Ch. 7.6 Squares, Squaring & Parabolas

Learning Intentions:

Learn about the squaring & square root function.

Graph parabolas.

Compare the squaring function with other functions.

Relate the squaring function to finding the area of a square.

4 3 2 1 0

In addition to Level 3, I go above & beyond what was taught in class, Examples: - Make connection with other concepts in math (i.e. System of Equations, Absolute Value Function, Other parent functions, etc. - Make connection with other content areas. - 7.6 Practice: GRAPH all given problems

Students will work with radicals and integer exponents. - Use square root symbols to solve equations in the form x2 = y . - Evaluate roots of small perfect square. - Evaluate roots of small cubes. - Apply square roots & cube roots as it relates to volume and area of cubes and squares.

Students will be able to: - Understand that taking the square root & squaring are inverse operations. - Understand that taking the cube root & cubing are inverse operations.

With help from the teacher, I have partial success with level 2 and 3.

Even with help, students have no success with the unit content.

Learning Goal: Students will work with radicals & solve quadratic equations using the absolute value function.

Perfect Squares

• 25, 16 and 81 are called perfect squares.

• This means that if each of these numbers were the area of a square, the length of one side would be a whole number.

5

5 4

4

9

9

Area = 81 𝒖𝒏𝒊𝒕𝒔𝟐 Area = 25 𝒖𝒏𝒊𝒕𝒔𝟐 Area =

16 𝒖𝒏𝒊𝒕𝒔𝟐

Perfect Squares - iff s is an integer, then A = a perfect square.

𝑠2 = Area • 12 = 1

• 22 = 4

• 32 = 9

• 42 = 16

• 52 = 25 • 62 = 36 • 72 = 49 • 82 = 64 • 92 = 81 • 102 = 100

Recursive Sequence: common b or d? A: {1, 4, 9, 16, 25, 36, 49, 64, 81, 100, … } 𝑫𝟏: 3 5 7 9 11 13 𝐷2: 2 2 2 2 2 …

Common Differences occur in the 2nd round: Thus, y = degree 2 polynomial y = a𝑥𝟐 + bx + c Since there is no transformation, this sequence

equals the parent function: y = 𝒙𝟐

x length y Area

1 1

2 4

3 9

4 16

x 𝒙𝟐

Non-Perfect Squares - numbers whose square root equals an irrational value.

• Why isn’t 20 a perfect square?

• 20 does NOT equal an Area with equivalent whole number dimensions.

w = 20 units

l = 1 unit

Area = 20 𝒖𝒏𝒊𝒕𝒔𝟐

Area = 20 𝒖𝒏𝒊𝒕𝒔𝟐

The square root of 20 must be an irrational number between 4 and 5.

l = 2 units

l = 4 units A = 20 𝒖𝒏𝒊𝒕𝒔𝟐

w = 5 units

w = 10 units

l = 𝟐𝟎 units

w = 𝟐𝟎 units

A = 20 𝒖𝒏𝒊𝒕𝒔𝟐

How to find the approximate square root of 20… 1. What two perfect squares does 20 lie between?

a) 16 and 25 b) The square root of 16 is 4, so the square root of 20 must be a

little more than 4.

2. How to find the “little more” a) Is the “non-perfect square” 20 closer to 16 or 25? b) It seems to be right in the middle. So pick a number in between

4 and 5. c) Multiply 4.4 times 4.4. What do you get?

i. 19.36 ii. 20 – 19.36 = 0.64

d) Lets see if we can get closer to 20. Multiply 4.5 times 4.5. What do you get?

i. 20.25 ii. 20 – 20.25 = -0.25

e) 4.5 is the best estimate for the square root of 20.

Area = 𝑥2 x

x

𝑨

𝑨

Recall: 𝑨· 𝑨 = A Because any same number x square rooted equals that given number.

Square Root (Radical) Function: Only gives positive solutions unless otherwise noted.

Simplifying 𝟗 vs. ± 𝟗

𝟗 = 3 vs. ± 𝟗 = ±3 Radical vs. Quadratic 𝑥 = y vs. 𝑥2 = y

Let x = 16 Let y = 16

16 = y vs. 𝑥2 = 16

4 = 𝒚 𝑥2 = 16

𝑥 = 16

x = 4 or x = -4

Vocabulary:

Squaring: the process of multiplying a number by itself.

𝒙𝟐 ‘x to the power of 2’ or ‘x squared’

Parabola: the graph of y = 𝒙𝟐

NOTE: the graph of the squaring function is a parabola. Although every positive number has TWO square roots, the square root function ( 𝑥 ) in your calculator gives only the POSITIVE square root.

Absolute Value vs. Quadratic Functions

y = 𝒙

y = 𝒙𝟐

Let y = 4 y = 𝒙 vs. y = 𝒙𝟐 4 = x 4 = x𝟐

4 = x𝟐 x = -4 or x = 4 2 = x

x = -2 or x = 2

Ex.)

0

0

Integers Square Roots

Ex.)

