Chapter 12 Notes
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Day 27 12.1 Functions Involving Square Roots 12.2 Operations with Radical Expressions 12.1 “The Square Root Function” 1. y = x
“Parent Function”
x y Domain: Range:
2. y =2 x
“Child Function” (just one of many possible)
x y Domain: Range: Q: How does this child compare with the parent?
3. y = x +1 Another “Child Function” x y Domain: Range: Q: How does this child compare with the parent?
1
4. y = x −3 Yet another “Child Function” x y Domain: Range: Q: How does this child compare with the parent?
General Equation for the Square Root Function
a:
h:
k:
Q: What happens if a is negative?
2
12.2 Operations with Radical Expressions Adding and Subtracting 1. 2 2 +5 − 6 2
2. 4 3 − 27
3. 3 7 − 5 7+ 2 7
4. 8 5 +125
Multiplying 1.
2• 8
2.
2 5− 3
(
)
(
)
(
)(
3. 1+ 5
2
4. a − b a + b
)
3
Dividing 1.
3 5
2.
1 c− d
3.
4 3+ 2
Checking Solutions 1. Check whether 2 + 3 is a solution of x 2 − 4x + 1= 0
2. Is
−5 − 33 a solution of x 2 + 5x − 2= 0? 2
4
(Day 27) 12.1 Square Root Funsheet
Finish for homework.
1-4 Fill in the table and sketch a graph of each function. State the domain and range for each. 1. y = x x 0 y
1
domain: ......................
4
9
range: ....................
0
1
domain: ......................
1 x 2
4
9
range: ....................
2. y =−
3. y =2 x x y
2. y =− x x 0 y
1
domain: ......................
4
9
range: ....................
x y
0
1
domain: ......................
4
9
range: ....................
5. The general shape of the graph of y = x looks like ............................................................................. 6. When the coefficient in front of
x is negative, the graph .....................................................................
7. When the coefficient in front of
x is greater than 1, the graph .............................................................
8. When the coefficient in front of
x is between 0 and 1, the graph ......................................................... 5
Now graph each of the following. Label at least 2 points, and state the domain and range of each. 9. y = x +4
D:
10. y = x −3
R:
D:
R:
12. y =2 x − 1
11. y = x − 1
D:
D:
R:
11. y = − 3 x +5
D:
R:
11. y = x +3
R:
D:
R:
15. How is the graph of y = x +k different from the graph of y = x ? 16. How is the graph of y = x −k different from the graph of y = x ? 17. How is the graph of y = x +h different from the graph of y = x ? 18. How is the graph of y = x −h different from the graph of y = x ?
1 x +3 −4 . State the domain and range. 2
19. Without a calculator, sketch the graph of y =−
6
(Day 27) 12.2 Handout “Operations with Radical Expressions”
Finish for homework.
Simplify the expression. 1. 32 + 2
2.
80 − 45
3.
147 − 7 3
4. 3 11 +176 +11
5.
243 −75 +300
6.
5⋅ 8
7.
10.
(
6 7 3 +6
2 2
)
8.
( a −b)
11.
2
6 10 + 2
(
)(
9. 1+ 13 1− 13
12.
)
3 3 −1
7
13. Write a radical expression and its conjugate.
14. Find the area of a rectangle with length
17 +9 and height
68 .
15. A pole-vaulter’s approach velocity v (in feet per second) and height reached h (in feet) are related by the following equation: ν=8 h . You are pole vaulting and reach a height of 20 feet, while your opponent reaches a height of 16 feet. How much faster were you running than your opponent?
16. From the falling object model, h = − 16t 2 + s , the distance an object falls after it is dropped is
d= 16t 2 . Solving this formula for t, yields t =
d . t gives the time in seconds it takes for an object to fall 4
a certain distance. To break clamshells, sea gulls drop them on rocks while flying. A gull drops a clam shell from a height of 96 feet. Find the time it takes to reach the ground.
