Sec 4 - Gwinnett County Public Schools...Sec 4.2 – Rational & Radical Relationships Solving...

Sec 4.1 – Rational & Radical Relationships Graphs of Radical Functions Name:

Consider the following EQUATIONS, make a table, plot the points, and graph what you think the graph looks like.

1. 2. 3. 4.

5. What happens to the graph as the number in front of x gets Larger? Close to Zero? Negative?

6. 7. 8. 9.

10. What happens to the graph as we add or subtract a number inside or outside the radical?

M. Winking Unit 4-1 page 57

2f x x f x x 0.5f x x 2f x x

x y

4

1

0

1

2

3

4

x y

4

1

0

1

2

3

4

x y

4

1

0

1

2

3

4

x y

4

1

0

1

2

3

4

4f x x f x x 1f x x 4 3f x x

x y

4

3

2

1

0

1

4

x y

5

4

3

2

0

3

5

x y

3

0

1

2

3

4

5

x y

5

4

3

2

0

3

5


11. 12. 13. 14.

15. Sketch a graph of 2 3 4f x x and answer the following questions. a. Explain the transformations of the parent function

p x x to create f x .

b. Domain of f(x):

c. Range of f(x):

d. End Behavior of f(x):

e. Extrema:

16. Sketch a graph of 33 4 2g x x and answer the following questions. a. Explain the transformations of the parent function

3p x x to create g x .

b. Domain of f(x):

c. Range of f(x):

d. End Behavior of f(x):

33f x x 3f x x 3 3f x x 3 3 4f x x

x y

8

4

1

0

1

4

8

x y

8

4

1

0

1

4

8

x y

8

4

3

2

1

5

8

x y

8

4

3

2

1

5

8


Sec 4.2 – Rational & Radical Relationships Solving Radical Equations & Inequalities Name:

Find the value for the variable that makes the statement true. (SHOW WORK NEATLY)

1. 8 2 3 10 30x 2. 2 4 9 1x

3. 3 4 5 13a 4. 12 2 2 1 4x

5. 3 2 1 2 5x 6. 3 2 10 6x

– 4 – 4

– 4 – 4

One Radical Basic Equation



7. 5 2 3 12x x 8. 2 5 3 2x x

9. 83452 xx 10. 33 635352 xx

11. xx 153



12. 951 xx 13. xx 21112

Find the values for the variable that make the inequality statement true. (SHOW WORK NEATLY)

14. 633 x 15. 812 x 16. 412 x

17. 259 x 18. 3143 x


Sec 4.3 – Rational & Radical Relationships Finding Inverses of Radical Functions Name:

1. Consider the function 𝑓(𝑥) shown below. Find the inverse of the function, sketch a graph of the inverse, and determine whether or not the inverse is a function. A.

B.

C.

D.


Is the Inverse a Function? YES NO

Graph of Inverse




Graph of Inverse

Graph of Inverse

Graph of Inverse

2. Consider the function 𝑓(𝑥) shown below. Find the inverse of the function, sketch a graph of the inverse, and determine whether or not the inverse is a function.

A.

B.

C.

D.


Is the Inverse a Function?

YES NO




Graph of Inverse

Graph of Inverse

Graph of Inverse

Graph of Inverse

Sec 4.4 – Rational & Radical Relationships Multiplying & Dividing Rational Expressions Name:

Multiply and Simply the following Rational Expressions (Hint: It helps to factor all of the numerators and denominators first.)

1. 𝑥+2

𝑥2−4𝑥−12 ∙

𝑥2−36

𝑥−2 2.

𝑥2+𝑥−2

𝑥2+5𝑥−6 ∙

𝑥+6

𝑥+5

3. 1

3𝑚+6 ∙

3

𝑚+3 4.

2𝑎+4

8𝑎2 ∙

12𝑎

𝑎+2

5. 𝑦2−2𝑦−15

𝑦2−3𝑦−10 ∙

𝑦2−4𝑦+3

𝑦2−9 6.

2𝑥2−3𝑥−2

3𝑥−6 ∙

6𝑥

4𝑥2−1


1. 2.

3. 4.

5. 6.

Divide and Simply the following Rational Expressions (Hint: It helps to factor all of the numerators and denominators first.)

7. 𝑥2+3𝑥−10

𝑥2−2𝑥−15 ÷

𝑥2+𝑥−6

𝑥2+6𝑥+9 8.

