Unit 1 Equations, Inequalities, Functions - Home - … · 2016-12-12 · Algebra 2, Pages 1 coordinate axes with labels and scales. MAFS.912.A MAFS.912.A Unit 1 Equations, Inequalities,

Unit 1

Equations, Inequalities, Functions Algebra 2, Pages 1-100

Overview:

This unit models real-world situations by using one- and two-variable linear

equations. This unit will further expand upon pervious knowledge of linear

relationships by way of inverse functions, composite functions, piecewise-

defined functions, operations on functions, and systems of linear equations

and inequalities.

Standards:

Standards in this Unit:

MAFS.912.A-CED.1.2 Create equations in two or more variables to represent relationships between quantities; graph equations on

coordinate axes with labels and scales.

MAFS.912.A-CED.1.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. MAFS.912.F-BF.2.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. MAFS.912.F-BF.2.4 Find inverse functions.

a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x³ or f(x) = (x+1)/(x–1) for x ≠ 1.

b. Verify by composition that one function is the inverse of another. c. Read values of an inverse function from a graph or a table, given that the function has an inverse. d. Produce an invertible function from a non-invertible function by restricting the domain.

MAFS.912.F-IF.3.7a Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

a. Graph linear and quadratic functions, and show intercepts, maxima and minima. b. Graph square root, cube root, and piece-wise functions, including step functions and absolute value functions.

MAFS.912.F-BF.1.1 Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. c. Compose functions.

MAFS.912.A-REI.3.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. (Tasks are limited to 3 by 3 systems in Algebra 2.)

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Embedded Assessment 1:

Equations, Inequalities, and Systems –

Gaming Systems (page 55)

Write an equation to represent a real-world scenario.

(Lesson 1-2)

Graph an equation to represent a real-world scenario.

(Lesson 2-1)

Evaluate an equation. (Lesson 1-2)

Write a system of inequalities to model given real-world

constraints. (Lesson 1-3)

Graph a system of inequalities to model constraints. (Lesson 2-2)

Shade the solution region on a graph that is common to all of the

inequalities. (Lesson 2-2)

Identify an ordered pair that satisfies constraints on a graph.

(Lesson 2-2)

Explain what an ordered pair represents in the context of a

situation. (Lesson 2-2)

Write an absolute value equation given real-world constraints.

(Lesson 1-3)

Solve an absolute value equation. (Lesson 1-3)

Interpret the solutions to an absolute value equation.

(Lesson 1-3)

Write a system of three equations to represent a real-world

scenario. (Lesson 3-2)

Solve a system of three equations. (Lesson 3-2)

Interpret the solution to the system. (Lesson 3-2)

Embedded Assessment 2:

Piecewise-Defined, Composite, and Inverse Functions

– Currency Conversion (page 99)

Define the units which represent the domain and range of a

real-world scenario. (Lesson 4-1)

Convert quantities. (Lesson 5-1)

Write and explain a composition of functions (Lesson 5-2)

Identify the domain and range of composed functions

(Lesson 5-2)

Use an inverse function. (Lesson 6-1)

Write a piecewise-defined function. (Lesson 4-1)

Write the domain of a piecewise-defined function using an

inequality, interval notation and set notation. (Lesson 4-1)

Write the range of a piecewise-defined function using set

notation. (Lesson 4-1)

Graph a piecewise-defined function (Lesson4-2)

Find the inverse of functions (Lesson 6-2)

Graph an absolute value function. (Lesson 4-3)

Describe the graph of a transformation of an absolute value

graph. (Lesson 4-3)

Interpret an absolute value graph (Lesson 4-3)

Vocabulary (Academic/Math):

Academic vocabulary: interpret, compare, contrast, feasible, confirm, prove

Math vocabulary: absolute value equation, absolute value inequality, constraints, consistent, inconsistent, independent, dependent,

ordered triple, Gaussian elimination, matrix, dimensions of a matrix, square matrix, multiplicative identity matrix, multiplicative

inverse matrix, matrix equation, coefficient matrix, variable matrix, constant matrix, piecewise-defined function, step function, parent

function, composition, composite function, inverse function

3

Support for Lesson1-1

Learning Targets for lesson are found on page 3.

Main Ideas for success in Lesson 1-1:

Create an equation in one variable from a real-world context.

Solve an equation in one variable.

