Lesson 5.4.2

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Lesson 5.4.2Lesson 5.4.2

More Graphs of y = nx2More Graphs of y = nx2

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Lesson

5.4.2More Graphs of y = nx2More Graphs of y = nx2

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 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

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More Graphs of y = nx2More Graphs of y = nx2Lesson

5.4.2

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.

x

y

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The Graph of y = nx2 is Still a Parabola if n is Negative

Lesson

5.4.2

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.

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5.4.2

Example 1

Solution follows…

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.

–2–3–4–5x –1 0

x2 491625 1 0

y = –x2 –4–9–16–25 –1 0

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…

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Solution continues…


5.4.2

Example 1


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.

–2–3–4–5x –1 0

x2 491625 1 0

y = –x2 –4–9–16–25 –1 0

321 4 5

941 16 25

–9–4–1 –16 –25

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5.4.2

Example 1


Solution (continued)

Now you can plot the points. 0 2 40

–10

–20

–30

–2–4

The graph of y = –x2 is also a parabola. But instead of being “u‑shaped,” it’s “upside down u-shaped.”

–2–3–4–5x –1 0

x2 491625 1 0

y = –x2 –4–9–16–25 –1 0

321 4 5

941 16 25

–9–4–1 –16 –25

y

x

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5.4.2

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.

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Example 2

Solution follows…

Lesson

5.4.2

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

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2Solution

As always, make a table and plot the points.

x –2x2

0

1

2

3

4

0

–2

–8

–18

–32

–3x2

0

–3

–12

–27

–48

–4x2

0

–4

–16

–36

–64

–½ x2

0

–0.5

–2

–4.5

–8 Solution continues…

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x –2x2

0

1

2

3

4

0

–2

–8

–18

–32

x

0

1

2

3

4

–3x2

0

–3

–12

–27

–48

x

0

1

2

3

4

–4x2

0

–4

–16

–36

–64

x

0

1

2

3

4

–½ x2

0

–0.5

–2

–4.5

–8


Example 2

Lesson

5.4.2

0 4

–100

–2–4 2 31–1–3

–80

–60

–40

–20

05–5Solution

y = –4x2

y = –3x2

y = –2x2

y = – x21

2

Dec

reas

ing

val

ues

of n

(n = –4)

(n = –3)

(n = –2)

(n = –½)

x

y

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5.4.2

This time, since n is negative, all the graphs are “upside down u-shaped” parabolas.

Also, the more negative the value of n, the steeper and narrower the parabola will be.

But all the graphs still have their vertex (the vertex is the highest point this time) at the same place, the origin.

0 4

–100

–2–4

2 31–1–3

–80

–60

–40

–20

05–5

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Guided Practice

Solution follows…

Lesson

5.4.2

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

y = – x21

2

0 4

–100

–2–4 2 31–1–3

–80

–60

–40

–20

05–5

y = –4x2

y = –3x2

y = –2x2

y = – x212

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Guided Practice

Solution follows…

Lesson

5.4.2

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)

y = –3x2

y = –3x2

all

y = – x21

2

0 4

–100

–2–4 2 31–1–3

–80

–60

–40

–20

05–5

y = –4x2

y = –3x2

y = –2x2

y = – x212

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Guided Practice

Solution follows…

Lesson

5.4.2

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

1

2

0 4

–100

–2–4 2 31–1–3

–80

–60

–40

–20

05–5

y = –4x2

y = –3x2

y = –2x2

y = – x212

x = 2 or x = –2

x = 3 or x = –3

x = 4 or x = –4

x

y

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Guided Practice

Solution follows…

Lesson

5.4.2

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

20 4

–100

–2–4 2 31–1–3

–80

–60

–40

–20

05–5

y = –4x2

y = –3x2

y = –2x2

y = – x212

x = 2 or x = –2

x = 3 or x = –3

x = 3.7 or x = –3.7

x

y

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Guided Practice

Solution follows…

Lesson

5.4.2

Plot the graphs in Exercises 15–16 for x between –4 and 4.

15. y = –5x2

16. y = – x2 1

3

0 40

–2–4 2 31–1–3

–20

–40

–60

–80

–100

y = –5x2

y = – x21

3

x

y

x

–5x2

±4

–80

±3

–45

±2

–20

±1

–5

0

0

x

– x2

±4

–

±3

–3

±2

–

±1

–

0

01

3

16

3

4

3

1

3

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Graphs of y = nx2 for n > 0 and n < 0 Are Reflections

Lesson

5.4.2

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

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Example 3

Solution follows…

Lesson

5.4.2

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

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.

0–1–2–3 1 2 3

–30

–20

–10

0

10

20

30

y = –3x2

y = –x2

y = 3x2

y = 2x2

y = x2

y = –2x2

x

y

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Guided Practice

Solution follows…

Lesson

5.4.2

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.

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Independent Practice

Solution follows…

Lesson

5.4.2

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. 0–1–2–3 1 2 3

–30

–20

–10

0

10

20

30

y = 1.5x2

y = –1.5x2

x

y

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Independent Practice

Solution follows…

Lesson

5.4.2

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 = r2. Describe what a graph of A against r would look like.Check your answer by plotting points for r = 1, 2, 3, and 4.

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4(0, 0)

Half a u‑shaped parabola above the x-axis with its vertex at (0, 0).

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Round UpRound Up

Lesson

5.4.2

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.

Documents

Lesson 5.4.2