Lesson 5.4.2

1. More Graphs of y = nx2 Lesson 5.4.2

2. Lesson 5.4.2 More Graphs of y = nx2 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 more about how to plot graphs of equations with squared variables in them, and how to use the graphs to solve equations. • Key words: • graph • vertex

3. y x Lesson 5.4.2 More Graphs of y = nx2 In the last Lesson you saw a lot of bucket-shaped graphs. These were all graphs of equations of the form y = nx2, where n was positive. The obvious next thing to think about is what happens when n is negative.

4. Lesson 5.4.2 More Graphs of y = nx2 The Graph of y = nx2 is Still a Parabola if n is Negative By plotting points, you can draw the graph of y = –x2. Don’t forget — y = –x2is justy = nx2 with n = –1, so like all y = nx2 graphs, it will be a parabola.

5. Example 1 x –5 –4 –3 –2 –1 0 25 16 9 4 1 0 x2 –25 –16 –9 –4 –1 0 y = –x2 Lesson 5.4.2 More Graphs of y = nx2 Plot the graph of y = –x2 for values of x from –5 to 5. Solution As always, first make a table of values, then plot the points. You don’t need a table for x = 1, 2, 3, 4, and 5, as it will contain the samevalues of y as above. Solution follows… Solution continues…

6. Example 1 1 2 3 4 5 1 4 9 16 25 x –5 –4 –3 –2 –1 0 –1 –4 –9 –16 –25 25 16 9 4 1 0 x2 –25 –16 –9 –4 –1 0 y = –x2 Lesson 5.4.2 More Graphs of y = nx2 Plot the graph of y = –x2 for values of x from –5 to 5. Solution (continued) However, if you find it easier to have all the values of x listed separately, then make a bigger table like the one below. Solution continues…

7. Example 1 x –5 –4 –3 –2 –1 0 1 2 3 4 5 25 16 9 4 1 0 1 4 9 16 25 x2 –25 –16 –9 –4 –1 0 –1 –4 –9 –16 –25 y = –x2 –4 –2 0 2 4 0 –10 –20 –30 Lesson 5.4.2 More Graphs of y = nx2 Plot the graph of y = –x2 for values of x from –5 to 5. Solution (continued) Now you can plot the points. x The graph of y = –x2 is also a parabola. But instead of being “u‑shaped,” it’s “upside down u-shaped.” y

8. Lesson 5.4.2 More Graphs of y = nx2 Nearly everything from the last Lesson about y = nx2 for positive values of n also applies for negative values of n. However, for negative values of n, the graphs are belowthe x-axis.

9. Plot the graphs of the following equations for values of x between –4 and 4.a)y = –2x2b)y = –3x2c)y = –4x2d)y = – x2 x –2x2 –3x2 –4x2 –½x2 0 0 0 0 0 1 –2 –3 –4 –0.5 2 –8 –12 –16 –2 1 3 –18 –27 –36 –4.5 2 4 –32 –48 –64 –8 Lesson 5.4.2 More Graphs of y = nx2 Example 2 Solution As always, make a table and plot the points. Solution continues… Solution follows…

10. x –4x2 0 0 x –2x2 y = – x2 1 –4 0 0 (n = –½) 2 –16 1 –2 3 –36 2 –8 Decreasing values of n 4 –64 (n = –2) 3 –18 4 –32 1 (n = –3) 2 x –½x2 x –3x2 0 0 0 0 (n = –4) 1 –0.5 1 –3 2 –2 2 –12 3 –4.5 3 –27 4 –8 4 –48 Lesson 5.4.2 More Graphs of y = nx2 Example 2 Solution –5 –4 –3 –2 –1 0 1 2 3 4 5 x 0 –20 –40 y = –2x2 –60 y = –3x2 –80 y = –4x2 –100 y

11. –5 –4 –3 –2 –1 0 1 2 3 4 5 0 –20 –40 –60 –80 –100 Lesson 5.4.2 More Graphs of y = nx2 This time, since n is negative, all the graphs are “upside down u-shaped” parabolas. But all the graphs still have their vertex (the vertex is the highest point this time) at the same place, the origin. Also, the more negative the value of n, the steeper and narrower the parabola will be.