Vocabulary used in this lesson includes: algebraic expression, evaluate, interpret

Standards for Mathematical Practice to be demonstrated are: Reason Abstractly And Quantitatively, Make Sense Of

Problems, Model With Mathematics

Practice Support for Lesson 1-1

This example is a model to help solve Practice problems #14-15.

EXAMPLE: When full, a water storage tank holds 17,000 gallons of water. The tank currently holds 9,000 gallons of water

and is being filled at a rate of 34 gallons per minute.

Part 1 - Write an equation that can be used to find h, the number of hours it will take to fill the tank from its current level.

Explain the steps you used to write your equation.

Solution:

The storage tank holds 17,000 gallons of water and is starting off at a level of 9,000 gallons of water. We must figure out

how many hours it will take to fill the reminder of the tank. Since the problem said that the tank is filled at a rate of 34

gallons per minute and we are told to use h to represent the number of hours we must take into account that there are

60 minutes in 1 hour. We will multiply 34, the number of gallons filled per minute, by 60, the number of minute in an

hour. 34(60h) + 9,000 = 17,000.

60 multiplied by h will represent the total number of minutes necessary to fill the remainder of the tank. This is

multiplied by 34 because for each minute that passes, 34 gallons of water are added to the tank.

9,000 is added to this value because it is an initial amount that is already sitting in the tank.

When we add both the product of 34(60h) and the 9,000 it will equal the total capacity (17,000) of the water storage

tank.

Part 2 - Solve your equation from above, and interpret the solution.

Solution:

Original

Distributive property

Subtraction property of equality

Combine like terms

Division property of equality

To interpret my answer I want to describe what my value of 3.9 represents in terms of my problem situation. It will take

approximately 3.9 hours to fill the remainder of the water storage tank to capacity.

4

Practice Support for Lesson 1-1 Continued:

This example is a model to help solve Practice problem 16.

EXAMPLE The movie theater is open 7 days a week. The theater has 10 ticket booths, and each booth has a ticket seller

from 11:00am to 10:00pm. On average, ticket sellers work 25 hours per week. This situation will be used for all

examples to follow.

Write an equation that can be used to find t, the minimum number of ticket sellers the theater needs. Explain the steps

you used to write your equation.

Solution:

The minimum number of ticket sellers, t, times 25 hours per week equals the total number of hours worked by the ticket

sellers per week. This needs to equal the total number of hours the theater is open. The problem stated that each of the

10 booths is manned at all times. If the theater is open from 11:00am to 10:00pm, it is open for 11 hours per day and

since it is open 7 days a week that means the theater is open for a total of 77 hours per week (11 times 7).


EXAMPLE Solve the equation from above and interpret the solution. Refer to the above example.

Solution:

Original

Distributive property

Division property of equality

Since t represents the minimum number of ticket sellers, t must be a whole number, it represents a number of people. It

is not okay to round down to 30 because then we would be short a ticket seller at some point throughout the week.

Therefore, we must round up to 31 to ensure that we will have enough. 31 is the fewest number of ticket sellers the

theater needs.


EXAMPLE The theater plans to hire 15% more than the minimum number of ticket sellers needed in order to account for

sickness, vacation, and lunch breaks. How many ticket sellers should the park hire? Explain. Refer to the above example.

Solution:

We found earlier that the minimum number of ticket sellers needed was 31. We must now find what 15% of 31 is. I must

convert 15% to its equivalent decimal by dividing 15 by 100 to get 0.15 and then multiply it by our 31 people.

Again, we want to round up to the next whole number since we are representing people. Therefore the theater will

want to add an additional 5 people. The theater will then be hiring a total of 36 ticket sellers (31 + 5).

5

Additional Practice:

Answers to additional practice:

6



Main Ideas for success in 1-2:

Create equations in two variables to represent relationships between quantities.

Graph two-variable equations.

Vocabulary used in this lesson includes: independent variable, dependent variable, linear equation, y-intercept, slope

Standards for Mathematical Practice to be demonstrated are: Reason Abstractly, Construct Viable Arguments, Reason

Quantitatively.



EXAMPLE: The water storage tank is periodically emptied and cleaned to prevent any algae growth. The label on the

chemical states to add 4 fluid ounces per 1,000 gallons of water.

Make a table that shows how much of the chemical to add for other water tanks holding 5,000; 10,000; 15,000; and

20,000 gallons of water.

Solution:

Gallons of water Process Fluid ounces of chemical

1,000

cross multiply

division property

4 fluid ounces

5,000

20 fluid ounces

10,000

40 fluid ounces

15,000

60 fluid ounces

20,000

80 fluid ounces


EXAMPLE: Write a linear equation in two variables that models the situation above. Tell what each variable in the

equation represents. Refer to the above example.