𝑥+5

2𝑥 ÷

𝑥+5

8

9. 𝑚2

𝑚+5 ÷

𝑚2+5𝑚

𝑚2+10𝑚+25 10.

𝑝2+2𝑝−3

𝑝2+2𝑝−8 ÷

𝑝2−1

𝑝−2

11.

2

2

4

8 1612

2 8

x x

x x

x

12. 2

2

3

6 92

9

b

b bb

b

7. 8.

9. 10.

11. 12.


Sec 4.5 – Rational & Radical Relationships Add & Subtract Rational Expressions Name:

ADD, SUBTRACT, and SIMPLIFY the following Rational Expressions.

1. 5𝑦

4𝑦2+

12

4𝑦2+

3𝑦

4𝑦2 2.

9𝑎3

3𝑎2−

6𝑎

3𝑎2

3. 2𝑡

3𝑡−12−

8

3𝑡−12 4.

𝑤2

𝑤2−9+

2𝑤−15

𝑤2−9

5. 𝑎2+10

𝑎2−4−

7𝑎

𝑎2−4 6.

𝑦2−13

𝑦2−25+

3(1−𝑦)

𝑦2−25


1. 2.

3. 4.

5. 6.

ADD, SUBTRACT, and SIMPLIFY the following Rational Expressions (need to first rewrite each with a common denominator).

7. 2

𝑦2+

1

3−

5

6𝑦2 8.

𝑥2−12

𝑥2−4+

2

𝑥−2

9. 𝑑

𝑑2−1−

𝑑

𝑑−1 10.

𝑥

𝑥+1+

8

𝑥−2

11. 3

𝑚+2+

𝑚2

𝑚2−4−

1

𝑚−2 12.

6

𝑎2−2𝑎−35−

2

𝑎2+9𝑎+20

7. 8.

9. 10.

11. 12.


Sec 4.6 – Rational & Radical Relationships Solving Rational Equations & Inequalities Name:

Find the value of the variable that makes each of the statements true.

1. 4

𝑥2=

1

9 2.

2𝑡

𝑡−2=

𝑡+4

𝑡−2

3. 𝑏+2

𝑏−3=

3𝑏−4

𝑏−3 4.

𝑤2

𝑤−4−

8

𝑤−4=

2𝑤

𝑤−4

5. 5

𝑥−4=

3

𝑥 6.

𝑝2

𝑝+2=

4𝑝+12

𝑝+2


1. 2.

3. 4.

5. 6.

Find the value of the variable that makes each of the statements true.

7. 4𝑎2−9

2𝑎−3= 9 8.

𝑥2−3𝑥+4

𝑥−4= −4

9. 2

3𝑥2=

1

𝑥−

1

3 10.

𝑥

𝑥−4=

𝑥+10

𝑥−2

11. 𝑝−1

𝑝+3−

2

𝑝−3=

7−3𝑝

𝑝2−9 12.

𝑥

𝑥−2+

2

𝑥+3=

3𝑥+4

𝑥2+𝑥−6

7. 8.

9. 10.

11. 12.


Find the set of values of the variable that makes each of the inequality statements true.

13. 3𝑎

𝑎−1≤ 4 14.

𝑚−3

𝑚+2≥ 0

15. 𝑦

𝑦+2< 3 16.

3

𝑥−2≥

1

𝑥+2


13. 14.

15. 16.

Sec 4.7 – Rational & Radical Relationships Graphing Rational Functions Name:

Characteristic Description Example

Hole (Point Discontinuity)

A hole usually occurs in the graph of a rational function when a linear factor in the numerator and denominator “divide out”. The result is the same as the graph of the simplified function but with a missing point in the graph.

Vertical Asymptote (Infinite Discontinuity)

A vertical asymptote occurs any time a linear factor of the denominator doesn’t “divide out” with a factor in the numerator.

Vertical Asymptote & Hole

To have both a hole and a vertical asymptote the rational function must have at least one linear factor that divides out and one linear factor that does not.

Sketch a graph of the following rational functions. Label any holes or vertical asymptotes. Use your calculator for additional assistance.

1. 𝒇(𝒙) =𝒙𝟐+𝟐𝒙−𝟑

𝒙−𝟏 2. 𝒇(𝒙) =

𝒙𝟐+𝟑𝒙−𝟒

𝒙𝟐+𝟐𝒙−𝟖


Potential Horizontal Asymptotes

Description Example

Case #1: A rational function that has a numerator polynomial with the same degree as the polynomial in the denominator creates a horizontal asymptote that passes through the y-axis at the quotient of the leading coefficients.

