SKILL REVIEW
Below are skills that you will need in order to do well on Chapter 2. You are responsible for the information on this sheet. If you have any questions you need to come see me in ac lab or before/after school.
Concept #1: Factoring!
Factor the following:
1. 2. 3.
4. 5. 6.
7. 8. 9.
10. 11. 12.
Concept #2: Quadratic Formula.
Use the quadratic formula to solve for x.
13. 14. 15.
Concept #3: Long Division
16. 17. 18.
Concept #4: Evaluating Functions
Evaluate the following functions
19.
a) f(1) b) f(-2) c) f(0)
20.
a) f(-1) b) f(3) c) f(y)
Chapter 2: Polynomial Functions
Essential Questions
What is a polynomial?
Why do we study polynomials as a group?
Why is it important to be able to find the roots/zeros/x-intercepts of a polynomial?
What is an imaginary number?
Why do we study imaginary numbers?
What does it mean for a function to have imaginary roots/zeros?
Why is it important to be able to model a set of data using a function?
Learning Targets
Determine whether a function is a polynomial.
Find the complex roots/zeros/x-intercepts of a polynomial.
Graph all polynomials.
Write all polynomials as a product of their linear factors.
Determine what degree polynomial should be used to model different sets of data.
Perform operations on complex numbers.
Homework
Section 2.1 – Quadratic Functions
Describe, in words, the transformations that are occurring to the graph of .
Sketch the graph of the quadratic function. Identify the vertex and x-intercept(s).
Write the equation in standard form of the quadratic function that has the indicated vertex and travels through the given point.
A rancher has 200 feet of fencing. He plans on enclosing a rectangular area and then dividing that area into two congruent pens by putting a fence in the middle. What are the dimensions of the enclosed rectangular area that will produce the maximum area?
The path of a diver is given by where y is the height in feet and x is the horizontal distance in feet from the end of the diving board. What is the maximum height of the diver?
Section 2.2 – Polynomial Functions of Higher Degree
For exercises 1-8, match the polynomial functions with its graph.
(a) (b)
(c) (d)
(e) (f) (g)
Graph the functions. Make sure to label their roots.
Find a polynomial function that has the given zeros.
An open box is to be made from a square piece of material 36 centimeters on a side by cutting equal squares with sides of length x from the corners and turning up the sides. Find the formula for the volume of the box and determine the dimensions of the box that will maximize the volume.
Section 2.3 – Real Zeros of Polynomial Functions
Use synthetic division to divide the polynomials.
Use synthetic division to find each function value.
Find all of the roots of the given polynomials. Write each function as the product of linear factors.
Find all of the solutions of the polynomial equations.
Section 2.4 – Complex Numbers
Perform the given operation and write your answer in standard form.
Section 2.5 – The Fundamental Theorem of Algebra
Find all zeros of the functions.
Find all of the zeros of the functions. Then graph them and write the functions as the product of linear factors.
Find a polynomial with real coefficients that has the given zeros.
Use the given root to find all of the roots of the function.
Chapter 2 Review
For 1-3, sketch the graph of the quadratic functions. Make sure to label the vertex and any intercepts.
Write the complex number in standard form.
Find all of the zeros of the given polynomials.
Sketch the graph of the polynomial functions. Make sure to label any intercepts and their multiplicities.
Find the polynomial function with the given zeros.
a) 1, -4, -3i b) -2i, 3
c) 1+3i, 0, 0, 2 d) -3-i, 4, 1, 1
Write an equation that represents the graph below.
Match the polynomial function to the given zeros and multiplicities.
a) b)
c) d)
-3 (multiplicity of 2), 2 (multiplicity of 3)
-3 (multiplicity of 3), 2 (multiplicity of 2)
-1 (multiplicity of 4), 3 (multiplicity of 3)
-1 (multiplicity of 3), 3 (multiplicity of 4)
22. A plumber’s total charge includes a fixed service charge plus an hourly rate for the job. If the total charge is $140 for a 3-hour job and $200 for a 5-hour job, what is the total charge for an 8-hour job?
