Topic 1: Linear Functions and Systems - Weebly

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Chapter 1 Note Packet

Algebra 2 Name: ________________________________________

Period: ______

Topic 1: Linear Functions and Systems

Date Section Topic HW Due Date

1.0A Domain and Range

1.1A Key Features of Graphs

1.1B Key Features of Graphs

1.0B Absolute Value Functions

1.3 Piecewise-Defined Functions

1.5 Solving Equations and Inequalities by Graphing

1.6A Linear Systems

1.6B Linear Systems

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1.0A Domain and Range Date:

Domain Range Function

Interval Notation Set Notation/Set Builder Notation

Discrete Graph Continuous Graph

Ex 1 Find the domain and range for each.

A. B.

Try It!

C. D.

Ex 2 Find the domain and range for each. Try It!

A. B. C. D.

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Ex 3 Find the domain and range for each continuous graph.

A. B. C.

Try It!

D. E. F.

G. H. I.

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Ex 4 Find the domain and range of each continuous graph.

A. B. C.

Try It!

D. E. F.

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1.1A Key Features of Graphs

Increasing Decreasing Intercepts

Finding using graph

Finding algebraically

Intervals where positive Intervals where negative

Ex 1 A. What are the domain and range of the function Try it! B. What are the domain and range of the defined by 𝑦 = 𝑥2 − 3? function defined by 𝑦 = |𝑥 − 4| ?

Interval Notation: Interval Notation:

Set-Builder NoTation: Set-Builder Notation:

Ex 2 A. What are the x- and y-intercepts of the graph of 𝑦 = |𝑥| − 3? Find algebraically.

Try It! B. What are the x- and y-intercepts of 𝑔(𝑥) = 4 − 𝑥2?

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C. What do the x- and y-intercepts represent about the situation graphed below?

Ex 3 For what intervals is the graph positive? For what intervals is the graph negative?

A. 𝑓(𝑥) = 𝑥2 − 9 B. 𝑔(𝑥) = 3𝑥 − 2

Try It!

For what interval(s) is the graph positive? For what interval(s) is the graph negative?

C. ℎ(𝑥) = 2𝑥 + 10 D. 𝑓(𝑥) = −𝑥2 + 4

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1.1B Key Features of Graphs

Ex 4 For what values of x is the graph increasing? For what values is it decreasing?

A. 𝑔(𝑥) = 2 − |𝑥| B. 𝑦 = 2𝑥 − 1

Try It!

For what values of x is each function increasing? For what values of x is it decreasing?

C. 𝑓(𝑥) = 𝑥2 − 4𝑥 D. 𝑓(𝑥) = −2𝑥 − 3

Ex 5 Find the function values given the graph. Try it!

A. find 𝑓(−2) D. find 𝑓(−5)

B. find 𝑓(0) E. find 𝑓(−3)

C. find 𝑓(1) F. find 𝑓(0)

Ex 6 A. Find the minimum and maximum values of Try It! B. Find the max and min values on [-2,3]

f(x) on the interval [-3, 2].

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1.1 Additional Practice

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1.0B Absolute Value Functions

Vertex Form

𝑦 = ±𝑎|𝑥 − ℎ| + 𝑘

Ex 1 Graph each absolute value function:

A. 𝑦 = |𝑥 − 1| − 4 B. 𝑦 = |𝑥 + 3| + 1 C. 𝑦 = |𝑥 + 2|

Try It!

D. 𝑦 = |𝑥 + 1| − 5 E. 𝑦 = |𝑥 − 1|

Ex 2 Graph each absolute value function

A. 𝑦 = −|𝑥 − 3| − 1 B. 𝑦 = −|𝑥 + 2| + 5 C. 𝑦 = −|𝑥| + 3

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TRY IT!

D. 𝑦 = −|𝑥 − 2| + 4 E. 𝑦 = −|𝑥 + 1| + 3

Ex 3 Graph each absolute value function.

A. 𝑦 =2

3|𝑥 − 1| − 4 B. 𝑦 =

1

2|𝑥 − 2| − 3 C. 𝑦 = −3|𝑥 − 1| + 5

TRY IT!

D. 𝑦 =5

3|𝑥 + 1| − 5 E. 𝑦 =

2

3|𝑥| − 3 F. 𝑦 = −

1

3|𝑥 − 1| + 4

Ex 4 Write the equation for the absolute value function show.

A. B. Try It! C.

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1.0B Additional Practice

A. B. C. D.

Write the equation for the graph shown. Also, write the range in interval notation.

E. F. G.

H. I. J.

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1.3 Piecewise-Defined Functions

Ex 1 Alani has a summer job as a lifeguard. She makes $8/h for up to 40 h each week. If she works more than 40 h,

she makes 1.5 times her hourly pay, or $12/h, for each hour over 40 h.

