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Lesson 5.4.3

Graphing y = nx 3. Lesson 5.4.3. Lesson 5.4.3. Graphing y = nx 3. California Standards: Algebra and Functions 3.1 Graph functions of the form y = nx 2 and y = nx 3 and use in solving problems . Mathematical Reasoning 2.3

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Lesson 5.4.3

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  1. Graphing y = nx3 Lesson 5.4.3

  2. Lesson 5.4.3 Graphing y = nx3 California Standards: Algebra and Functions 3.1 Graph functions of the form y = nx2and y = nx3and 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. y x ±4 ±3 ±2 ±1 0 y (= x2) 16 9 4 1 0 x 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 tableof values, then plotthe points.

  5. 60 40 20 0 x –4 –3 –2 –1 0 1 2 3 4 –4 –2 0 2 4 –20 y (= x3) –64 –27 –8 –1 0 1 8 27 64 –40 –60 Lesson 5.4.3 Graphing y = nx3 Example 1 Draw the graph of y = x3 for x between –4 and 4. y Solution First make a table of values: x Then plot the points on a graph. Solution continues… Solution follows…

  6. y 60 40 20 y = x2 0 –4 –2 0 2 4 x –20 –40 –60 Lesson 5.4.3 Graphing y = nx3 Example 1 Draw the graph of y = x3 for x between –4 and 4. 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.” 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 negativevalues of y.

  7. 60 40 20 0 –4 –2 0 2 4 –20 y = x3 –40 –60 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. y x

  8. 60 x –4 –3 –2 –1 –0.5 40 y –64 –27 –8 –1 –0.125 20 x 0 0.5 1 2 3 4 0 y 0 0.125 1 8 27 64 –4 –2 0 2 4 –20 –40 –60 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 x Solution follows…

  9. 60 40 20 x = 1 0 y = x2 –4 –2 0 2 4 –20 –40 –60 y = x3 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. x

  10. 60 x y (= x3) 50 0 0 40 0.5 0.125 30 1 1 20 2 8 10 3 27 0 4 64 0 1 2 3 4 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? y Solution 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. 60 50 40 30 y = x2 20 10 0 0 1 2 3 4 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? 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. x Solution continues…

  12. 1.5 y y 1 60 50 y = x2 0.5 40 y = x3 30 x y = x2 0 20 0 0.5 1 1.5 10 x 0 0 1 2 3 4 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? Solution (continued) y = x3 But if you could zoom in really close near the origin, you’d see that the graph of y = x3 is belowthe 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).

  13. 30 20 10 0 –4 –2 0 2 4 –10 –20 –30 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. y x x3 = 10 Þx» 2.2 y = x3

  14. Example 3 20 0 –4 –2 0 2 4 –40 –60 y = x3 Lesson 5.4.3 Graphing y = nx3 Use the graph in Example 1 to solve the equation x3 = –20. y x –2.7 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. 60 40 20 0 –4 –2 0 2 4 –20 –40 –60 y = x3 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 = 64 3.x3 = 1 4.x3 = –1 5.x3 = –27 6.x3 = 30 7.x3 = –50 y x = 4 x = 1 x = –1 x = –3 x x» 3.1 x» –3.7 Solution follows…

  16. 60 40 20 0 –4 –2 0 2 4 –20 –40 –60 y = x3 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 One — since the graph of y = x3 takes each value of y just once. x Solution follows…

  17. 60 40 20 0 –4 –2 0 2 4 –20 –40 –60 y = x3 Lesson 5.4.3 Graphing y = nx3 The Graph of y = nx3 is Stretched or Squashed The exact shape of the graph of y = nx3depends on the value of n. y Don’t forget — the value of n for the graph of y = x3 is one. x n = 1

  18. Example 4 x x3 2x3 3x3 ½x3 –3 –27 –54 –81 –13.5 Plot points to show how the graph of y = nx3 changes as n takes the values 1, 2, 3, and . –2 –8 –16 –24 –4 –1 –1 –2 –3 –0.5 0 0 0 0 0 1 1 1 2 3 0.5 2 2 8 16 24 4 3 27 54 81 13.5 Lesson 5.4.3 Graphing y = nx3 Solution 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. 80 60 x x3 2x3 3x3 ½x3 40 –3 –27 –54 –81 –13.5 20 Plot points to show how the graph of y = nx3 changes as n takes the values 1, 2, 3, and . –2 –8 –16 –24 –4 0 –1 –1 –2 –3 –0.5 –3 0 3 –20 0 0 0 0 0 1 –40 1 1 2 3 0.5 2 2 8 16 24 4 –60 3 27 54 81 13.5 –80 Lesson 5.4.3 Graphing y = nx3 Example 4 y y = 3x3 (n = 3) Solution (continued) y = 2x3 (n = 2) y = x3 (n = 1) y = ½x3 (n = ½) x Solution continues…

  20. 80 y 60 y = 3x3 (n = 3) 40 y = 2x3 (n = 2) 20 Plot points to show how the graph of y = nx3 changes as n takes the values 1, 2, 3, and . y = x3 (n = 1) 0 –3 0 3 y = ½x3 (n = ½) –20 x 1 –40 2 –60 –80 Lesson 5.4.3 Graphing y = nx3 Example 4 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. y = 3x3 80 60 y = 2x3 40 y = x3 20 y = ½x3 0 –3 0 3 –20 1 1 –40 2 2 –60 –80 Lesson 5.4.3 Graphing y = nx3 Guided Practice y 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 x» –2.7 x» 2.5 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. Example 5 1 2 Lesson 5.4.3 Graphing y = nx3 Plot points to show how the graph of y = nx3 changes as n takes the values –1, –2, –3, and – . 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. 80 60 x –x3 –2x3 –3x3 –½x3 40 –3 27 54 81 13.5 20 –2 8 16 24 4 0 –1 1 2 3 0.5 –3 0 3 –20 0 0 0 0 0 –40 1 –1 –2 –3 –0.5 2 –8 –16 –24 –4 –60 3 –27 –54 –81 –13.5 –80 Lesson 5.4.3 Graphing y = nx3 Example 5 y Solution x y = –½x3 (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. 80 60 40 20 0 –3 0 3 y = –½x3 –20 y = –x3 1 –40 2 y = –2x3 –60 –80 y = –3x3 Lesson 5.4.3 Graphing y = nx3 Guided Practice y Use the graphs shown to solve the equations in Exercises 16–18. 16. –3x3 = –50 17. –3x3 = 50 18.– x3 = 10 x» 2.6 x x» –2.6 x» –2.7 Solution follows…

  26. 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 80 60 y = – x3 y = 1.5x3 40 20 0 –3 0 3 –20 1 1 –40 3 3 –60 –80 y = –4x3 Lesson 5.4.3 Graphing y = nx3 Independent Practice y x Solution follows…

  27. 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.

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