a. $430 b. $290 c. $260 d. $250 e. $240
23. A function that is defined by the set of ordered pairs {(2,1), (4,2), (6,3)} has domain {2, 4, 6}. What is the domain of the function defined by the set of ordered pairs {(0,2), (2,2), (3, -2)}?
a. {2} b. {-2, 2} c. {-2, 0, 3} d. {0, 2, 3} e. {-2, 0, 2, 3}
24. Which of the following expressions is equivalent to 3x4 + 6x2 – 45?
a. (x2+5)(x2-3) b. (3x2-15)(x2+3) c. 3(x4+6x-45)
d. 3(x2-5)(x2+3) e. 3(x2+5)(x2-3)
25. What is the sum of the 2 solutions of the equation x2+x-6=0?
a. -6 b. -3 c. -1 d. 0 e. 2
SKILL REVIEW
Concept #1:Solving Systems
Solve the following system using the given method:
4x – 3y = 8
8x – 6y = 16
1. Solve using graphing 2. Solve using substitution 3. Solve using linear combinations
Chapter 7: Solving Systems of Equations
Essential Questions
What is a system of equations?
What does a solution to a system of equation represent in terms of its equations? Its graphs?
Which method do you prefer when trying to solve a system of equations? Why would you choose one method over the other?
Learning Targets
Solve systems of equations using substitution, linear combination, and graphing.
Homework
Section 7.1 – Solving Systems of Equations
Solve the systems of equations using substitution.
A small fast-food restaurant invests $5000 to produce a new food item that will sell for $3.49. Each item can be produced for $2.16. How many items must be sold to break even?
Section 7.2 – Systems of Linear Equations in Two Variables
Solve the systems of equations using elimination.
Find a system of equations that has (3, -4) as a solution.
600 tickets were sold at the last 5FD show. The tickets for adults cost $5 and the tickets for students cost $3. If the receipts for the show totaled $2330, how many of each type of ticket was sold?
Section 7.3 – Multivariable Linear Systems
Solve the systems of equations.
An object moving vertically is at the given heights at the given times. Find the position equation for the object.
Chapter 7 Review
Solve the systems of equations.
11. Mr. Wu spent his three day weekend selling lemonade on the streets. He spent $30 on signs, a chair, and a table for his stand. If he sold each cup of lemonade for $.25, but it cost $.09 total for the cup, ice, lemons, and sugar, how many cups of lemonade would Mr. Wu have to sell to break even?
12. Mr. Coulson has $5000 to invest. He splits the money into an IRA and a 403b, which have a return of 3% and 6% respectively. If he earns $240 total after one year, how much did Mr. Coulson invest into each account?
13. Mrs. Hopkins, Mr. Coulson, and Mr. Wu went to six flags together this summer. They bought three tickets, six cheeseburgers, and two jumbo ice cream cones for a total of $140. One ticket cost as much as all of the cheeseburgers, and you could buy 2 cheeseburgers for the cost of one jumbo ice cream cone. How much does a jumbo ice cream cone cost?
14. A plumber’s total charge includes a fixed service charge plus an hourly rate for the job. If the total charge is $140 for a 3-hour job and $200 for a 5-hour job, what is the total charge for an 8-hour job?
a. $430 b. $290 c. $260 d. $250 e.$240
15. As a fund-raiser, a local youth group sold boxes of regular popcorn for $5 each and boxes of caramel popcorn for $8 each. Altogether, they sold 160 boxes for $1,100. How many boxes of caramel popcorn did they sell?
a. 20 b. 32 c. 60 d. 80 e. 100
16. On a recent test, some questions were worth 2 points each and the rest were worth 3 points each. Tuan answered correctly the same number of 2-point questions as 3-point questions and earned a score of 60 points. How many 3-point questions did he answer correctly?
a. 36 b. 30 c. 20 d. 12 e. 10
17. Becky has 76 solid-colored disks that are either red, blue, or green. She lines them up on the floor and finds that there are 4 more red disks than green and 6 more green disks than blue. How many red disks does she have?
a. 10 b. 20 c. 24 d. 26 e. 30
Midterm Review
1) Describe and graph the transformation of the function f(x)= -1/3 (x-4)2 + 5.
