general 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 general form of a quadratic function
f(x) ax2 bx c is changed to the vertex
form f(x) 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
f(x) 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
f(x) 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 general form f(x) ax2 bx
c into the vertex 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 which is a little different than
completing the square to solve a quadratic
equation.
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Graphing Quadratic Functions by Completing the
Square
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Remember about Perfect Square Trinomials
Factor x2 6x 9
(x 3)(x 3) or (x 3)2
Perfect Square Trinomial
The factors are in the form (x a)2 or (x - a)2.
Note the relationship between the middle term and
the last term.
The last term is one-half the middle term squared.
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Find the value of the last term that will make
the following perfect square trinomials.
(x 7)2
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x2 14x _______
x2 7x _______
x2 - 3x _______
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Changing from Standard Form to Vertex Form
Write y x2 10x 23 in the form y a(x - h)2
k. Sketch the graph.
y x2 10x 23
(
)
1. Bracket the first two terms.
y (x2 10x ____ - ____) 23
25
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2. Add a value within the brackets to make
a perfect square trinomial. Whatever
you add must be subtracted to keep the value
of the function the same.
y (x2 10x 25) - 25 23
y (x 5)2 - 2
3. Group the perfect square trinomial.
4. Factor the trinomial and simplify.
(-5, -2)
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Changing from Standard Form to Vertex Form
Write y 2x2 - 12x -11 in the form y a(x - h)2
k. Sketch the graph.
1. Bracket the first two terms.
y 2x2 - 12x - 11
(
)
y 2(x2 - 6x) - 11
2. Factor out the coefficient of the x2-
term.
y 2(x2 - 6x ____ - ____) - 11
9
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y 2(x2 - 6x 9) - 18 - 11
3. Add a value within the brackets to make
a perfect square trinomial. Whatever
you add must be subtracted to keep the value
of the function the same.
y 2(x - 3)2 - 29
Multiply, when you remove this term from the
brackets.
4. Group the perfect square trinomial. When
grouping the trinomial, remember to
distribute the coefficient.
5. Factor the trinomial and simplify.
(3, -29)
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Completing the Square
y -3x2 5x - 1
y -3x2 5x - 1
( )
y -3(x2 - x) - 1
y -3(x2 - x ______ - ______ ) - 1
y -3(x2 - x ) - 1
y -3(x - )2 - 1
y -3(x - )2
Vertex is
y -3(x - )2
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The Vertex
Formula
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Completing the Square - The General Case
Using the general form, y ax2 bx c,
complete the square
y ax2 bx c
y (ax2 bx ) c
The vertex is
This IS the vertex BUT it is easier just to
remember that the x-value is and then plug
that in to the equation to get the y-value for
the vertex.
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Using the Vertex Formula
Find the vertex and the maximum or minimum value
of
f(x) -4x2 - 12x 5
using the axis of symmetry, the vertex is
Find the x-value of the vertex
Find the y-value of the vertex
Therefore there is a maximum of y 14, when x
The vertex is
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Direction of the Parabola

• If the coefficient of x2 is positive the parabola
  will open up.
• ?

• If the coefficient of x2 is negative the parabola
  will open down.
• ?

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CHARACTERISTICS OF THE GRAPH OF A QUADRATIC
FUNCTION
f(x) ax2 bx c
Parabola opens up and has a minimum value if a gt
0. Parabola opens down and has a maximum value 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 AND Y INTERCEPTS OF A QUADRATIC FUNCTION

• Find the x-intercepts by setting the quadratic
  function equal to zero and solve by whatever
  method is easiest.
• 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.
• 3. 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.
• 4. 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.
• 5. Find the y-intercept by substituting x0 into
  function.

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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