1. Graphing y = nx3
Lesson 5.4.3

2. Graphing y = nx3 5.4.3 California Standards: What it means for you:
Lesson
5.4.3
Graphing y = nx3
California Standards:
Algebra and Functions 3.1
Graph functions of the form y = nx2 and y = nx3 and use in solving problems.
Mathematical Reasoning 2.3
Estimate unknown quantities graphically and solve for them by using logical reasoning and arithmetic and algebraic techniques.
What it means for you:
You’ll learn about how to plot graphs of equations with cubed variables in them, and how to use the graphs to solve equations.
Key words:
parabola
plot
graph

3. Lesson
5.4.3
Graphing y = nx3
For the last two Lessons, you’ve been drawing graphs of y = nx2. y
y = nx2 x
Graphs of y = nx3 are very different, but the method for actually drawing the graphs is exactly the same.

4. Graphing y = nx3 5.4.3 The Graph of y = x3 is Not a Parabola
Lesson
5.4.3
Graphing y = nx3
The Graph of y = x3 is Not a Parabola
You can always draw a graph of an equation by plotting points in the normal way.
First make a table of values, then plot the points. x y ±1 ±2 ±3 ±4 x
y (= x2) 1 4 9 16

5. Lesson
5.4.3
Graphing y = nx3
Example 1
Draw the graph of y = x3 for x between –4 and 4. y –4 –2 2 4 60 40 20
–20
–40
–60
Solution
First make a table of values: –1 –2 –3 –4 x 1 2
y (= x3) –8
–27
–64 8 3 4 27 64 x
Then plot the points on a graph.
Solution continues…
Solution follows…

6. Lesson
5.4.3
Graphing y = nx3
Example 1
Draw the graph of y = x3 for x between –4 and 4. –4 –2 2 4 60 40 20
–20
–40
–60 x y
Solution (continued)
The graph of y = x3 is completely different from the graph of y = x2. It isn’t “u-shaped” or “upside down u-shaped.”
y = x2
The graph still goes steeply upward as x gets more positive, but it goes steeply downward as x gets more negative.
The graph of y = x3 passes through all positive and negative values of y.

7. Lesson
5.4.3
Graphing y = nx3
The shape of the graph of y = x3 is not a parabola — it is a curve that rises very quickly after x = 1, and falls very quickly below x = –1. –4 –2 2 4 60 40 20
–20
–40
–60
y = x3 y x

8. Graphing y = nx3 5.4.3 Guided Practice
Lesson
5.4.3
Graphing y = nx3
Guided Practice
1. Draw the graph of y = –x3 by plotting points with x-coordinates –4, –3, –2, –1, –0.5, 0, 0.5, 1, 2, 3, and 4. y –1 –2 –3 –4 x
–0.5 4 y –8
–27
–64
–0.125 64 2 1
0.5 3 8
0.125 27 –4 –2 2 4 60 40 20
–20
–40
–60 x
Solution follows…

9. Graphing y = nx3 5.4.3 The Graph of y = x3 Crosses the Graph of y = x2
Lesson
5.4.3
Graphing y = nx3
The Graph of y = x3 Crosses the Graph of y = x2 y
If you look really closely at the graphs of y = x3 and y = x2 you’ll see that they cross over when x = 1. –4 –2 2 4 60 40
–20
–40
–60
x = 1
y = x3
y = x2 20 x

10. Lesson
5.4.3
Graphing y = nx3
Example 2
Draw the graph of y = x3 for x values between 0 and 4. Plot the points with x-values 0, 0.5, 1, 2, 3, and 4.
How does the curve of y = x3 differ from that of y = x2? 1 2 3 4 10 20 30 40 50 60 y
Solution 2 1
0.5 x 3 4
y (= x3) 8
0.125 27 64
y = x3
Plotting the points with the coordinates shown in the table gives you the graph on the right. x
Solution continues…
Solution follows…