2) Find the polynomial of degree 4 with zeros at 4, 0, and -2i. Multiply and simplify your answer.
3) Determine the intervals on which the function is increasing and decreasing, and find the relative maximum and minimum.
4) Find an equation of the line that passes through the point (0, 4) and is (a) parallel to and (b) perpendicular to the line
5) Find the domain of: .
For problems 6-8 (a) identify the parent function of f, (b) describe in words the transformations from f to g, and (c) sketch a graph of g.
6) 7) 8)
9) Determine whether the function has an inverse, and if so, find the inverse function.
a) b)
10) Divide x4 + 3x2 – 9x + 5 by (x + 3).
11) List the possible rational zeros of the function. Use a graphing calculator to graph the function and find all the real zeros.
a) b)
12) Perform the operation and write the result in standard form.
a) b) c) d)
13) Write an equation for a polynomial function with zeros at 6, 2(multiplicity of 2), 0, and -3. Draw a graph of the polynomial. State the degree of the polynomial.
14) Find the zeros of the graph and state their multiplicity, then write an equation for the graph. Multiply and simplify your answer.
Solve the following problems without a calculator. Check your answer by solving with a calculator.
15) 16) 17)
18) 19) 20)
21) 22) 23)
24) 25)
26) Mr. Coulson likes to mix up his own fertilizer to keep his yard looking green and pristine. As we all know, fertilizer is made up of Nitrogen (N), Phosphorus (P) and Potassium (K). The proportion of each component for a given type of fertilizer is shown on the label as three numbers in N-P-K order. For example, a 5-10-15 fertilizer has 5/30 or 1/6 N, 10/30 or 1/3 P and 15/30 or 1/2 K. Farmer C. wants to put down 120 lbs of 12-5-7 fertilizer, but only has bags containing 10-10-10 from Abernathy Seed Company, 30-5-5 from Badlands Lawn Care, and 20-10-20 from Crabgrass Corporation. How much of these fertilizers does he need to blend to get the proportion of N-P-K he desires?
SKILL REVIEW
Below are skills that you will need in order to do well on Chapter 3. You are responsible for the information on this sheet. If you have any questions you need to come see me in ac lab or before/after school.
Concept #1: Properties of Exponents
1. 2. 3.
4. 5. 6.
7. 8. 9.
10. 11. 12.
Concept #2: Logarithmic and Exponential form
Write each log function in exponential form and write each exponent in logarithmic form.
13. log2 8 = 3 14. 15. 43 = 64
Concept #3: Evaluate
16. log232 17. log27 3 18. Log5125
Concept #4: Expanding/Condensing Logs
Expand the following logs
19. log3 4x 20. 21. 22.
Condense the following logs
23. log47 – log410 24. 6logx + 4 log y
25. 5log2 +7logx -4logy 26. 6log2 – 4 logy
Chapter 3: Exponential & Logarithmic Functions
Essential Questions
What is an exponential function?
What is exponentiating?
What is a logarithmic function?
How are exponential and logarithmic functions similar to other functions that we have studied?
How are exponential and logarithmic functions different from other functions?
Learning Targets
Determine whether a function is exponential or logarithmic.
Graph all exponential and logarithmic functions.
Solve exponential and logarithmic equations.
Determine whether a set of data can best be modeled using a linear, polynomial, rational, exponential, or logarithmic function.
Homework
Sections 3.1 - Exponential Functions & Graphs
Graph the function. Identify any asymptotes.
1. 2. 3. 4.
For problems 5-8, match the function to the graph.
(a) (b) (c) (d)
5. 6. 7. 8.
Sketch the graph of the function.
9. 10. 11. 12.
13. 14. 15.
16. Compound Interest
Complete the table for balance A using the appropriate compound interest formula.
P = $1000 r = 6% t = 10 years
A
17. Radioactive Decay
Let Q represent a mass of Carbon 14 (14C), in grams, whose half-life is 5730 years. The quantity present after t years is given by .
(a) Determine the initial quantity (t = 0)
(b) Determine the quantity present after 2000 years.
(c) Graph the function Q over the interval from t = 0 to t = 10,000
(d) When will this quantity of 14C be 0 grams? Explain your answer.
18. Population Growth