17. Challenge: The sum of three consecutive perfect squares is 110. What is the product of the square roots of these numbers?
8
Day 28 12.3 Solving Radical Equations Solve and check your solution(s). 1.
x −7 =0
2.
2x − 3+ 3= 4
3.
x +2 =x
4.
x+ 13 = 0
Caution: When squaring both sides of an equation, you may get .................................................. Thus, you should always ................................ Definition: Geometric Mean of a and b
1. If the geometric mean of a and 6 is 12, find a.
2. You work for United Airlines and remove ice from airplanes. The relationship among the flow rate r (in gallons per minute) of the antifreeze for de-icing, the nozzle diameter d (in inches), and the nozzle pressure P (in lbs per square inch) is r = 30d 2 P . You want a flow rate of 250 gallons per minute. Find the nozzle pressure for a nozzle that has diameter 1.25 inches.
9
Practice. Yay! Solve the equation. 1. 2x − 3 =x + 6
3.
5.
3
2. 3w − 19 w + 20 = 0
x2 − 1 =2
4. 3 +3x + 1= x
2x − 3 −x + 7+ 2= 0
6.
6x + 12 −4 x + 9= 1
10
(Day 28) 12.3 Handout “Fun with Radicals”
Finish for homework.
1 3 −x − = 2 2
1.
x− 10 = 0
2.
3x + 9= 12
3.
4.
3x − 4= 6
5.
4 −x =6
6. − 7 =6x + 7
7.
1 7 x +1 = 9 3
8. 6 −7x − 9= 3
9. x 2 = 6− x
11
10. x 2 = 100 − 15x
13.
1 2 x =x +3 4
16. 30 = 2(32)h
11. 2x =− 13x − 10
2
14. ( x −2) =2x − 1
17. 3 =2π
L 32
12. 1+ x =1− 2x
15. 4 x 2 = 4x + 15
18.
5a =15
12
Day 29 12.4 Using Completing the Square to Solve Quadratic Equations “Completing the square” means to create a ...................................................... example of a PST:
Complete the square for x2 – 8x.
In general: Complete the square for the expression x2 + bx.
What is “completing the square” used for?
Solve for x using completing the square. Don’t forget to balance your equation! Notice in #1-2, a = .........., 1. x2 + 6x + 8 = 0
2. x2 + 10x = 24
13
Notice in 4-8, a ................. 3. x2 – x – 3 = 0
4. –x2 + 3x – 3 = 0
5. 2x2 – x = 2
6. 0 = –2x2 + 8x – 5
7. ½x2 + x = 7
8. –(1/3)x2 + 2x + 4 = 0
14
(Day 29) 12.4 Handout “Completing the Square”
Finish for homework.
Write the trinomial as the square of a binomial. (factor) 1. x2 – 6x + 9
2. x2 – 18x +81
3. x2 + x + 1/4
Find the term that should be added to the expression to create a perfect square trinomial. 4. x2 – 16x 5. x2 + 18x 6. x2 – 13x 7. x2 – ¾ x
8. x2 + 11x
9. x2 +2x/9
Finding the term that creates a perfect square trinomial is called Completing the Square. This method can be used to solve quadratic equations. Here is an example. Solve x2 + 10x = 24 Solution:
x2 + 10x = 24
Write the quadratic equation with the C term on the right-hand side.
x2 + 10x + 52 = 24 + 52
Add 5 2 to each side. Note that 5 is half the coefficient of x in
(x + 5) = 49
the equation. Write the left side as a perfect square.
x + 5 = ±7
Find the square root of each side.
x = -5 ± 7
Subtract 5 from each side.
x=2 or x=12
Simplify.
2
Solve the equation by completing the square. 10. x2 + 6x = -5 11. x2 + 8x – 4 = 5
12. x2 – 6x + 5 = -4
13. x2 + 4x – 7 = 0
14. x2 + 6x – 1 = 0
15. x2 – 6x + 7 = 0
16. x2 + 8x + 13 = 0
17. 2x2 – 4x – 5 = 0
18. 3x2 – 6x – 1 = 0
15
“Parent Function”
x y Domain: Range:
2. y =2 x
“Child Function” (just one of many possible)
x y Domain: Range: Q: How does this child compare with the parent?
