Algebra 2/Trig
5.1 Graphing Cubic Functions
Name _____________________
Graphing Cubic Functions
Complete the table, graph the ordered pairs and then pass a smooth curve through
the plotted
y
3
points to obtain the graph of 𝑓(𝑥) = 𝑥
8
𝑥
−2
−1
0
1
2
𝑓(𝑥) = 𝑥
7
6
5
4
3
2
1
3
Use the graph to analyze the function and complete the table
–8 –7 –6 –5 –4 –3 –2 –1
–1
–2
–3
–4
–5
–6
–7
–8
1 2 3 4 5 6 7 8
x
Domain
Range
End Behavior
Zeros
Where the function has positive values
Where the function has negative values
Where the function is increasing
Where the function is decreasing
Is the function even (𝑓(𝑥) = 𝑓(−𝑥)), odd
(𝑓(−𝑥) = −𝑓(𝑥)) or neither?
Graphing Transformations of 𝑓(𝑥) = 𝑥 3
𝑓(𝑥) = 𝑎(𝑥 − ℎ)3 + 𝑘
“Starting point”: (ℎ, 𝑘) (it goes through this point horizontally for a bit)
If 𝑎 > 0, it goes up from left to right
If 𝑎 < 0, it goes down from left to right
To find the zeros, let 𝑓(𝑥) = 0 and solve
for 𝑥
The graph is increasing when it is going up
from left to right
The graph is positive when it is above the xaxis
The graph is decreasing when it is going
down from left to right
The graph is negative when it is below the
x-axis
Algebra 2/Trig
5.1 Graphing Cubic Functions
Name _____________________
Ex: Graph 𝒇(𝒙) = 𝟐(𝒙 − 𝟏)𝟑 + 𝟐. Determine where the graph is positive and negative.
y
Starting point: (1, 2)
5
4
𝑎 > 0 so it goes up from left to right
3
2
1
The graph is positive for all values of 𝑥 > 0
–5
The graph is negative for all values of 𝑥 < 0
–4
–3
–2
–1
–1
1
2
3
4
5
x
–2
–3
–4
–5
y
8
7
6
5
4
3
2
1
Ex: Graph 𝒇(𝒙) = −(𝒙 + 𝟐)𝟑 + 𝟏 and find the end behavior
Starting point:
a ____ 0 so it goes ________ from left to right
End behavior: 𝑎𝑠 𝑥 → −∞, 𝑦 → ___________
𝑎𝑠 𝑥 → +∞, 𝑦 → ___________
–8 –7 –6 –5 –4 –3 –2 –1
–1
–2
–3
–4
–5
–6
–7
–8
1 2 3 4 5 6 7 8
y
𝟏
Ex: Graph 𝒇(𝒙) = 𝟐 (𝒙 − 𝟑)𝟑 and find the zero
Starting point:
a ____ 0 so it goes ________ from left to right
Zero:
8
7
6
5
4
3
2
1
–8 –7 –6 –5 –4 –3 –2 –1
–1
–2
–3
–4
–5
–6
–7
–8
1 2 3 4 5 6 7 8
x
x
Algebra 2/Trig
5.1 Graphing Cubic Functions
Name _____________________
Ex: Graph 𝒇(𝒙) = −𝟑𝒙𝟑 + 𝟑. Find the zeros, end behavior, and where the graph is
increasing/decreasing
Starting point:
a ____ 0 so it goes ________ from left to right
Zeros:
End Behavior:
Increasing:
Decreasing:
y
8
7
6
5
4
3
2
1
–8 –7 –6 –5 –4 –3 –2 –1
–1
–2
–3
–4
–5
–6
–7
–8
1 2 3 4 5 6 7 8
x
Ex: Graph 𝒇(𝒙) = 𝟒(𝒙 − 𝟏)𝟑 + 𝟒. Find the zeros, end behavior, and where the graph is
increasing/decreasing and positive/negative.
y
Starting point:
a ____ 0 so it goes ________ from left to right
Zeros:
End Behavior:
Increasing:
Decreasing:
Positive:
Negative:
8
7
6
5
4
3
2
1
–8 –7 –6 –5 –4 –3 –2 –1
–1
–2
–3
–4
–5
–6
–7
–8
1 2 3 4 5 6 7 8
x
Algebra 2/Trig
5.1 Graphing Cubic Functions
Name _____________________
Classwork
𝟏
3. Graph 𝒇(𝒙) = −𝟐(𝒙 + 𝟏)𝟑 − 𝟐 and
find the zeros, and where the graph
is increasing and decreasing.
Zeros:
Increasing:
Decreasing:
1. Graph 𝒇(𝒙) = 𝟐 (𝒙 − 𝟑)𝟑 and find
the end behavior
End behavior:
y
8
7
6
5
4
3
2
1
y
–8 –7 –6 –5 –4 –3 –2 –1
–1
–2
–3
–4
–5
–6
–7
–8
1 2 3 4 5 6 7 8
8
7
6
5
4
3
2
1
x
–8 –7 –6 –5 –4 –3 –2 –1
–1
–2
–3
–4
–5
–6
–7
–8
2. Graph 𝒇(𝒙) = (𝒙 − 𝟐)𝟑 + 𝟏 and
find the zeros
Zeros:
x
4. Graph 𝒇(𝒙) = 𝟒𝒙𝟑 − 𝟒 and find the
zeros, and where the graph is
positive and negative
Zeros:
Positive:
Negative:
y
8
7
6
5
4
3
2
1
–8 –7 –6 –5 –4 –3 –2 –1
–1
–2
–3
–4
–5
–6
–7
–8
1 2 3 4 5 6 7 8
y
1 2 3 4 5 6 7 8
x
8
7
6
5
4
3
2
1
–8 –7 –6 –5 –4 –3 –2 –1
–1
–2
–3
–4
–5
–6
–7
–8
1 2 3 4 5 6 7 8
x