11. Lesson
5.4.3
Graphing y = nx3
Example 2
Draw the graph of y = x3 for x values between 0 and 4. Plot the points with x-values 0, 0.5, 1, 2, 3, and 4.
How does the curve of y = x3 differ from that of y = x2? 1 2 3 4 10 20 30 40 50 60 y
Solution (continued)
y = x3
You can see that the graph of y = x3 rises much more steeply as x increases than the graph of y = x2 does.
y = x2 x
Solution continues…

12. Lesson
5.4.3
Graphing y = nx3
Example 2
Draw the graph of y = x3 for x values between 0 and 4. Plot the points with x-values 0, 0.5, 1, 2, 3, and 4.
How does the curve of y = x3 differ from that of y = x2? 1 2 3 4 10 20 30 40 50 60 x y
Solution (continued) x y
0.5 1
y = x2
y = x3
1.5
y = x3
But if you could zoom in really close near the origin, you’d see that the graph of y = x3 is below the graph of y = x2 between x = 0 and x = 1.
y = x2
The two graphs cross over at the point (1, 1), and cross again at (0, 0).

13. Graphing y = nx3 5.4.3 Use the Graphs of y = x3 to Solve Equations
Lesson
5.4.3
Graphing y = nx3
Use the Graphs of y = x3 to Solve Equations
If you have an equation like x3 = 10, you can solve it using a graph of y = x3. –4 –2 2 4 30 20 10
–10
–20
–30 y x
x3 = 10 Þ x » 2.2
y = x3

14. Lesson
5.4.3
Graphing y = nx3
Example 3
Use the graph in Example 1 to solve the equation x3 = –20. –4 –2 2 4 20
–40
–60
y = x3 y
–2.7 x
Solution
–20
–20
First find –20 on the vertical axis.
Then find the corresponding value on the horizontal axis — this is the solution to the equation.
So x = –2.7 (approximately).
Solution follows…

15. Graphing y = nx3 5.4.3 Guided Practice
Lesson
5.4.3
Graphing y = nx3
Guided Practice
Use the graph of y = x3 to solve the equations in Exercises 2–7.
2. x3 = x3 = 1
4. x3 = –1 5. x3 = –27
6. x3 = x3 = –50 y –4 –2 2 4 60 40 20
–20
–40
–60
y = x3
x = 4
x = 1
x = –1
x = –3 x
x » 3.1
x » –3.7
Solution follows…

16. Graphing y = nx3 5.4.3 Guided Practice
Lesson
5.4.3
Graphing y = nx3
Guided Practice
8. How many solutions are there to an equation of the form x3 = k? Use the graph in Example 1 to justify your answer. y –4 –2 2 4 60 40 20
–20
–40
–60
y = x3
One — since the graph of y = x3 takes each value of y just once. x
Solution follows…

17. Graphing y = nx3 5.4.3 The Graph of y = nx3 is Stretched or Squashed
Lesson
5.4.3
Graphing y = nx3
The Graph of y = nx3 is Stretched or Squashed
The exact shape of the graph of y = nx3 depends on the value of n. y –4 –2 2 4 60 40 20
–20
–40
–60
y = x3
Don’t forget — the value of n for the graph of y = x3 is one. x
n = 1

18. Lesson
5.4.3
Graphing y = nx3
Example 4
Plot points to show how the graph of y = nx3 changes as n takes the values 1, 2, 3, and . 1 2
Solution x x3 –3 –2 –1 1 2
–27 –8 8
2x3
–54
–16 16
3x3
–81
–24 3 24
½ x3
–13.5 –4
–0.5
0.5 4 27 54 81
13.5
Using values of x between –3 and 3 should be enough for any patterns to emerge.
So make a suitable table of values, then plot the points.
Solution continues…
Solution follows…

19. Lesson
5.4.3
Graphing y = nx3
Example 4
Plot points to show how the graph of y = nx3 changes as n takes the values 1, 2, 3, and . 1 2 3 20 40 60 80
–80
–60
–40
–20 –3 y
y = 3x3
(n = 3)
Solution (continued)
y = 2x3
(n = 2) x x3 –3 –2 –1 1 2
–27 –8 8
2x3
–54
–16 16
3x3
–81
–24 3 24
½ x3
–13.5 –4
–0.5
0.5 4 27 54 81
13.5
y = x3
(n = 1)
y = ½ x3
(n = ½) x
Solution continues…

