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Graphing quadratic functions:
- the only method I will use to graph quadratic functions is
transformations
o remember when using transformations that whatever changes
happen OUT side the parentheses, do exactly what you see to the
OUT puts; whatever changes take place IN side the parentheses,
do the IN verse operation to the IN puts.
- to graph using transformations, I will use standard form of a
quadratic function to transform the parent function 𝑓
2
A parent function is the simplest function of a family of functions. For
quadratic functions, the simplest function is 𝑓
2
Example 1 : Graph the quadratic function 𝑔
2
− 3 by
transforming the parent function 𝑓
2
The quadratic function 𝑔 is already in standard form, so we don’t need to
change it at all to sketch its graph using transformations. I will simply
take the three points that are given from the graph of the parent function
2
, and
, and transform them.
Inputs, outputs, and ordered pairs
for the parent function 𝑓(𝑥) = 𝑥
2
Inputs Outputs
Ordered
Pairs
2
= 0 Vertex
Inputs
Outputs
𝑓(𝑥) = 𝑥
2
Inside the parentheses of the function 𝑔
2
− 3 we have
𝑥 − 1 , which indicates that we will take the inputs of the parent function 𝑓
and add 1 to them (inputs + 1 ). Remember that when changes take place
in side the parentheses of a function, we do the in verse operation to the
in puts.
Outside the parentheses of the function 𝑔
2
− 3 we have a
factor of 2 and a term of − 3. This indicates that we will take the outputs
of the parent function 𝑓, multiply them by 2 first, and then subtract 3
outputs
. Remember that when changes take place out side the
parentheses, we do exactly what we see to the out puts. Also remember
that order of operation says that we multiply/divide first, and add/subtract
second.
𝟐
(inputs + 1 , 2 (outputs) − 3 )
𝒈
( 𝒙
) = 𝟐
( 𝒙 − 𝟏
)
𝟐
− 𝟑
Old Vertex
New Vertex ( 1 , − 3 )
Transforming the points from the parent function 𝑓
2
to get the
new points for the function 𝑔
2
− 3 results in the graph on
the following page:
First LON-CAPA Problem:
𝟐
(inputs + 1 , 2 (outputs) − 3 )
Old Vertex
→ New Vertex
𝑓
( 𝑥
) = 𝑥
2
Outputs
Inputs
Example 2 : Graph the quadratic function 𝑗
2
transforming the parent function 𝑓
2
Since the quadratic function 𝑗 is in polynomial form, I will convert it to
standard form first before transforming the graph of the parent function
2
. To convert 𝑗
2
- 6 𝑥 − 7 to standard form, I will
start by finding its vertex.
2
Inputs, outputs, and ordered pairs
for the parent function 𝑓
2
Inputs Outputs
Ordered
Pairs
2
= 0 Vertex
Inputs
Outputs
𝑓
( 𝑥
) = 𝑥
2
Students who don’t like or don’t understand transformations may use other
methods such as making an input/output table and/or using intercepts.
However making an input/output table may require more work, and not
every quadratic function has 𝑥-intercepts, so using intercepts may not be a
viable option at all.
Next I will go through another problem from LON-CAPA, this time one
that is similar to Example 2.
𝑓(𝑥) = 𝑥
2
𝑗
( 𝑥
) = −𝑥
2
or
𝑗(𝑥) = −(𝑥 − 3 )
2
Outputs
Inputs
Second LON-CAPA Problem:
a.
𝟐
( inputs + 1 , 2
( outputs
) − 3
)
Old Vertex ( 0 , 0 ) → ( 0 + 1 , − 2 ( 0 ) − 3 ) → New Vertex
𝑓(𝑥) = 𝑥
2
Outputs
Inputs
(hint: on these two problems, find the vertex of each quadratic function first, then express
each quadratic function in standard form (𝑓(𝑥) = 𝑎(𝑥 − ℎ)
2
c. 𝑗
2
2
d.
𝑓
( 𝑥
) = 𝑥
2
𝑓
( 𝑥
) = 𝑥
2
Outputs Outputs
Inputs
Third LON-CAPA Problem:
𝟐
(inputs + 1 , 2 (outputs) − 3 )
Old Vertex
→ New Vertex
𝑓
( 𝑥
) = 𝑥
2
Outputs
Inputs