Solution:

Since the problem states that we need to pour 4 fluid ounces of the chemical in for every 1,000 gallons that is equivalent

to 0.004 gallons of the chemical for every 1 gallon of water. This is from dividing 4 by 1,000. X represents the number of

gallons the water tank holds, input. Y represents the total number of fluid ounces of the chemical to be added, output.

7

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EXAMPLE: Graph the equation. Be sure to include titles and use an appropriate scale on each axis. Refer to the example

above.

Solution:

Using the table from earlier, we designated x as being the amount of water in a storage tank and y as being the amount

of chemical added in fluid ounces to the water tank. We can take these values from our table and plot them as ordered

pairs. (x, y). (1,000, 4), (5,000, 20), (10,000, 40), (15,000, 60), (20,000, 80). This will represent the line of our equation.


EXAMPLE: What is the slope and y-intercept of the graph? What do they represent in the situation? Refer to the above

example.

Solution: Our equation from above is in slope-intercept form therefore we can easily identify the slope and y-intercept.

Slope intercept form is where m is the slope and b is the y-intercept. Since our equation is

we know that the slope is 0.004 and the y-intercept is 0, because there is no constant term at the end of our

equation. A slope of 0.004 means that for each additional gallon of water the tank holds we need to add in 0.004

fluid ounces of the chemical. A y-intercept of 0 means that when the amount of water in the tank is 0 gallons that

we need to add in 0 fluid ounces of the chemical.


EXAMPLE: An employee adds 62 fluid ounces of the chemical to a water storage tank that holds 16,000 gallons of water.

Did the employee add the correct amount? Explain.

Solution: No, substituting 16,000 for x in the equation and solving for y shows that the employee should have added 64

fluid ounces of the chemical.

equation

evaluate when x = 16,000

amount of fluid ounces of the chemical to be added

Chemical Label

8


Answers to Additional Practice:

1-5

1

.

2

3

4

5

1

2

3

4

5

9




Write, solve, and graph absolute value equations.

Solve and graph absolute value inequalities.

Vocabulary used in this lesson includes: absolute value equation, absolute value inequality, compare, contrast

Standards for Mathematical Practice to be demonstrated are: Reason Abstractly, Reason Quantitatively, Make Sense

Of Problems, Critique The Reasoning Of Others, Model With Mathematics

Practice Support for Lesson1-3

This example is a model to help solve Practice problems 6 and 7.

EXAMPLE: Solve the absolute value equation

original (the absolute value expression is already isolated)

or write as two equations using the definition of absolute value

or add 8 to both sides

or

divide both sides by 4

There are two solutions:

and

.


EXAMPLE: Solve the absolute value equation

original

isolate the absolute value expression by subtracting 10 on both sides

No solution

Since the absolute value of any expression will equal a positive value, there is no solution. There is no value to plug in for

x that will provide us with an output of -5.

This example is a model to help solve Practice problems 10.

EXAMPLE The Little River Dam has a flow rate that can vary up to 74 gallons per minute from the target flow rate of 1,200

gallons per minute. Write and solve an absolute value equation to find the extreme values of the flow rate for the Little

River Dam.

original

or use the definition of absolute value to make two equations.

or add 1,200 to both sides.

The two extreme values that represent the flow rate for the Little River Dam are 1,126 and 1,274 gallons per minute.

10

Practice Support for Lesson Continued1-3


EXAMPLE: Solve the absolute value inequality. Graph the solution on a number line.

original (the absolute value expression is already isolated)

or write two inequalities (notice the change in inequalities and signs)


or divide both sides by 3



original

add 4 to both sides to isolate the absolute value expression

or write two inequalities (note sign changes)


or divide both sides by 3



original

add 10 to both sides to isolate the absolute value expression

write the compound inequality

add 4 to all parts

divide by 2 on all parts

4 0

-7 11

-1 11

11


1.

2.

3.

4.


1a.

1b.

2.

3.

a

b

c

.

c.

4.

1

Should there be a period

after 43? Take the extra

space out between b and

values. Define what y

intercept is

Support for Lesson 2-1


Main Ideas for success in 2-1

Write equations in two variables to represent relationships between quantities.

Graph equations on coordinate axes with labels and scales.

Vocabulary used in this lesson includes function, rate of change, and slope intercept form.