12. –5 –4 –3 –2 –1 0 1 2 3 4 5 0 y = – x2 y = – x2 –20 –40 y = –2x2 1 1 –60 2 2 y = –3x2 –80 y = –4x2 –100 Lesson 5.4.2 More Graphs of y = nx2 Guided Practice On which of the graphs in Example 2 do the points in Exercises 1–4 lie? Choose from y = –x2, y = –2x2, y = –3x2, and y = ½ x2. 1. (1, –3) 2. (–3, –4.5) 3. (4, –32) 4. (–5, –75) y = –3x2 y = –2x2 y = –3x2 Solution follows…

13. –5 –4 –3 –2 –1 0 1 2 3 4 5 0 y = – x2 –20 –40 y = –2x2 1 1 –60 y = – x2 2 2 y = –3x2 –80 y = –4x2 –100 Lesson 5.4.2 More Graphs of y = nx2 Guided Practice On which of the graphs in Example 2 do the points in Exercises 5–8 lie? Choose from y = –x2, y = –2x2, y = –3x2, and y = ½x2. 5. (–3, –27) 6. (2, –2) 7. (5, –75) 8. (0, 0) y = –3x2 y = –3x2 all Solution follows…

14. –5 –4 –3 –2 –1 0 1 2 3 4 5 0 y = – x2 –20 –40 y = –2x2 1 1 –60 2 2 y = –3x2 –80 y = –4x2 –100 Lesson 5.4.2 More Graphs of y = nx2 Guided Practice Solve the equations in Exercises 9–11 using the graphs in Example 2. There are two possible answers in each case. 9. –3x2 = –12 10. – x2 = –4.5 11. –2x2 = –32 x x = 2 or x = –2 x = 3 or x = –3 x = 4 or x = –4 y Solution follows…

15. –5 –4 –3 –2 –1 0 1 2 3 4 5 0 y = – x2 –20 –40 y = –2x2 1 1 –60 2 2 y = –3x2 –80 y = –4x2 –100 Lesson 5.4.2 More Graphs of y = nx2 Guided Practice Solve the equations in Exercises 12–14 using the graphs in Example 2. There are two possible answers in each case. 12. – x2 = –2 13. –3x2 = –27 14. –3x2 = –40 x x = 2 or x = –2 x = 3 or x = –3 x = 3.7 or x = –3.7 y Solution follows…

16. –4 –3 –2 –1 0 1 2 3 4 0 –20 y = – x2 –40 –60 1 16 1 1 1 4 x ±4 ±3 ±2 ±1 0 3 3 3 3 3 3 – x2 – –3 – – 0 x ±4 ±3 ±2 ±1 0 y = –5x2 –80 –5x2 –80 –45 –20 –5 0 –100 Lesson 5.4.2 More Graphs of y = nx2 Guided Practice Plot the graphs in Exercises 15–16 for x between –4 and 4. 15.y = –5x2 16. y = – x2 x y Solution follows…

17. Lesson 5.4.2 More Graphs of y = nx2 Graphs of y = nx2 for n > 0 and n < 0 Are Reflections The graphs you’ve seen in this Lesson (of y = nx2 for negativen) and those you saw in the previous Lesson (of y = nx2 for positiven) are very closely related. negativen positiven

18. y = 3x2 30 20 y = 2x2 10 y = x2 0 –3 –2 –1 0 1 2 3 y = –x2 –10 y = –2x2 –20 –30 y = –3x2 Lesson 5.4.2 More Graphs of y = nx2 Example 3 y By plotting the graphs of the following equations on the same set of axes for x between –3 and 3, describe the link between y = kx2 and y = –kx2.y = x2, y = –x2, y = 2x2, y = –2x2, y = 3x2, y = –3x2. x Solution Plotting the graphs gives the diagram shown on the right. For a given value of k, the graphs of y = kx2and y = –kx2 are reflectionsof each other. One is a “u‑shaped” graph above the x-axis, while the other is an “upside down u‑shaped” graph below the x-axis. Solution follows…

19. Lesson 5.4.2 More Graphs of y = nx2 Guided Practice 17. The point (5, 100) lies on the graph of y = 4x2. Without doing any calculations, state the y-coordinate of the point on the graph of y = –4x2 with x-coordinate 5. 18. Without plotting any points, describe what the graphs of the equations y = 100x2 and y = –100x2 would look like. –100 The first is a steep u‑shaped parabola above the x-axis with its vertex at (0, 0). The second is a reflection of this across the x-axis. Solution follows…

20. 30 20 y = 1.5x2 10 0 –3 –2 –1 0 1 2 3 –10 y = –1.5x2 –20 –30 Lesson 5.4.2 More Graphs of y = nx2 Independent Practice y 1. Draw the graph of y = –1.5x2 for values of x between –3 and 3. 2. Without calculating any further y-values, draw the graph of y = 1.5x2for values of x between –3 and 3. x Solution follows…

21. 1 4 Lesson 5.4.2 More Graphs of y = nx2 Independent Practice 3. What are the coordinates of the vertex of the graph of y = – x2? 4. If a circle has radius r, its area A is given by A = pr2. Describe what a graph of A against r would look like.Check your answer by plotting points for r = 1, 2, 3, and 4. (0, 0) Half a u‑shaped parabola above the x-axis with its vertex at (0, 0). Solution follows…

22. Lesson 5.4.2 More Graphs of y = nx2 Round Up Well, there were lots of pretty graphs to look at in this Lesson. The graphs of y = nx2 are important in math, and you’ll meet them again next year. But next Lesson, it’s something similar... but different.