3. y = x +1 Another “Child Function” x y Domain: Range: Q: How does this child compare with the parent?
1
4. y = x −3 Yet another “Child Function” x y Domain: Range: Q: How does this child compare with the parent?
General Equation for the Square Root Function
a:
h:
k:
Q: What happens if a is negative?
2
12.2 Operations with Radical Expressions Adding and Subtracting 1. 2 2 +5 − 6 2
2. 4 3 − 27
3. 3 7 − 5 7+ 2 7
4. 8 5 +125
Multiplying 1.
2• 8
2.
2 5− 3
(
)
(
)
(
)(
3. 1+ 5
2
4. a − b a + b
)
3
Dividing 1.
3 5
2.
1 c− d
3.
4 3+ 2
Checking Solutions 1. Check whether 2 + 3 is a solution of x 2 − 4x + 1= 0
2. Is
−5 − 33 a solution of x 2 + 5x − 2= 0? 2
4
(Day 27) 12.1 Square Root Funsheet
Finish for homework.
1-4 Fill in the table and sketch a graph of each function. State the domain and range for each. 1. y = x x 0 y
1
domain: ......................
4
9
range: ....................
0
1
domain: ......................
1 x 2
4
9
range: ....................
2. y =−
3. y =2 x x y
2. y =− x x 0 y
1
domain: ......................
4
9
range: ....................
x y
0
1
domain: ......................
4
9
range: ....................
5. The general shape of the graph of y = x looks like ............................................................................. 6. When the coefficient in front of
x is negative, the graph .....................................................................
7. When the coefficient in front of
x is greater than 1, the graph .............................................................
8. When the coefficient in front of
x is between 0 and 1, the graph ......................................................... 5
Now graph each of the following. Label at least 2 points, and state the domain and range of each. 9. y = x +4
D:
10. y = x −3
R:
D:
R:
12. y =2 x − 1
11. y = x − 1
D:
D:
R:
11. y = − 3 x +5
D:
R:
11. y = x +3
R:
D:
R:
15. How is the graph of y = x +k different from the graph of y = x ? 16. How is the graph of y = x −k different from the graph of y = x ? 17. How is the graph of y = x +h different from the graph of y = x ? 18. How is the graph of y = x −h different from the graph of y = x ?
1 x +3 −4 . State the domain and range. 2
19. Without a calculator, sketch the graph of y =−
6
(Day 27) 12.2 Handout “Operations with Radical Expressions”
Finish for homework.
Simplify the expression. 1. 32 + 2
2.
80 − 45
3.
147 − 7 3
4. 3 11 +176 +11
5.
243 −75 +300
6.
5⋅ 8
7.
10.
(
6 7 3 +6
2 2
)
8.
( a −b)
11.
2
6 10 + 2
(
)(
9. 1+ 13 1− 13
12.
)
3 3 −1
7
13. Write a radical expression and its conjugate.
14. Find the area of a rectangle with length
17 +9 and height
68 .
15. A pole-vaulter’s approach velocity v (in feet per second) and height reached h (in feet) are related by the following equation: ν=8 h . You are pole vaulting and reach a height of 20 feet, while your opponent reaches a height of 16 feet. How much faster were you running than your opponent?
16. From the falling object model, h = − 16t 2 + s , the distance an object falls after it is dropped is
d= 16t 2 . Solving this formula for t, yields t =
d . t gives the time in seconds it takes for an object to fall 4
a certain distance. To break clamshells, sea gulls drop them on rocks while flying. A gull drops a clam shell from a height of 96 feet. Find the time it takes to reach the ground.
17. Challenge: The sum of three consecutive perfect squares is 110. What is the product of the square roots of these numbers?