Standards for Mathematical Practice to be demonstrated are reason quantitatively, model with mathematics, and

make sense of problems


This example is a model to help solve Practice problems 15-20.

EXAMPLE:


1. rite the equation of the line with y-intercept -2 and a slope of

. Graph the equation.

Since you are given the slope and the y-intercept you can plug the values directly into slope intercept form:

Remember m represents the slope and b represents the y-intercept, so

. Plug the m and b values

into the equation.

.

2. When graphing BEGIN with the b value. Since -2 represents the y-intercept, I will move down two units on the y

axis and plot a point. Slope (m) is what we MOVE from the y-intercept. Slope represents the rise over the run, so

the numerator is what you move up or down (rise), and the denominator is what you move left or right (run).

Since our slope is positive, we will move up four and right 3 from our y-intercept of -2.

A quick check to make sure your answer is reasonable: positive slopes should rise from left to right, and negative

slopes should fall from left to right.

EXAMPLE:

2

On page 2 include an

explanation of y

intercept (slope) form

Practice Support for Lesson Continued 2-1


EXAMPLE

Write the equation of the line that through the point (-1,-4) with a slope of 2. Graph the equation.

Use the coordinate and the slope to find the y-intercept. In the coordinate (-1,-4), -1 is an x value and -4 is a y value.

Remember from the previous example that the slope 2 will be m.

Start with slope-intercept form

Plug in all given values: y=-4, m=2, and x=-1. The only unknown value in the equation is b or the y-intercept.

Isolate b

Substitute b back into slope-intercept form with the given slope.

Graph

BEGIN at y-intercept: -2 (b)

MOVE

(m) check to make sure your line and your slope match. (Positive slope should rise from left to right)


EXAMPLE:

Graph the function

. F(x) is the function notation for y.

So replace f(x) with y.

Get into slope-intercept form

Distribute the

3

Lesson support heading

continued should be at

the top of the page.

Should that 13 be there

with a period before the

word “make”

Move this bullet &

example 19 to page 4

Combine like terms

Slope (m) =

and y-intercept (b)= 3

BEGIN at the y-intercept 3 and MOVE the slope

. Make sure when you are moving the slope from the y-intercept that

you only move ONE direction negative, not both numbers in the negative direction. For example you could move up one

and left three or down one and right three but NOT down one and left three. Two negatives make a positive.

Graph

This example is a model to help solve Practice problems 18

EXAMPLE:

Rachel already has 8,524 frequent flyer miles, and she will earn 1,776 more miles from her round-trip flight to Seattle. In

addition, she earns 1 frequent flyer mile for each dollar he charges on her credit card.

Write the equation of a function f(d) that represents the total number of frequent flyer miles Rachel will have after her

trip if she charges d dollars on her credit card.

What does Rachel already have? 8,524 frequent flier miles and she is about to get 1,776 more for her Seattle trip. So

those will be added together 8,524+1,776= 10,300 frequent flier miles

How can she get additional miles? 1 mile per dollar charged on her credit card. The amount of miles is dependent on the

number of dollars charged, so $1d or just d.

So the function modeling her frequent flier miles after her trip would be:

This example is a model to help solve Practice problems 19

EXAMPLE:

Graph which is the same as . Identify the slope and the y-intercept so you can

graph the function. The slope is 1, the coefficient of d, and the y-intercept is 10,300.

4

Practice Support for Lesson Continued:

This example is a model to help solve Practice problems

EXAMPLE

The numbers on the x-axis represent the dollars Rachel

charged on her credit card. The dollars could be counted in

several different increments, but it would be good to count by

at least $30.

5

Practice Support for Lesson Continued 2-1:

EXAMPLE:

Reason quantitatively. How many dollars will Rachel need to charge on her credit card to have a total of 13,000

frequent flyer miles? Explain how you determined your answer.

Use the equation created in number 18: to solve this question.

d is the number of dollars charged on Rachel’s credit card and f(d) is the number of frequent flier miles given

the dollars spent. So for this problem 13,000 will be substituted for f(d) and d will be our unknown value.

Solve for d

This means that when Rachel charges $2700 on her credit card, she will have 13,000 frequent flier miles.

6


Additional Practice Answers:

7

8



Represent constraints by equations or inequalities.

Use a graph to determine solutions of a system of inequalities.

Vocabulary used in this lesson includes inequalities, and constraints

Standards for Mathematical Practice to be demonstrated are construct viable arguments, attend to precision, use

appropriate tools strategically, and model with mathematics.



