

Quadratic Equations and Quadratic
Function 
Graphing a quadratic
function 
Transformations
of the graph of the quadratic function 

Quadratic Inequalities

Solving quadratic inequality graphically






Graphing a quadratic
function 
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 the given quadratic using the property of the
polynomial explored under the title, 
'
Source
or original polynomial function '. 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 Inequalities

To solve a quadratic inequality we can examine the sign of the
equivalent quadratic function. 
The
xintercepts
or roots are the points where a quadratic function changes the sign. The
xintercepts determine
the three intervals on the xaxis in which the function is above or
under the xaxis, that is, where the function is positive or negative. 

Example:
Solve the inequality

x^{2}
+ 2x
+
3 ≤
0. 
Solution:
Solve the quadratic
equation ax^{2}
+ bx
+
c
= 0
to get the boundary points. 
The zeroes
or roots of equivalent function (see the graph
below) are the endpoints of the intervals and are included in the solution. 
The turning point V(x_{0},
y_{0}), 

The roots, 
x^{2}
+ 2x
+
3 =
0 



Solution: 












Intermediate
algebra contents 



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