1.41

1

Integers Square Roots

SOLUTION:

10 9

81 100 𝟖𝟓

𝟖𝟓 ≈ 9.219544457

2

1 4 2

1.73

3

Since:

81 < 𝟖𝟓 < 100

𝟗 < 𝟖𝟓 < 𝟏𝟎

Ex.) Find the side of the square whose area is 6.25 𝑐𝑚2. Use a graph to check your answer.

6.25𝑐𝑚2 x

x

SOLUTION: Ex.) Find the side of the square whose area is 6.25 𝑐𝑚2. Use a graph to check your answer.

6.25𝑐𝑚2 x

x

Let x = side length of the square (cm.) Solve the equation: 𝑥2 = 6.25

𝑥2 = 6.25

𝑥2 = 6.25 𝑥 = 2.5

x = -2.5 or x = 2.5 The equation has two solutions, but because the side of the square must be POSITIVE, the only realistic solution is 2.5cm.

Graph: 𝒚 = 𝟔. 𝟐𝟓

𝒚 = 𝒙𝟐

Absolute Value Functions vs. Quadratic Equations

1.) Graph the function f(x) = 𝑥2 What other equation produces the same graph?

2.) Solve each equation for x.

a.) 𝒙 = 6 b.) 𝒙𝟐 = 36 c.) 𝒙 = 3.8 d.) 𝒙𝟐 = 14.44 3.) Solve each equation, if possible.

a.) 4.7 = 𝒙 - 2.8 b.) -41 = 𝒙𝟐 – 28 c.) 11 = 𝒙𝟐 – 14 4.) Solve each equation for x. Sketch a graph of each.

a.) 𝒙 − 𝟐 = 4 b.) (𝒙 − 𝟐)𝟐 = 16 c.) 𝒙 + 𝟑 = 7 d.) (𝒙 + 𝟑)𝟐 = 49

SOLUTIONS: Absolute Value Functions vs. Quadratic Equations

1.) Graph the function f(x) = 𝑥2 What other equation produces the same graph? The absolute value function (y = 𝒙 )

2.) Solve each equation for x.

a.) 𝒙 = 6 b.) 𝒙𝟐 = 36 c.) 𝒙 = 3.8 d.) 𝒙𝟐 = 14.44 𝒙𝟐 = 36 𝒙𝟐 = 14.44 x = -6 or x = 6 𝒙 = 6 x = -3.8 or x = 3.8 𝒙 = 3.8

x = -6 or x = 6 x = -3.8 or x = 3.8 3.) Solve each equation, if possible.

a.) 4.7 = 𝒙 - 2.8 b.) -41 = 𝒙𝟐 – 28 c.) 11 = 𝒙𝟐 – 14

7.5 = 𝒙 -13 = 𝒙𝟐 25 = 𝒙𝟐 x = -7.5 or x = 7.5 x = no solution x = -5 or x = 5

y = 𝒙𝟐 AND y = 𝒙

SOLUTIONS: Absolute Value Functions vs. Quadratic Equations

4.) Solve each equation for x. Sketch a graph of each.

a.) 𝒙 − 𝟐 = 4 b.) (𝒙 − 𝟐)𝟐 = 16 c.) 𝒙 + 𝟑 = 7 d.) (𝒙 + 𝟑)𝟐 = 49

x – 2 = -4 or x – 2 = 4 (𝒙 − 𝟐)𝟐 = 16 x + 3 = -7 or x + 3 = 7 (𝒙 + 𝟑)𝟐 = 49

x = -2 or x = 6 𝒙 − 𝟐 = 4 x = -10 or x = 4 𝒙 + 𝟑 = 7

x – 2 =-4 or x – 2 = 4 x + 3 =-7 or x + 3 = 7

x = -2 or x = 6 x = -10 or x = 4

y = 𝒙𝟐 AND y = 𝒙

y = 4

y = 𝑥 − 2

Absolute Value & Quadratic Function Notation Ex.) Consider the function f(x) = 𝒙 . Each point: (x, y) = (x, f(x))

a.) What is f(-3)? b.) What is f(-2)? c.) Solve f(x) = 10

Solutions:

Absolute Value & Quadratic Function Notation Ex.) Consider the function f(x) = 𝒙 . Each point: (x, y) = (x, f(x))

a.) What is f(-3)? b.) What is f(-2)? c.) Solve f(x) = 10

f(x) = 𝑥 f(-3) = −3 f(-3) = 3

(x, f(x)) = (-3, 3)

f(x) = 𝑥 f(-2) = −2 f(-2) = 2

(x, f(x)) = (-2, 2)

f(x) = 𝑥 f(x) = 10 10 = 𝒙 x = -10 or x = 10

(x, f(x)) = (-10, 10) or (10, 10)

f(x) = 10

f(-3)

f(10)

-3

SOLUTIONS: Remember – Square Root Function ≠ Square Roots of a Number (e.) (d.)

Ex.) Equations of Parabolas

SOLUTION: Equations of Parabolas

y = a(x – h)𝟐 + k

(h, k) = vertex h = horizontal shift k = vertical shift a = size change