Quadratic Functions

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Quadratic Functions
Classroom Lecture Algebra 1 Anne Walton - IUSD
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Introduction
We have already learned about linear functions,
for example f(x) 4x 1 g(x) -6x -2
h(x) (-2/3)x 2
The graphs of these functions are straight lines
and each equation gives us clues as to what
each line looks like f(x) 4x 1
Slope of 4
y-intercept of 1
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Question
What do the functions f(x) 2x2 4x 3
g(x) x2 4x

h(x) -x2 5x 6 have in common?
Answer
They are all QUADRATIC FUNCTIONS that can be
written in standard form f(x) ax2 bx c
where a ? 0
Pay close attention these coefficients a, b
and c will be clues that help us to know what the
graphs of these quadratic functions look like
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Real Life Connection
Remember, quadratic functions represent real-life
situations such as
the height of a kicked soccer ball f(x)
-1/2(-9.8)t2 20t
the motion of falling objects pulled by gravity
height -16t2 100
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Graphs of Quadratic Functions
Quadratic Functions graph into a shape called a
Parabola
If the coefficient of the x2 term, a, is
positive, the parabola opens upward
If the coefficient of the x2 term, a, is
negative, the parabola opens downward
Maximum Point
Vertex
Minimum Point
f(x) x2 - 4
f(x) -x2 5
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The Line of Symmetry
The graph of every quadratic function has a line
of symmetry. This is a vertical line that goes
through the vertex and divides the graph into two
symmetrical halves.
Remember, that our standard form for a quadratic
function is f(x) ax2 bx c
X -2
The equation for the line of symmetry for a
quadratic function is x -b/(2a)
In our example here, we substitute a1 and b4
and get the equation x - 4/2
x - 2
x2 4x 4
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Finding the Vertex for our graph
You can see from the graph that the vertex is on
the line of symmetry
so it follows that the value for the x
coordinate of the vertexs ordered pair (x,y) is
b/(2a) which in this case is -2.
X -2
We then substitute this value for x into the
equation and discover that the value for y
coordinate is y (-2)2 4(-2) 4 y 4 8
4 y 0
(-2,0)
Our vertex has the coordinates (-2,0)
f(x) x2 4x 4
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Roots of Quadratic Functions
The roots of quadratic functions are the values
of x where the function equals zero ( f(x) 0 ).
When the function is graphed, the roots are
depicted as the places where the graph crosses
the x axis (since at this location, f(x) y 0
).
Quadratic functions can have two, one or zero
real-number roots.
Equivalently, their parabolas can have two
one
or zero real-number roots.
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Discriminant
Is there a way that we can know how many roots a
quadratic function has without having to graph it?
Yes! We can use the Discriminant. The
discriminant is a unique value for a function,
determined by the expression b2 4ac, that
tells us how many real-number roots the function
has!
If the value of the discriminant is positive, it
means that there are two real-number roots. If
the value is zero, it means that there is one
real-root and if the value is negative, it means
that there are zero real-number roots for this
function.
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Putting it all together
Heres a problem that lets us use everything
weve learned today

• Given the function f(x) 5x x2 6
• Tell whether the function graphs into a parabola
  that opens upwards or downwards.
• Find the equation of the line of symmetry for the
  graph
• Find the coordinates of the vertex of the
  parabola
• Determine how many real-number roots the function
  has
• Graph the function

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Putting it all together

• Given the function f(x) 5x x2 6
• Tell whether the function graphs into a parabola
  that opens upwards or downwards.

Since our best clues are found in the
coefficients a , b and c of our equation in
standard form, first lets put our function into
standard form. f(x) -x2 5x 6 Our
coefficients are thus a -1, b 5 and c
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We know that the parabola opens upwards because
the coefficient of the x2 term, a , is negative.
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Putting it all together
f(x) -x2 5x 6 a -1, b 5 and c 6
Given the function f(x) 5x x2 6 b) Find
the equation of the line of symmetry for the graph
The equation of the line of symmetry is defined
as x -b/(2a) . In this example that translates
into x -5 / (2 (-1)) X -5 / (-2) X 5/2 X
2.5
X 2.5
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Putting it all together
Given the function f(x) 5x x2 6 c) Find
the coordinates of the vertex of the parabola
f(x) -x2 5x 6 a -1, b 5 and c 6
The x-coordinate of the vertex is going to lie on
the line of symmetry. Remember, this lines
equation is x 2.5 So, the vertex is ( 2.5 , y )
( 2.5 , 12.25 )
And the y coordinate can be calculated by
substituting 2.5 for x and solving for y.
F(x) y -(2.5)2 5(2.5) 6 y
-6.25 12.5 6 y 12.25 So the
vertex is at ( 2.5 , 12.25 )
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Putting it all together
Given the function f(x) 5x x2 6 d)
Determine how many real-number roots the function
has
f(x) -x2 5x 6 a -1, b 5 and c 6
We answer this question by figuring out the value
of the Discriminant.
Discriminant b2 4ac D (5)2 (4)(-1)(6) D
25 (-24) D 25 24 D 49
Since the value of the discriminant is positive,
the function must have two real-number roots.
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Putting it all together
Given the function f(x) 5x x2 6 e) Graph
the function
f(x) -x2 5x 6 a -1, b 5 and c 6
We already know the line of symmetry and the
vertex of our function.
(2.5,12.25)
(1,10)
In order to determine 4 other points on this
graph, we will make a chart, choosing our vertex
as our middle value.
(4,10)
(0,6)
(5,6)
roots
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Summary
Quadratic functions are functions in the form
f(x) ax2 bx c.
Quadratic functions graph into a shape called a
Parabola. If a lt 0, the parabola opens
downwards. If a gt 0, the parabola opens upwards.
The graph of every quadratic function has a line
of symmetry. The equation for the line of
symmetry for a quadratic function is x
-b/(2a). The vertex of the parabola will lie on
this line of symmetry.
Quadratic functions can have two, one or zero
real-number roots.
The discriminant is a unique value for a
function, determined by the expression b2
4ac, that tells us how many real-number roots the
function has.