EXAMPLE:

Graph these inequalities on the same grid, and shade the solution region that is common to all of the inequalities:

. The solutions to these inequalities lie where the three shadings overlap.

1. Graph each inequality on the same graph and shade.

A.

graph shade the side of the line where the y values are greater than one. The line is solid because

the coordinates on the line are also solutions to the inequality.

B.

Use the slope and the y-intercept to graph (as described in lesson 2-1)

Shade where the y values are less than the graphed line.

9


C.

Graph the line and shade where the x values are less than 4

The final graph should look like this; the solutions lie where the shading overlaps in the highlighted triangle.


EXAMPLE

Using the final graph from question 19, choose two points from the highlighted region where all three shadings overlap.

The coordinates in this region will make all three inequalities true. Remember since the lines are solid, that coordinates

on the lines can be used. So (3,1) and (4,2) can be solutions.


EXAMPLE

A snack company plans to package a mixture of almonds and peanuts. The table shows information about these

types of nuts. The company wants the nuts in each package to have at least 30 grams of protein and to cost no more

than $2. Use this information for Items 21–23.

Model with mathematics. Write inequalities that model the constraints in this situation. Let x represent the number

of ounces of almonds in each package and y represent the number of ounces of peanuts. Start by writing an

inequality to represent the constraints on each of the following: protein, cost, x, and y. Remember almonds are x

and peanuts are y. Protein: , Cost , Almonds (can’t have negative almonds or

peanuts) Peanuts

10

Practice Support for Lesson Continued:


EXAMPLE

Graph the constraints. Shade the solution region that is common to all of the inequalities.

Apply the same process as number 19. Graph and shade each inequality on the same graph and see where the shading

overlaps. Replace inequality with an equal sign and get into slope-intercept form so you can graph line.

a. b. c. d.

Solve for y

This example is a model to help solve Practice problem 23

EXAMPLE:

a. Identify two ordered pairs that satisfy the constraints. Pick any two point from the region where the shading

overlaps. (2,6) and (4,4)

b. Reason quantitatively. Which ordered pair represents the more expensive mixture? Which ordered pair

represents the mixture with more protein? Explain your answer.

Plug both points into both the cost and protein equation from number 21 to determine higher amount.

Protein: (2,6) (4,4)

g g

Cost: (2,6) (4,4)

11

1

After plugging in both points look to see which point produced the larger number. For protein, the coordinate

(2,6) has 44 grams compared to (4,4) which only has 40 grams. So (2,6) represents the mixture with more protein.

Looking at cost, (4,4) is one dollar while (2,6) is only 90 cents. Therefore (4,4) is the more expensive mixture.

12



1



Main Ideas for success in 3-1

Use graphing, substitution, elimination to solve systems of linear equations in two variables.

Formulate systems of linear equations in two variables to model real-world situations.

Vocabulary used in this lesson includes solutions of a system of linear equations, consistent, inconsistent,

independent, dependent, substitution method, and elimination method.

Standards for Mathematical Practice to be demonstrated are reason abstractly and quantitatively, make sense of

problems and persevere in solving them, model with mathematics, construct viable arguments, and make use of

structure.



EXAMPLE:

Solve the system by graphing.

Get both equations into slope-intercept form in order to graph them.

Use the slope and the y-intercept to graph each line on the same coordinate plane. Remember to begin by

plotting the y-intercept, and move the slope (

from the y-intercept for each line.


EXAMPLE:

Solve the system using substitution.

This method is good for solving a system when there is a variable with a coefficient of one. The top equation is

already solved for x, which has a coefficient of one. Substitute the into the bottom equation where

the is.

distribute negative

combine like terms

solve for y

The solution to any system is the point that is on both

lines. The two lines intersect at (-1,1) making it the

solutions to this system of equations.

2


Remember the solution to any system is a coordinate that satisfies both equations or is on both lines. So the answer

should always be an (x,y) coordinate. Now that the y value is known, it can be used to solve for x. Plug the y value

into either equation in order to find the value for x.

Write the answer as an ordered pair: (-40,-9) Could not remove box


EXAMPLE:

Solve the system by elimination

In elimination method, the goal is to eliminate one of the variables. This means that the coefficient must become

zero; this occurs when the coefficients of one variable become the same number with the opposite sign. It does not

matter which variable is chosen to eliminate. In this case both equations will have to be multiplied by a value in

order to get a coefficient of zero. Let’s eliminate x in this example.