8
Day 28 12.3 Solving Radical Equations Solve and check your solution(s). 1.
x −7 =0
2.
2x − 3+ 3= 4
3.
x +2 =x
4.
x+ 13 = 0
Caution: When squaring both sides of an equation, you may get .................................................. Thus, you should always ................................ Definition: Geometric Mean of a and b
1. If the geometric mean of a and 6 is 12, find a.
2. You work for United Airlines and remove ice from airplanes. The relationship among the flow rate r (in gallons per minute) of the antifreeze for de-icing, the nozzle diameter d (in inches), and the nozzle pressure P (in lbs per square inch) is r = 30d 2 P . You want a flow rate of 250 gallons per minute. Find the nozzle pressure for a nozzle that has diameter 1.25 inches.
9
Practice. Yay! Solve the equation. 1. 2x − 3 =x + 6
3.
5.
3
2. 3w − 19 w + 20 = 0
x2 − 1 =2
4. 3 +3x + 1= x
2x − 3 −x + 7+ 2= 0
6.
6x + 12 −4 x + 9= 1
10
(Day 28) 12.3 Handout “Fun with Radicals”
Finish for homework.
1 3 −x − = 2 2
1.
x− 10 = 0
2.
3x + 9= 12
3.
4.
3x − 4= 6
5.
4 −x =6
6. − 7 =6x + 7
7.
1 7 x +1 = 9 3
8. 6 −7x − 9= 3
9. x 2 = 6− x
11
10. x 2 = 100 − 15x
13.
1 2 x =x +3 4
16. 30 = 2(32)h
11. 2x =− 13x − 10
2
14. ( x −2) =2x − 1
17. 3 =2π
L 32
12. 1+ x =1− 2x
15. 4 x 2 = 4x + 15
18.
5a =15
12
Day 29 12.4 Using Completing the Square to Solve Quadratic Equations “Completing the square” means to create a ...................................................... example of a PST:
Complete the square for x2 – 8x.
In general: Complete the square for the expression x2 + bx.
What is “completing the square” used for?
Solve for x using completing the square. Don’t forget to balance your equation! Notice in #1-2, a = .........., 1. x2 + 6x + 8 = 0
2. x2 + 10x = 24
13
Notice in 4-8, a ................. 3. x2 – x – 3 = 0
4. –x2 + 3x – 3 = 0
5. 2x2 – x = 2
6. 0 = –2x2 + 8x – 5
7. ½x2 + x = 7
8. –(1/3)x2 + 2x + 4 = 0
14
(Day 29) 12.4 Handout “Completing the Square”
Finish for homework.
Write the trinomial as the square of a binomial. (factor) 1. x2 – 6x + 9
2. x2 – 18x +81
3. x2 + x + 1/4
Find the term that should be added to the expression to create a perfect square trinomial. 4. x2 – 16x 5. x2 + 18x 6. x2 – 13x 7. x2 – ¾ x
8. x2 + 11x
9. x2 +2x/9
Finding the term that creates a perfect square trinomial is called Completing the Square. This method can be used to solve quadratic equations. Here is an example. Solve x2 + 10x = 24 Solution:
x2 + 10x = 24
Write the quadratic equation with the C term on the right-hand side.
x2 + 10x + 52 = 24 + 52
Add 5 2 to each side. Note that 5 is half the coefficient of x in
(x + 5) = 49
the equation. Write the left side as a perfect square.
x + 5 = ±7
Find the square root of each side.
x = -5 ± 7
Subtract 5 from each side.
x=2 or x=12
Simplify.
2
Solve the equation by completing the square. 10. x2 + 6x = -5 11. x2 + 8x – 4 = 5
12. x2 – 6x + 5 = -4
13. x2 + 4x – 7 = 0
14. x2 + 6x – 1 = 0
15. x2 – 6x + 7 = 0
16. x2 + 8x + 13 = 0
17. 2x2 – 4x – 5 = 0
18. 3x2 – 6x – 1 = 0
15
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