lesson 5.4.3

1

Lesson 5.4.3Lesson 5.4.3

Graphing y = nx3Graphing y = nx3

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Lesson

5.4.3Graphing y = nx3Graphing y = nx3

California Standards:Algebra and Functions 3.1Graph functions of the form y = nx2 and y = nx3 and use in solving problems.

Mathematical Reasoning 2.3Estimate 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

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Graphing y = nx3Graphing y = nx3Lesson

5.4.3

For the last two Lessons, you’ve been drawing graphs of y = nx2.

Graphs of y = nx3 are very different, but the method for actually drawing the graphs is exactly the same.

y = nx2

x

y

4


The Graph of y = x3 is Not a Parabola

Lesson

5.4.3

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.

±1±2±3±4x 0

y (= x2) 14916 0

x

y

5


Example 1

Solution follows…

Lesson

5.4.3

Draw the graph of y = x3 for x between –4 and 4.

Solution

First make a table of values:

Then plot the points on a graph.

–4 –2 0 2 4

60

40

20

0

–20

–40

–60

–1–2–3–4x 0 1 2

y (= x3) –1–8–27–64 0 1 8

3 4

27 64

Solution continues…

x

y

6

The graph of y = x3 is completely different from the graph of y = x2. It isn’t “u-shaped” or “upside down u-shaped.”

The graph of y = x3 passes through all positive and negative values of y.


Example 1

Lesson

5.4.3

Draw the graph of y = x3 for x between –4 and 4.

Solution (continued)

The graph still goes steeply upward as x gets more positive, but it goes steeply downward as x gets more negative.

–4 –2 0 2 4

60

40

20

0

–20

–40

–60

x

y

y = x2

7


5.4.3

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 0 2 4

60

40

20

0

–20

–40

–60y = x3

x

y

8


Guided Practice

Solution follows…

Lesson

5.4.3

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.

–4 –2 0 2 4

60

40

20

0

–20

–40

–60

x

y–1–2–3–4x –0.5

4

y –1–8–27–64 –0.125

64

210.50x 3

y 810.1250 27

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The Graph of y = x3 Crosses the Graph of y = x2

Lesson

5.4.3

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 0 2 4

60

40

0

–20

–40

–60

x = 1

y = x3

y = x2

20

x

y

10


Example 2

Solution follows…

Lesson

5.4.3

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?

Solution

Plotting the points with the coordinates shown in the table gives you the graph on the right. 2

1

0.5

0

x

3

4

y (= x3)

8

1

0.125

0

27

640 1 2 3 4

0

10

20

30

40

50

60 y = x3

x

y


11Solution continues…


Example 2

Lesson

5.4.3



0 1 2 3 40

10

20

30

40

50

60 y = x3

x

y

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

12


Example 2

Lesson

5.4.3


Solution (continued)y = x3

0 1 2 3 40

10

20

30

40

50

60

x

y

y = x2

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.

The two graphs cross over at the point (1, 1), and cross again at (0, 0).

x

y

00 0.5 1

0.5

1

y = x2

y = x3

1.5

1.5

13


5.4.3

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 0 2 4

30

20

10

0

–10

–20

–30

y = x3

x

y

x3 = 10 x 2.2

14

–4 –2 0 2 4

20

0

–40

–60y = x3


Example 3

Solution follows…

Lesson

5.4.3

Use the graph in Example 1 to solve the equation x3 = –20.

Solution

Then find the corresponding value on the horizontal axis — this is the solution to the equation.

First find –20 on the vertical axis.

So x = –2.7 (approximately).

–20 –20

–2.7 x

y

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Guided Practice

Solution follows…

Lesson

5.4.3

Use the graph of y = x3 to solve the equations in Exercises 2–7.

2. x3 = 64 3. x3 = 1

4. x3 = –1 5. x3 = –27

6. x3 = 30 7. x3 = –50–4 –2 0 2 4

60

40

20

0

–20

–40

–60y = x3

x = 4 x = 1

x = –1 x = –3

x 3.1 x –3.7

x

y

16


Guided Practice

Solution follows…

Lesson

5.4.3

8. How many solutions are there to an equation of the form x3 = k? Use the graph in Example 1 to justify your answer.

One — since the graph of y = x3 takes each value of y just once.