Multiply both equations in order to eliminate x:

Add new equations together:

+

Solve for y:

Plug y into one of the original equations:

Solve for x:

Write the answer as coordinate:

3

Practice Support for Lesson Continued:3-1


EXAMPLE

Make sense of problems and persevere in solving them. At one company, a level I engineer receives a salary of

$30,000, and a level II engineer receives a salary of $42,000. The company has 9 level I engineers. Next year, it

can afford to pay $282,000 for their salaries. Write and solve a system of equations to find how many of the

engineers the company can afford to promote to level II.

Define variables and create two equations

Let x= level I engineers Let y= level II engineers

Equation representing engineers:

Equation representing salaries:

Solve the system of equations using the method of your choice

(substitution method)

Remember we defined y as the number of level II engineers, so the company can promote 1 level one engineer.


EXAMPLE:

Answers may vary as there is no one correct way to solve a system. Substitution was the chosen method due to

the fact that the coefficients were already one in the engineer’s equation.

4


5


6




Solve systems of three linear equations in three variables using Gaussian elimination

Formulate systems of three linear equations in three variables to model a real-world situation.

Vocabulary used in this lesson includes ordered triple, Gaussian elimination

Standards for Mathematical Practice to be demonstrated are make sense of problems, and use appropriate tools

strategically



EXAMPLE:

Solve the system using substitution.

Solve the first equations for the variable with a coefficient of one.

Substitute the expression for z into the second equation, and solve for y.

Substitute the expression for z into the third equation, and solve for y.

Plug y into the previous equation to find the value of x

7


Plug x and y into the first equation to find z.

Write answer as an ordered pair (x,y,z)


EXAMPLE:

Solve the system using Gaussian elimination.

When solving a three variable system using the elimination method, the goal is to get the system to only have

two variables. Then proceed to solve the two-variable system using the method of your choice.

Group the first two equations and the second two equations and eliminate the same variable in each group

For this example eliminate x

Add equations together

+

+

Group the two resulting equations and solve the two-variable system

Plug z into the top equation in order to find the value of y.

8


Plug y and z into one of the first equations in order to find the value of x.

–

Write solution as an ordered pair (x,y,z).


EXAMPLE

Write a system of equations that can be used to determine small, medium, and large, the cost in dollars of each.

Define the variables: x= small cups sold, y= medium cups sold, z= large cups sold

Create three different equations to represent the sales; this will become your system.

Morning sales:

Afternoon sales:

Evening sales:


EXAMPLE

Solve your equation and explain what it means in the context of this situation.

Choose a method to solve the system. This will model elimination.

Remember to group the first two equations and the last two equations and eliminate the same variable in each

group. The goal is to get this three variable system down to a two variable system so it is easier to solve.

+

+

9


Choose a method to solve the two variable system

Plug z into one of the two-variable equations in order to find the value of y.

Plug y and z into one of the three-variable equations to find the value of x.

The solution to this system is . In this context this solution means that a small cup sold for $4.26, a

medium cup sold for $2.94, and a large cup sold for $5.33.


EXAMPLE

Which method did you use to solve the system? Explain why you used that method.

Answers here may vary. In the example, elimination was used in order to reduce the three-variable system to a

two-variable system so it was easier to solve.

10


11


1




Add, subtract, and multiply matrices.

Use a graphing calculator to perform operations on matrices.

Vocabulary used in this lesson includes: matrix, dimensions of a matrix, entries

Standards for Mathematical Practice to be demonstrated are: Make Use Of Structure, Express Regularity In Repeated

Reasoning, Critique The Reasoning Of Others



EXAMPLE: Use these matrices to answer items 12-17. What are the dimensions of A?

2 x 3 number of rows x number of columns


EXAMPLE: Use these matrices from above, what is the entry with the address ?

3. Look at matrix C for the entry on the second row and the first column


EXAMPLE: Use these matrices from above, find .

Both matrices have the same dimensions, they can be added together. Add the entries that share the same address.

and


EXAMPLE: Use these matrices from above, find

Both matrices have the same dimensions, they can be subtracted together. Subtract the entries that share the same

address. and

2



EXAMPLE: Use these matrices from above, find if it is defined.

The dimensions of are 2 x 3 and the dimensions of are 2 x 2. In order to multiply matrices, the number of columns in

is equal to the number of rows in . Since there are 3 columns in and only 2 rows in , we cannot multiply the

matrices and . Therefore it is not defined.


EXAMPLE: Use these matrices from above, find if it is defined.

