

Polynomial and/or Polynomial
Functions and Equations 
Quadratic function
and equation 
The
graph of quadratic function 
Translated form of
quadratic function 
The sum and product of
the roots 
Vertex (the turning
point, maximum or minimum)  coordinates of translations 
Transformations
of the graph of the quadratic function 
Quadratic equation
word problems 





Quadratic function
and equation 
The
polynomial function of the second degree, f
(x)
=
a_{2}x^{2}
+ a_{1}x
+ a_{0}, is called a
quadratic function. 


y
= f (x)
=
a_{2}x^{2}
+ a_{1}x
+ a_{0}
or y

y_{0}
= a_{2}(x

x_{0})^{2}, 
where 


coordinates
of translations of the quadratic function. 

By
setting x_{0}
=
y_{0}
= 0, we get 
y
=
a_{2}x^{2},
the source quadratic function.
The turning point V
(x_{0},
y_{0}). 

The
real zeros of the
quadratic function: 



y
= f (x) =
a_{2}x^{2}
+ a_{1}x
+ a_{0} = a_{2}(x
 x_{1})(x
 x_{2})
= a_{2}[x^{2}

(x_{1} +
x_{2})x
+
x_{1}x_{2}] 

The graph of a quadratic function is curve called a parabola. The parabola is symmetric with respect to a vertical line
called the axis of symmetry. 
As
the axis of symmetry passes through the vertex of the parabola
its equation is x
= x_{0}. 

Transformations
of the graph of the quadratic function 
How
changes in the expression of the quadratic function affect its
graph is shown in the figures below. 



The graph of quadratic polynomial will intersect the
xaxis in two
distinct points if its leading coefficient
a_{2} and the vertical translation
y_{0}
have different signs, i.e., if a_{2}
· y_{0
}<
0. 


Example:
Find zeros and vertex of
the quadratic function y
=

x^{2}
+ 2x
+
3
and sketch its graph. 
Solution:
A quadratic function can
be rewritten into translatable form y

y_{0}
= a_{2}(x

x_{0})^{2}
by completing the square, 
y
=

x^{2}
+ 2x
+ 3 
Since a_{2}
· y_{0
}<
0 given
quadratic function must have two different real zeros. 
y
=

(x^{2}

2x)
+ 3 
To find zeros of a function, we set
y
equal to zero and solve for x.
Thus, 
y
=

[(x

1)^{2}

1]
+ 3 

4
=

(x

1)^{2} 
y 
4
=

(x

1)^{2} 
(x

1)^{2}
=
4 
y

y_{0}
= a_{2}(x

x_{0})^{2} 
x

1
=
+ sqrt(4) 
V(x_{0},
y_{0})
=> V(1,
4) 
x_{1,2}
=
1
+
2, => x_{1}
= 
1
and x_{2}
= 3. 


We
can deal with given quadratic using the property of the
polynomial explored under the title, 'Transformations of the polynomial function
applied to the quadratic and cubic functions' above. 
Thus, 
1)
calculate the coordinates of translations of the quadratic
y
=
f (x) =

x^{2}
+ 2x
+
3 

2)
To
get the source quadratic function, plug the coordinates
of translations (with changed signs) 
into the general form
of the quadratic, i.e., 
y
+ y_{0}
= a_{2}(x
+ x_{0})^{2}
+ a_{1}(x
+ x_{0})
+ a_{0}
=>
y
+ 4
= 
(x
+ 1)^{2}
+ 2(x
+ 1)
+
3 
y
=

x^{2
}the source quadratic function 
3)
Inversely, by plugging the coordinates of translations into the source quadratic function 
y
 y_{0}
= a_{2}(x
 x_{0})^{2}
=>
y
 4
= 
(x 
1)^{2}

obtained is given quadratic in general form
y
=

x^{2}
+ 2x
+
3. 

Quadratic equation
word problems 
Example:
A train made up for delay of 12 minutes after 60 km of way by running 10 km/h faster then regular speed.
What is the regular speed of the train? 


Example:
In a theater each row has the same number of seats. Number of rows equals number of seats. By doubling number
of rows and decreasing number of seats 10 per row, total number of seats in the theater increases by 300. How
many rows are in the theater? 
Solution:
Taking x
as the number of rows (or
seats per row),
then 









College
algebra contents C




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