A. How could you make a graph and write a function that shows Alani’s weekly earnings based on the number of hours

she worked?

Try It! How much will Alani earn if she works

B. 37 hours? C. 43 hours?

Ex 2 A. How do you graph a piecewise defined function?

𝑓(𝑥) = {4𝑥 + 11 − 10 ≤ 𝑥 < −2

𝑥 + 1 2 < 𝑥 ≤ 10

What are the domain and range?

Over what intervals is the function increasing or decreasing?

Try It! Graph the piecewise-defined function. State the domain and range. State the intervals over which the

function is increasing and decreasing.

B. 𝑓(𝑥) = {2𝑥 + 5 − 6 ≤ 𝑥 ≤ −2−𝑥 − 4 1 ≤ 𝑥 ≤ 3

C. 𝑓(𝑥) = {3 − 4 < 𝑥 ≤ 0

−𝑥 0 ≤ 𝑥 ≤ 23 − 𝑥 2 < 𝑥 < 4

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Ex 3 What is the rule that describes the piecewise-defined function shown in the graph?

A. Try it! B. C.

Ex 5 A. The shipping cost of items purchased from an online store is dependent on the weight of the items. The

table below represents shipping costs y based on the weight x. Graph the function. What are the domain and range of

the function? What are the maximum and minimum values?

Try It! B. The table below represents fees for a parking lot. Graph the function. What are the domain and the

range of the function? What are the maximum and minimum values?

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1.3 Additional Practice

1)

2)

3)

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1.5 Solving Equations and Inequalities by Graphing

Ex 1 How can you use a graph to solve an equation?

A. Solve −3𝑥 + 20 = 5 by graphing. B. Solve |𝑥 − 4| =1

2𝑥 + 1 by graphing.

Try It! C. 5𝑥 − 12 = 3 D. −|𝑥 − 2| = −1

2𝑥 − 2

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Ex 2 How can you use a graph to solve an inequality?

A. 𝑥2 − 4 > 0

B. A motorcycle is 40 mi ahead of a car. The motorcycle travels at an average rate of 40 mph. The car travels at an

average rate of 60 mph. When will the car be ahead of the motorcycle?

Try It! C. 𝑥2 + 6𝑥 + 5 ≥ 0 D. 𝑥 + 3 > 7 − 3𝑥

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1.5 Additional Practice

1. −3|𝑥 + 3| + 2 = −1 2. −2

3|𝑥 − 4| − 1 = 3𝑥 − 2 3. 𝑥 + 3 > −4𝑥 − 2

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1.6A Linear Systems

Ex 1 What is the solution of the system of linear equations?

A. Use substitution to find a solution for x and y.

{𝑥 + 2𝑦 = 3𝑥 − 2𝑦 = 4

Try It! B. {2𝑥 + 𝑦 = −15𝑦 − 6𝑥 = 7

C. {3𝑥 + 2𝑦 = 56𝑥 + 4𝑦 = 3

Ex 2 A. Malcolm earns $20 per hour mowing lawns and $10 per hour waling dogs. His goal is to earn at least $200

each week, but he can work a maximum of 20 h per week. Malcolm must spend at least 5 h per week walking his

neighbors’ dogs. For how many hours should Malcolm work at each job in order to meet his goals?

Try It! B. Graph the set of all points that solve this system of linear inequalities.

{

2𝑥 + 𝑦 ≤ 14𝑥 + 2𝑦 ≤ 10

𝑥 ≥ 0𝑦 ≥ 0

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More Practice:

Solve each system.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

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1.6B Linear Systems

Solving a System of THREE Equations

Ex 3 What is the solution of the following systems?

A. {

2𝑥 + 𝑦 − 𝑧 = −10−𝑥 + 2𝑦 + 𝑧 = 3𝑥 + 2𝑦 + 3𝑧 = 13

B. {

2𝑥 − 𝑦 + 𝑧 = 3𝑥 + 𝑦 + 𝑧 = 5

−4𝑥 + 2𝑦 − 2𝑧 = 0

Try It!

C. {

𝑥 + 𝑦 + 𝑧 = 3𝑥 − 𝑦 + 𝑧 = 1𝑥 + 𝑦 − 𝑧 = 2

D. {

2𝑥 + 𝑦 − 2𝑧 = 3𝑥 − 2𝑦 + 7𝑧 = 123𝑥 − 𝑦 + 5𝑧 = 10

More Practice:

1. 2. 3.

4. 5. 6.

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1.6 Additional Practice

Solve the system of inequalities.

7. 8. 9.