The dimensions of are 2 x 2 and the dimensions of are 2 x 2. Since there are 2 columns in and 2 rows in , we can

multiply the matrices and because it is defined.

Find the entry in row 1, column 1 of . Use row 1 of and column 1 of . Multiply the first entries and the second

entries. Then add the products.





Find the entry in row2, column 2 of . Use row 2 of and column 2 of . Multiply the first entries and the second


Solution:

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Solve systems of two linear equations in two variables by using graphing calculators with matrices.

Solve systems of three linear equations in three variables by using graphing calculators with matrices.

Vocabulary used in this lesson includes: square matrix, multiplicative identity matrix, multiplicative inverse matrix,

matrix equation, coefficient matrix, variable matrix, constant matrix

Standards for Mathematical Practice to be demonstrated are: Construct Viable Arguments, Make Sense Of Problems,

Model With Mathematics, Use Appropriate Tools Strategically, Critique The Reasoning Of Others.



EXAMPLE: Use a graphing calculator to find the inverse of the matrix

.

Input the matrix into the graphing calculator. 2nd

MATRIX EDIT ENTER2X2

ENTER4ENTER8ENTER5ENTER2ENTER2nd QUIT2

nd MATRIX ENTER

ENTER

This is the inverse of the given matrix. To change from decimals to fractions press

MATHENTERENTER to get

This example is a model to help solve Practice problems 20-21.

EXAMPLE: Write a matrix equation to model the system. Then use the matrix equation to solve the system.

First write the system as a matrix equation.

=

Matrix A represents the coefficients in each equation, Matrix X represents the variables in each equation, Matrix B

represents the constants in each equation.

To solve the matrix equation, we can multiply both sides of the equation by the multiplicative inverse matrix of A. This

will give us . Use a graphing calculator to find as you did in the earlier problem.

Now you can find the matrix product of .

=

(Use the graphing calculator to multiply)

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EXAMPLE David has 3 euros and 9 British pound worth a total of $75. Daniel has 8 euros and 5 British pounds worth a

total of $67.

a.) Write a system of equations to model this situation, where x represents the value of 1 euro in dollars and y represents

the value of 1 British pound in dollars.

b.) Write the system of equations as a matrix equation.

=

c.) Use the matrix equation to solve the system. Then interpret the solution.

As in the earlier examples, use the graphing calculator to input matrix A (the coefficient matrix) and matrix B (the

constant matrix).

=

=

(4, 7); A euro is worth $4, and a British pound is worth $7.

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Graph piecewise-defined functions.

Write the domain and range of functions using interval notation, inequalities, and set notation.

Vocabulary used in this lesson includes: piecewise-defined function, domain, range

Standards for Mathematical Practice to be demonstrated are: Make Use Of Structure, Model With Mathematics,

Critique The Reasoning Of Others



EXAMPLE: Graph the piecewise-defined function. Then write its domain and range using inequalities, interval notation,

and set notation.

The domain refers to all defined x-values in the function. The function is defined for all x-values from .

Domain: Inequalities , Interval Notation , Set Notation

The range refers to all defined y-values in the function. The function is defined for all y-values that are 0 and greater.

Range: Inequalities , Interval Notation , Set Notation


EXAMPLE: The range of a function is all real numbers greater than or equal to -12 and less than or equal to 3. Write the

range of the function using and inequality, interval notation, and set notation.

Range: Inequalities , Interval Notation , Set Notation

Since the description says “or equal to” we are sure to use brackets, not parentheses, and include the ‘or equal to’ part

under our inequality signs.

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EXAMPLE: Evaluate the piecewise function for .

To evaluate we must first figure out where it is defined (which of the 3 parts has a domain that includes -1). We

see it is included in the middle piece. We can now plug in -1 for x and evaluate to get

Similarly, is defined for the middle piece. We can then evaluate

Lastly, is defined for the third piece of the function, when evaluated,

Note that the first piece of the function was not used at all in this case, that is okay.


EXAMPLE A cell phone company charges customers a $15 fee plus $0.10 per text for the first 1,000 texts and $0.20 per

text for each text sent after the 1000th

text.

a.) Write a piecewise function that can be used to determine a customer’s monthly bill for using texts per month.

This function will have two pieces to it since there are two possible pricing scenarios. Either the customer sent fewer

than (or exactly) 1,000 texts, or more than 1,000 texts in the month.