Case #2: A rational function that has a polynomial in the numerator that has a smaller degree than the degree of the polynomial in the denominator creates a horizontal asymptote at y =0.

Case #3: A rational function that has a polynomial in the numerator that has a larger degree than the degree of the polynomial in the denominator does not have a horizontal asymptote.

Sketch a graph of the following rational functions. Label any vertical asymptotes, horizontal asymptotes, or holes. Use your calculator for additional assistance.

3. 𝒇(𝒙) =𝒙−𝟒

𝒙𝟐−𝟐𝒙−𝟖 4. 𝒇(𝒙) =

𝟐𝒙𝟐−𝟖

𝒙𝟐+𝒙−𝟔


Sec 4.8 – Rational & Radical Relationships Graphs of Absolute, Step, and Partial Functions Name:


1. 2. 3. 4.

5. What happens to the graph as the number in front of x gets Larger? Close to Zero? Negative?

6. 7. 8. 9.

10. What happens to the graph as we add or subtract a number inside or outside the absolute value?

2f x x f x x 0.5f x x 2f x x

x y

4

2

1

0

1

2

4

x y

3

2

1

0

1

2

3

x y

4

2

0

1

2

3

4

x y

4

1

0

1

2

3

4

2f x x 3f x x 2 1f x x 2 3f x x

x y

4

2

1

0

1

2

4

x y

5

4

3

2

1

0

1

x y

2

1

0

1

2

3

4

x y

2

1

0

1

2

3

4



11. 12. 𝑓(𝑥) = ⌊𝑥⌋ or 𝑓(𝑥) = ⟦𝑥⟧ 13. 14. 𝑓(𝑥) = 2⟦𝑥 − 1⟧

15. Graph the following partial functions (piece-wise). a. b. c.

16. Given the following function evaluate the following. 17. Describe the following graph using a partial function

f x x 0.5 1f x x x y

0.5

0

0.5

1

1.2

2

2.4

x y

0.5

0

0.5

1

1.2

2

2.4

x y

0.5

0

0.5

1

1.2

2

2.4

x y

0.5

0

0.5

1

1.2

2

2.4

𝑓(𝑥) = {𝑥2 − 3 𝑖𝑓 𝑥 ≤ 1

−2𝑥 + 3 𝑖𝑓 𝑥 > 1 𝑓(𝑥) = {

−𝑥 + 5 𝑖𝑓 𝑥 ≤ 22𝑥 − 1 𝑖𝑓 𝑥 > 2

𝑓(𝑥) = {

3 𝑖𝑓 𝑥 < −2−2𝑥 − 1 𝑖𝑓 −2 ≤ 𝑥 ≤ 2

𝑥 + 1 𝑖𝑓 𝑥 > 2

𝑔(𝑥) = {𝑥2 − 2𝑥 𝑖𝑓 𝑥 ≤ −13𝑥 + 2 𝑖𝑓 𝑥 > −1

a. 𝑔(−4) =

b. 𝑔(−1) =

c. 𝑔(3) =

ℎ(𝑥) = {


17. A city water company charges homeowners based on how much water

they use in thousands of gallons. The company progressively charges at a higher rate the more water that is used.

a. Based on the graph at the right how much does the city charged when a home owner uses the following number of gallons of water:

1700 gallons of water costs .



b. Finish filling in the following piece-wise equation below that describes the chart based on the graph:

𝑦 = {

50 𝑖𝑓 0 ≤ 𝑥 < 2

5𝑥 + 50 𝑖𝑓 6 ≤ 𝑥 < 8

30𝑥 − 140

Water Usage (in thousands of gallons)

Co

st o

f W

ate

r B

ill

(do

lla

rs)


Sec 4.9 – Rational & Radical Relationships Rational Functions in Context Name:

1. Ron rode 8 miles on his bike in the same amount of time it took Harmony to ride 6 miles. Ron rides his bike at a rate of 4 miles per hour (mph) faster than Harmony. How fast was Harmony riding on her bike?

2. Jerry drove his car a total of 40 miles in the same time it took his wife Katrina to drive 50 miles. Jerry drives his car at a rate of 10 miles per hour (mph) slower than Katrina. How fast was Katrina driving?

