L1.1 A1 Radical Functions Warm Up - Part 1
Completion requirements
Unit 1
Functions
Warm Up
To prepare for work with radical functions, you will first review functions in general, followed by a review of basic radical notation. A relation is a mathematical relationship between variables, coefficients, and constants. A function is a subset of a relation such that there is a single output value for every input value. In other words, every time a value is substituted in a function, it generates only one result.
Recall that functions are often represented as equations, as you will see in the activity that follows. Function notation identifies the independent variable in the equation. The output value is called the dependent variable. The function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»T«/mi»«mfenced»«mi»n«/mi»«/mfenced»«mo»=«/mo»«mn»4«/mn»«mi»n«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math» is written in function notation, where «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»n«/mi»«/mstyle»«/math» is the independent variable, and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»T«/mi»«mfenced»«mi»n«/mi»«/mfenced»«/mrow»«/mstyle»«/math» is the output. «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»T«/mi»«mfenced»«mi»n«/mi»«/mfenced»«/mrow»«/mstyle»«/math» is read as ‘«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»T«/mi»«/mstyle»«/math» of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»n«/mi»«/mstyle»«/math»’ or ‘«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»T«/mi»«/mstyle»«/math» at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»n«/mi»«/mstyle»«/math»’. «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»T«/mi»«mfenced»«mi»n«/mi»«/mfenced»«/mrow»«/mstyle»«/math» is a single variable (not to be confused with ‘«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»T«/mi»«/mstyle»«/math» times «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»n«/mi»«/mstyle»«/math»’) and can be treated like «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math».
In this activity, you will review how the same function can be represented using a variety of methods.
- The representations A, D, G, J, M, and P show closely related functions. Verify the list of ordered pairs satisfies each of the other representations in this group.
- Make two other groups of related functions using the representations.
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Lists of Ordered Pairs
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Equations
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A.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mfenced»«mrow»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mn»2«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»,«/mi»«mspace
width=¨0.33em¨/»«mfenced»«mrow»«mn»2«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»4«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mfenced»«mrow»«mn»3«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace
width=¨0.33em¨/»«mn»8«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«mn»4«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»16«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mfenced»«mrow»«mn»5«/mn»«mi
mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»32«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mfenced»«mrow»«mn»6«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»64«/mn»«/mrow»«/mfenced»«/mrow»«/mfenced»«/mstyle»«/math»
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D.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»d«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«/mstyle»«/math»
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B.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mfenced»«mrow»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»2«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»,«/mi»«mspace
width=¨0.33em¨/»«mfenced»«mrow»«mn»2«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»1«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mfenced»«mrow»«mn»3«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace
width=¨0.33em¨/»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mfenced»«mrow»«mn»4«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mrow»«/mfenced»«mi
mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mfenced»«mrow»«mn»5«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mfrac»«mn»2«/mn»«mn»5«/mn»«/mfrac»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»,«/mi»«mspace
width=¨0.33em¨/»«mfenced»«mrow»«mn»6«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«/mrow»«/mfenced»«/mrow»«/mfenced»«/mstyle»«/math»
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E.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»e«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mn»2«/mn»«msqrt»«mi»x«/mi»«/msqrt»«/mstyle»«/math»
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C.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mfenced»«mrow»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»2«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»,«/mi»«mspace
width=¨0.33em¨/»«mfenced»«mrow»«mn»2«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»2«/mn»«msqrt»«mn»2«/mn»«/msqrt»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mfenced»«mrow»«mn»3«/mn»«mi
mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»2«/mn»«msqrt»«mn»3«/mn»«/msqrt»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mfenced»«mrow»«mn»4«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace
width=¨0.33em¨/»«mn»4«/mn»«/mrow»«/mfenced»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mfenced»«mrow»«mn»5«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»2«/mn»«msqrt»«mn»5«/mn»«/msqrt»«/mrow»«/mfenced»«mi
mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mfenced»«mrow»«mn»6«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»2«/mn»«msqrt»«mn»6«/mn»«/msqrt»«/mrow»«/mfenced»«/mrow»«/mfenced»«/mstyle»«/math»
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F.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mfrac»«mn»2«/mn»«mi»x«/mi»«/mfrac»«/mstyle»«/math»
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Statements
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G.
If the number of participants in a club starts at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math» and doubles each year, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» is the number of participants after «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» years.
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H.
If two litres of tea are equally divided among «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» people, each person receives «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» litres.
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I.
If a square has an area of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» units squared, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» is double the side length of the square.
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Mapping Diagrams
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J.
K.
L.
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Graphs
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Tables of Values
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M.
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P.
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N.
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Q.
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O.
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R.
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- Explain how the different representations of a function are related.
- Describe an advantage or a disadvantage you see with each type of function representation.
- Each representation shows that an input of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math» leads to an output of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math», an input of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math» leads to an output of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»4«/mn»«/mstyle»«/math», and so on.
- B, F, H, K, O, and R represent a second group of related functions, and C, E, I, L, N, and Q represent a third group of related functions.
- Each representation shows how a particular «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-value is related to a particular output.
- Responses will vary, but may include descriptions such as some are more visual than others, some account for more points than others, or some are easier to manipulate algebraically than others.
If you were unable to confidently complete the activities in the Warm Up,
- watch the videos on the next page,
- enter “math functions” into a search engine to research functions, or
- contact your teacher.