The first piece, comes from the initial $15 fee applied monthly to everyone plus the charge of $0.10 per

text. This piece is only used when the total number of monthly texts is 1,000 or fewer.

The second piece, comes from the scenario when the customer sends more than 1,000 texts in

a given month. The $115 represents the total charge for the initial $15 fee applied to all customers monthly plus the

$100 they are charged for sending their first 1,000 texts. The second half represents the cost charged for the

additional texts (after the first 1,000). Take the total number of texts sent, x, and subtract 1,000 (since we are already

paying for those in our $115). The number of texts left over after the subtraction is the amount that will be charged at

the $0.20 rate.

b.) Graph the piecewise function.

c.) A customer sends 1234 texts in one month. How much will the customer owe the cell phone company? Explain

how you determined your answer.

Since 1234 is more than 1,000, I need to evaluate in the bottom piece of my function.

The customer will owe the company $161.80

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Graph step functions and absolute value functions.

Describe the attributes of these functions.

Vocabulary used in this lesson includes: step function

Standards for Mathematical Practice to be demonstrated are: Reason Abstractly, Make Sense Of Problems, Reason

Quantitatively, Construct Viable Arguments, Make Sense Of Problems And Persevere In Solving Them



EXAMPLE: A step function known as the ceiling function, written , yields the value that is the least integer

greater than or equal to .

a.) Graph this step function. (See to the right)

b.) Find .

The smallest integer that is larger than1.7 is 2. Therefore .

Similarly, the smallest integer that is larger than -4.1 is -4. Therefore .


EXAMPLE: Use the information below to aid in answering questions 14 and 15.

A day ticket for admission to a water park is $45 for children at least 5 years old and less than 13 years old. A day ticket for

children at least 13 years old and less than 18 years old costs $55. A day ticket for adults, anyone 18 years or older, costs $65.

Write the equation of a step function that can be used to determine the cost in dollars of a day ticket for the water park

for a person who is years old.


EXAMPLE: Graph the step function you wrote above.

Ticket Prices By Age

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EXAMPLE Use the absolute value function for questions 16-19.

Graph the absolute value function.


EXAMPLE: What are the domain and range of the function?

Domain: ; Range:

This example is a model to help solve Practice problem18.

EXAMPLE What are the coordinates of the vertex of the function’s graph?

(4, 0) It is the parent function shifted to the right 4 units.


EXAMPLE: Write the equation for the function using piecewise notation.

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Main Ideas for success in:

Identify the effect on the graph of replacing by .

Find the value of , given these graphs.

Vocabulary used in this lesson includes: vertical/horizontal translation, reflection, vertical/horizontal stretch, shrink

Standards for Mathematical Practice to be demonstrated are: Model With Mathematics, Reason Abstractly And

Quantitatively, Express Regularity In Repeated Reasoning, Attend To Precision

Practice Support for Lesson


EXAMPLE: The graph of is the graph of translated 4 units to the left, write the equation of .

Since translating left and right appears in the equation on the inside of the absolute value bars and where

shifts the graph nits to the left, our equation must be .


EXAMPLE: Describe the graph of as one or more transformations of the graph of .

A negative sign in the front of the equation means that the graph is reflected across the x-axis. The 3 in the front of the

equation in front of the absolute value bars will cause the function to be vertically stretched by a factor of 3.


EXAMPLE: What are the domain and range of ? Explain.

Since the two transformations applied to the parent function are a horizontal shift right 2 units and a vertical shift up 3

units. The graph will open upward and have a vertex of (2, 3). Domain of the parent function would be the

domain of remains the same it is still defined for all values of x. Range of the parent function would be

however since we have shifted our function up 3 units it is no longer defined from and the

new range becomes .


EXAMPLE: Graph each transformation.

a.) b.) c.)

Reflected across the x-axis. Horizontally shrink by a factor of ½ Reflected across the x-axis.

Horizontal translation left 3 Vertical translation down 1 Horizontal translation right 2

Vertical translation down 2 Vertical stretch by a factor of 4

Vertical translation up 3

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EXAMPLE: Write the equation for each transformation of described below.

a.) Translate right 7 units, stretch vertically by a factor of 3, and translate up 17 units.

b.) Translate left 3 units, stretch horizontally by a factor of 8, and reflect over the x-axis.

c.)

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Documents

Unit 1 Equations, Inequalities, Functions - Home - … · 2016-12-12 · Algebra 2, Pages 1 coordinate axes with labels and scales. MAFS.912.A MAFS.912.A Unit 1 Equations, Inequalities,