3. A passenger jet flew from Atlanta to Miami (650 miles) in the same time it took a propeller plane to fly from Atlanta to Savanah (250 miles). The jet was flying at a rate of 400 miles per hour (mph) faster than the propeller plan. How fast is the propeller plane flying?

4. A motorboat is traveling on the Mississippi River. The boat travels at 20 miles per hour (mph) in still water. The boat went 6 miles downriver with the current in the same amount of time it took the boat to travel 5 miles upstream against the current. How fast is the current?


5. Two planes left from Nashville at the same rate of speed. The first plane flew to Boston for a total of 1100 miles. The second plane flew to Milwaukee for a total of 600 miles. It took the first plane flying to Boston an hour longer to reach its destination. How long did it take the second plane to travel to Milwaukee?

6. A child went to a winter fair outside and got a balloon at the fair. The temperature outside was 280˚ Kelvin (about 44˚ F). When the child brought the balloon inside the temperature increased to 300˚ Kelvin (about 80˚ F). The balloon increased in size by 0.5 liters of air. How many liters of air did the balloon contain originally at the Fair?

Use Charles Law: 𝑉1

𝑇1=

𝑉2

𝑇2

7. Consider the Compound Interest Formula: 𝑨 = 𝑷 (𝟏 +𝒓

𝒏)

𝒏𝒕

a. Determine the Value of an account in which a person invested $4000 for

8 years at an annual rate of 11% compounded quarterly (n = 4).

b. Create a sketch of a graph showing the value of an account over t years given an initial investment of $500 at an annual rate of 12% compounded monthly (n = 12).

c. Create a sketch of a graph showing the value of an account for different numbers of compounds per year given the initial investment of $100 at an annual rate of 15% compounded n times over the course of 20 years.

A = Value of Account after Compounding P = Original Amount Invested r = Annual Interest Rate as a decimal n = Compounds per Year t = Number of Years Interest is Accrued

Years Value

1

2

4

8

10

12

14

16

20

Compound Value

1

2

3

4

6

8

10

12

14


8. Describe the domain and range of each function below as DISCRETE or CONTINUOUS

9. A postal company delivers packages based on their weight but will not ship anything over 50 pounds. The company charges $0.50 per pound to deliver the package anywhere in the United States. If we consider this situation a function where the number of pounds, x, is the independent variable and the cost in dollars, y, is the dependent variable determine the domain and range.

10. A limousine company rents their limousine by the hour. The company charges $85

per hour. The minimum time is 2 hours and a maximum of 12 hours. If we consider this situation a function where the number of hours, x, is the independent variable and the cost in dollars of renting the limousine, y, is the dependent variable determine the domain and range.

11. A student is growing a bean plant outside for a science project. The plants grow for 12 weeks

before reaching their maximum height. The student consider the week she started growing the plant to be week 0 and then realized that the plant closely followed the function model ℎ(𝑥) = 1.5 ∙ (1.2)𝑥, where x represents the number of weeks grown and 𝒉(𝒙) represents the height of the plant in inches. Using the function model describe the appropriate domain and range.

12. A vending company realized a relationship between the number of people present at the stadium during a Braves game and the number of hot dogs they sold. The minimum attendance due to players and support staff is 361 people and the maximum people that could be at the stadium is 86,436 people. The relationship that describes the number of hot

dogs sold very closely followed the function model ℎ(𝑥) = 15 ∙ √𝑥 where x represents the number of people at the stadium and 𝒉(𝒙) represents the number of hot dogs sold. What is the domain and range of the model?

Domain:

Range:

Domain:

Range:

Domain:

Range:

Domain: Range:

Domain:

Range:

Domain:

Range:

Domain:

Range:


04-00-Title04-01-GraphRadicalFunctions-Blank04-02-SolvingRadicalEqIneq-Blank04-03-InverseOfRadicalFunctions-Blank04-04-MultDivRatnlExprssns-Blank04-05-AddSubtractRatnlExprssns-Blank04-06-SolveRtnlEqutnsIneqlts-Blank04-07-GraphingRationalFunctions-Blank04-08-GraphingAbsStepPartialFunctions-Blank04-09-RationalFunctionsInContext-Blank

Sec 4 - Gwinnett County Public Schools...Sec 4.2 – Rational & Radical Relationships Solving...

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Transcript of Sec 4 - Gwinnett County Public Schools...Sec 4.2 – Rational & Radical Relationships Solving...