20. Lesson
5.4.3
Graphing y = nx3
Example 4
Plot points to show how the graph of y = nx3 changes as n takes the values 1, 2, 3, and . 1 2 3 20 40 60 80
–80
–60
–40
–20 –3
(n = 1)
(n = 3)
(n = 2)
(n = ½)
y = 3x3
y = 2x3
y = x3
y = ½ x3 x y
Solution (continued)
As n increases, the curves get steeper and steeper.
However, the basic shape remains the same.
All the curves have rotational symmetry about the origin.

21. Graphing y = nx3 5.4.3 Guided Practice
Lesson
5.4.3
Graphing y = nx3
Guided Practice 3 20 40 60 80
–80
–60
–40
–20 –3 y
Use the graphs shown to solve the equations in Exercises 9–14.
9. 3x3 = – x3 = 30
x3 = – x3 = 10
13. 3x3 = x3 = –35
y = 3x3
y = 2x3
x » –2.7
x » 2.5
y = x3 1 2 1 2
y = ½ x3 x
x » –2.7
x » 2.7
x » 2.4
x » –2.6
15. How many solutions are there to an equation of the form nx3 = k, where n and k are positive?
one
Solution follows…

22. Lesson
5.4.3
Graphing y = nx3
For n < 0, the Graph of y = nx3 is Flipped Vertically
If n is negative, the graph of y = nx3 is “upside down.” y y x x
y = x3
y = – x3
n is positive
n is negative

23. Lesson
5.4.3
Graphing y = nx3
Example 5
Plot points to show how the graph of y = nx3 changes as n takes the values –1, –2, –3, and – . 1 2
Solution
The table of values looks very similar to the one in Example 4. The only difference is that all the numbers switch sign — so all the positive numbers become negative, and vice versa.
Solution continues…
Solution follows…

24. Graphing y = nx3 5.4.3 Solution
Lesson
5.4.3
Graphing y = nx3
Example 5 3 20 40 60 80
–80
–60
–40
–20 –3 y
Solution x
–x3 –3 –2 –1 1 2 27 8 –8
–2x3 54 16
–16
–3x3 81 24 3
–24
–½ x3
13.5 4
0.5
–0.5 –4
–27
–54
–81
–13.5 x
y = –½ x3
(n = –½)
y = –x3
(n = –1)
y = –2x3
(n = –2)
This change in sign of all the values means the curves all do a “vertical flip.”
y = –3x3
(n = –3)

25. Graphing y = nx3 5.4.3 Guided Practice
Lesson
5.4.3
Graphing y = nx3
Guided Practice 3 20 40 60 80
–80
–60
–40
–20 –3 y
Use the graphs shown to solve the equations in Exercises 16–18.
16. –3x3 = –50
17. –3x3 = 50
18. – x3 = 10
y = –3x3
y = –2x3
x » 2.6
y = –x3
y = –½ x3 x
x » –2.6 1 2
x » –2.7
Solution follows…

26. Graphing y = nx3 5.4.3 Independent Practice
Lesson
5.4.3
Graphing y = nx3
Independent Practice 3 20 40 60 80
–80
–60
–40
–20 –3
Using a table of values, plot the graphs of the equations in Exercises 1–3 for values of x between –3 and 3.
1. y = 1.5x3
2. y = –4x3
3. y = – x3 1 3
y = –4x3 y
y = 1.5x3
y = – x3 1 3 x
Solution follows…

27. Graphing y = nx3 5.4.3 Independent Practice
Lesson
5.4.3
Graphing y = nx3
Independent Practice
4. If the graph of y = 8x3 goes through the point (6, 1728), what are the coordinates of the point on the graph of y = –8x3 with x-coordinate 6?
(6, –1728)
Solution follows…

28. Lesson
5.4.3
Graphing y = nx3
Round Up
That’s the end of this Section, and with it, the end of this Chapter. It’s all useful information.
You need to remember the general shapes of the graphs, and how they change when the n changes.