–4 –2 0 2 4

60

40

20

0

–20

–40

–60y = x3

x

y

17


The Graph of y = nx3 is Stretched or Squashed

Lesson

5.4.3

The exact shape of the graph of y = nx3 depends on the value of n.

–4 –2 0 2 4

60

40

20

0

–20

–40

–60y = x3

x

y

n = 1

Don’t forget — the value of n for the graph of y = x3 is one.

18Solution continues…


5.4.3

Example 4

Solution follows…

Plot points to show how the graph of y = nx3 changes

as n takes the values 1, 2, 3, and . 1

2Solution

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.

x x3

–3

–2

–1

0

1

2

–27

–8

–1

0

1

8

2x3

–54

–16

–2

0

2

16

3x3

–81

–24

–3

0

3

24

½ x3

–13.5

–4

–0.5

0

0.5

4

3 27 54 81 13.5

19



2

0 30

20

40

60

80

–80

–60

–40

–20–3


Example 4

Lesson

5.4.3


(n = 1)

(n = 3)

(n = 2)

(n = ½)

y = 3x3

y = 2x3

y = x3

y = ½ x3


x

y

x x3

–3

–2

–1

0

1

2

–27

–8

–1

0

1

8

2x3

–54

–16

–2

0

2

16

3x3

–81

–24

–3

0

3

24

½ x3

–13.5

–4

–0.5

0

0.5

4

3 27 54 81 13.5

20



2

All the curves have rotational symmetry about the origin.


Example 4

Lesson

5.4.3


As n increases, the curves get steeper and steeper.

However, the basic shape remains the same.

0 30

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

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Guided Practice

Solution follows…

Lesson

5.4.3

Use the graphs shown to solve the equations in Exercises 9–14.

9. 3x3 = –60 10. 2x3 = 30

11. x3 = –10 12. x3 = 10

13. 3x3 = 40 14. 2x3 = –35

0 30

20

40

60

80

–80

–60

–40

–20–3

y = 3x3

y = 2x3

y = x3

y = ½ x3

x –2.7 x 2.5

x –2.7 x 2.7

x 2.4 x –2.6

1

2

1

2

15. How many solutions are there to an equation of the form nx3 = k, where n and k are positive? one

x

y

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For n < 0, the Graph of y = nx3 is Flipped Vertically

Lesson

5.4.3

If n is negative, the graph of y = nx3 is “upside down.”

y = x3

x

y

y = – x3

x

y

n is positive n is negative

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5.4.3

Example 5

Solution follows…


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.


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This change in sign of all the values means the curves all do a “vertical flip.”

0 30

20

40

60

80

–80

–60

–40

–20–3


Example 5

Lesson

5.4.3

Solution

(n = –1)

(n = –3)

(n = –2)

(n = –½)

x –x3

–3

–2

–1

0

1

2

27

8

1

0

–1

–8

–2x3

54

16

2

0

–2

–16

–3x3

81

24

3

0

–3

–24

–½ x3

13.5

4

0.5

0

–0.5

–4

3 –27 –54 –81 –13.5

y = –3x3

y = –2x3

y = –x3

y = –½ x3

x

y

25


Guided Practice

Solution follows…

Lesson

5.4.3

Use the graphs shown to solve the equations in Exercises 16–18.

16. –3x3 = –50

17. –3x3 = 50

18. – x3 = 10

0 30

20

40

60

80

–80

–60

–40

–20–3

y = –3x3

y = –2x3

y = –x3

y = –½ x3

x 2.6

x –2.6

x –2.7

1

2

x

y

26


Independent Practice

Solution follows…

Lesson

5.4.3

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

0 30

20

40

60

80

–80

–60

–40

–20–3

y = 1.5x3

y = –4x3

y = – x31

3

x

y

27


Independent Practice

Solution follows…

Lesson

5.4.3

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)

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Round UpRound Up

Lesson

5.4.3

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.

lesson 5.4.3

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