Topic 1: Linear Functions and Systems - Weebly
Transcript of 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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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 π¦ = π₯2 β 3? function defined by π¦ = |π₯ β 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 π¦ = |π₯| β 3? Find algebraically.
Try It! B. What are the x- and y-intercepts of π(π₯) = 4 β π₯2?
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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. π(π₯) = π₯2 β 9 B. π(π₯) = 3π₯ β 2
Try It!
For what interval(s) is the graph positive? For what interval(s) is the graph negative?
C. β(π₯) = 2π₯ + 10 D. π(π₯) = βπ₯2 + 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. π(π₯) = 2 β |π₯| B. π¦ = 2π₯ β 1
Try It!
For what values of x is each function increasing? For what values of x is it decreasing?
C. π(π₯) = π₯2 β 4π₯ D. π(π₯) = β2π₯ β 3
Ex 5 Find the function values given the graph. Try it!
A. find π(β2) D. find π(β5)
B. find π(0) E. find π(β3)
C. find π(1) F. find π(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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1.0B Absolute Value Functions
Vertex Form
π¦ = Β±π|π₯ β β| + π
Ex 1 Graph each absolute value function:
A. π¦ = |π₯ β 1| β 4 B. π¦ = |π₯ + 3| + 1 C. π¦ = |π₯ + 2|
Try It!
D. π¦ = |π₯ + 1| β 5 E. π¦ = |π₯ β 1|
Ex 2 Graph each absolute value function
A. π¦ = β|π₯ β 3| β 1 B. π¦ = β|π₯ + 2| + 5 C. π¦ = β|π₯| + 3
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TRY IT!
D. π¦ = β|π₯ β 2| + 4 E. π¦ = β|π₯ + 1| + 3
Ex 3 Graph each absolute value function.
A. π¦ =2
3|π₯ β 1| β 4 B. π¦ =
1
2|π₯ β 2| β 3 C. π¦ = β3|π₯ β 1| + 5
TRY IT!
D. π¦ =5
3|π₯ + 1| β 5 E. π¦ =
2
3|π₯| β 3 F. π¦ = β
1
3|π₯ β 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 Alaniβs 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?
π(π₯) = {4π₯ + 11 β 10 β€ π₯ < β2
π₯ + 1 2 < π₯ β€ 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. π(π₯) = {2π₯ + 5 β 6 β€ π₯ β€ β2βπ₯ β 4 1 β€ π₯ β€ 3
C. π(π₯) = {3 β 4 < π₯ β€ 0
βπ₯ 0 β€ π₯ β€ 23 β π₯ 2 < π₯ < 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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1.5 Solving Equations and Inequalities by Graphing
Ex 1 How can you use a graph to solve an equation?
A. Solve β3π₯ + 20 = 5 by graphing. B. Solve |π₯ β 4| =1
2π₯ + 1 by graphing.
Try It! C. 5π₯ β 12 = 3 D. β|π₯ β 2| = β1
2π₯ β 2
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Ex 2 How can you use a graph to solve an inequality?
A. π₯2 β 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. π₯2 + 6π₯ + 5 β₯ 0 D. π₯ + 3 > 7 β 3π₯
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1.5 Additional Practice
1. β3|π₯ + 3| + 2 = β1 2. β2
3|π₯ β 4| β 1 = 3π₯ β 2 3. π₯ + 3 > β4π₯ β 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.
{π₯ + 2π¦ = 3π₯ β 2π¦ = 4
Try It! B. {2π₯ + π¦ = β15π¦ β 6π₯ = 7
C. {3π₯ + 2π¦ = 56π₯ + 4π¦ = 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
neighborsβ 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.
{
2π₯ + π¦ β€ 14π₯ + 2π¦ β€ 10
π₯ β₯ 0π¦ β₯ 0
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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. {
2π₯ + π¦ β π§ = β10βπ₯ + 2π¦ + π§ = 3π₯ + 2π¦ + 3π§ = 13
B. {
2π₯ β π¦ + π§ = 3π₯ + π¦ + π§ = 5
β4π₯ + 2π¦ β 2π§ = 0
Try It!
C. {
π₯ + π¦ + π§ = 3π₯ β π¦ + π§ = 1π₯ + π¦ β π§ = 2
D. {
2π₯ + π¦ β 2π§ = 3π₯ β 2π¦ + 7π§ = 123π₯ β π¦ + 5π§ = 10
More Practice:
1. 2. 3.
4. 5. 6.