1. More Graphs of y = nx2
Lesson 5.4.2

2. More Graphs of y = nx2 5.4.2 California Standards:
Lesson
5.4.2
More Graphs of y = nx2
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 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. Lesson
5.4.2
More Graphs of y = nx2 x y
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 = –x2 is just y = nx2 with n = –1, so like all y = nx2 graphs, it will be a parabola.

5. Lesson
5.4.2
More Graphs of y = nx2
Example 1
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. –2 –3 –4 –5 x –1 x2 4 9 16 25 1
y = –x2 –9
–16
–25
You don’t need a table for x = 1, 2, 3, 4, and 5, as it will contain the same values of y as above.
Solution continues…
Solution follows…

6. Lesson
5.4.2
More Graphs of y = nx2
Example 1
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. –2 –3 –4 –5 x –1 x2 4 9 16 25 1
y = –x2 –9
–16
–25 3 2 5
Solution continues…

7. Lesson
5.4.2
More Graphs of y = nx2
Example 1
Plot the graph of y = –x2 for values of x from –5 to 5.
Solution (continued) –2 –3 –4 –5 x –1 x2 4 9 16 25 1
y = –x2 –9
–16
–25 3 2 5
Now you can plot the points. 2 4
–10
–20
–30 –2 –4 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 below the x-axis.

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

10. More Graphs of y = nx2 5.4.2 Solution Lesson Example 2 –5 –4 –3 –2 –11 2 3 4 5 x
Decreasing values of n
(n = –4)
(n = –3)
(n = –2)
(n = –½) x 1 2 3 4
–3x2 –3
–12
–27
–48 x
–2x2 1 2 3 4 –2 –8
–18
–32 x 1 2 3 4
–4x2 –4
–16
–36
–64 x 1 2 3 4
–½ x2
–0.5 –2
–4.5 –8
y = – x2 1 2
–20
–40
y = –2x2
–60
y = –3x2
–80
–100
y = –4x2 y

11. Lesson
5.4.2
More Graphs of y = nx2
This time, since n is negative, all the graphs are “upside down u-shaped” parabolas. 4
–100 –2 –4 2 3 1 –1 –3
–80
–60
–40
–20 5 –5
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. More Graphs of y = nx2 5.4.2 Guided Practice
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) 4
–100 –2 –4 2 3 1 –1 –3
–80
–60
–40
–20 5 –5
y = –4x2
y = –3x2
y = –2x2
y = – x2
y = –3x2
y = – x2 1 2
y = –2x2
y = –3x2
Solution follows…

13. More Graphs of y = nx2 5.4.2 Guided Practice
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 = ½ x2.
5. (–3, –27)
6. (2, –2)
7. (5, –75)
8. (0, 0) 4
–100 –2 –4 2 3 1 –1 –3
–80
–60
–40
–20 5 –5
y = –4x2
y = –3x2
y = –2x2
y = – x2
y = –3x2
y = – x2 1 2
y = –3x2
all
Solution follows…

14. More Graphs of y = nx2 5.4.2 Guided Practice
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 4
–100 –2 –4 2 3 1 –1 –3
–80
–60
–40
–20 5 –5
y = –4x2
y = –3x2
y = –2x2
y = – x2 x
x = 2 or x = –2 1 2
x = 3 or x = –3
x = 4 or x = –4 y
Solution follows…

15. More Graphs of y = nx2 5.4.2 Guided Practice
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 1 2 4
–100 –2 –4 2 3 1 –1 –3
–80
–60
–40
–20 5 –5
y = –4x2
y = –3x2
y = –2x2
y = – x2 x
x = 2 or x = –2
x = 3 or x = –3
x = 3.7 or x = –3.7 y
Solution follows…

16. More Graphs of y = nx2 5.4.2 Guided Practice
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 4 –2 –4 2 3 1 –1 –3
–20
–40
–60
–80
–100 x
y = – x2 1 3
y = –5x2 x
–5x2 ±4
–80 ±3
–45 ±2
–20 ±1 –5 1 3 x
– x2 ±4 – ±3 –3 ±2 ±1 1 3 16 4 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 negative n) and those you saw in the previous Lesson (of y = nx2 for positive n) are very closely related.
negative n
positive n

18. Lesson
5.4.2
More Graphs of y = nx2
Example 3 y –1 –2 –3 1 2 3
–30
–20
–10 10 20 30
y = –3x2
y = –x2
y = 3x2
y = 2x2
y = x2
y = –2x2
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.
Solution x
Plotting the graphs gives the diagram shown on the right.
For a given value of k, the graphs of y = kx2 and y = –kx2 are reflections of 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. More Graphs of y = nx2 5.4.2 Guided Practice
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. More Graphs of y = nx2 5.4.2 Independent Practice
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 = 1.5x2 for values of x between –3 and 3. –1 –2 –3 1 2 3
–30
–20
–10 10 20 30
y = 1.5x2 x
y = –1.5x2
Solution follows…

21. More Graphs of y = nx2 5.4.2 Independent Practice
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. 1 4
(0, 0)
Half a u‑shaped parabola above the x-axis with its vertex at (0, 0).
Solution follows…

22. More Graphs of y = nx2 5.4.2 Round Up
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.