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Transcript of Job 1 Pays £10 per day. Total Earnings £300 Pays 1p per day but daily pay increases by 50% each...
![Page 1: Job 1 Pays £10 per day. Total Earnings £300 Pays 1p per day but daily pay increases by 50% each day. Total Earnings £3835.00 Summer Jobs - 30 days work.](https://reader035.fdocuments.in/reader035/viewer/2022062801/56649e7c5503460f94b7de86/html5/thumbnails/1.jpg)
Job 1• Pays £10 per day.
• Total Earnings• £300
• Pays 1p per day but daily pay increases by 50% each day.
• Total Earnings• £3835.00
Summer Jobs - 30 days work
Job 2
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Exponential Functions
Aims: To know the general formula for an exponential graph.
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What You Need To Be Able To Do
• To know the general formula for an exponential graph: We have seen the functions for linear, quadratic and some general polynomial functions.
• After this we will take a first look at logarithms.
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Maths
• Ex 7A is very weird but maybe worth a look especially Q5
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Geometric Sequence Graph
• The function of a basic exponential function f(x)=bx
• There are some limitations however…
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Limits of b Values
• b must be positive (since –b would not be defined for fractional values of x with even denominators)
• b cannot be 1 since that would result in every value being 1 and therefore a line.
• So given this information lets see some exponential graphs…
• On a graphical calculator with a standard viewing window plot…
• y=2x y=4x y= (½)x
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Look
–4 –2 2 4 6
2
4
6
x
y
Equation 2: y=4Ì
Equation 1: y=2Ì
Equation 3: y=(½)Ì
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Look
–4 –2 2 4 6
2
4
6
x
y
Equation 2: y=0.5(4Ì)
Equation 1: y=3(2Ì)
Equation 3: y=4(½)Ì
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Summary
• You are expected to be familiar with the shapes of exponential graphs.
• You are expected to know the effect of having different values of b for the graph y=bx.
• To know where these graphs cross the y axis and that this is the value of k in y=kbx.
• Lets have a quick go…
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Sketch Showing Points of Intersection with the Axes.
xy 3
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Sketch Showing Points of Intersection with the Axes.
xy 65
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Sketch Showing Points of Intersection with the Axes.
xy 21
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Sketch Showing Points of Intersection with the Axes.
xy 315
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Sketch Showing Points of Intersection with the Axes.
xy 375
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Sketch Showing Axes Intercepts and Asymptote
32 xy
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Sketch Showing Points of Intersection with the Axes.
5102 xy
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Sketch Showing Axes Intercepts
12 xy
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Sketch Showing Points of Intersection with the Axes.
1b0 and 0agiven xbay
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Logarithms
Aims: To know what logarithms are.
To be able to evaluate logarithms including solving equations
involving logarithms.
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What You Need To Be Able To Do
• Name: What is a logarithm• Describe: The relationship between
an = x and Logax• Explain: How to solve basic missing
value type equations that include logs.
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Maths
• Ex 7B p 356
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Inversing
• What is the inverse of +?• What is the inverse of x?• What is the inverse of √?• But what is the inverse of taking 2 to
the power of a number e.g. How can you make x the subject of y=2x?
• The answer is that we do not currently have an inverse for an exponential…
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Napier• John Napier of Merchiston (1550
– 4 April 1617) – also signed as Neper, Nepair – named Marvellous Merchiston, was a Scottish mathematician, physicist, astronomer & astrologer, and also the 8th Laird of Merchistoun.
• John Napier is most renowned as the inventor of the logarithm, and of an invention called "Napier's bones".
• Napier also made common the use of the decimal point in arithmetic and mathematics.
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The Logarithm
• The logarithm is an inverse of an exponential base it is a function that must have a base corresponding to the base it is inversing.
• E.g. To inverse 2x one would have to use log base 2, written log2.
• We say that we apply logn to a value (a) and the answer is the power of n that gives you a.
• E.g. log216 = 4
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The Logarithm Equivalence
• So IF 53=125 then Log5125=3
• Can you write this statement in general?
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Solving Problems
• You will be expected to be able to write logarithm statements as indices and vice versa.
• E.g. If log61296 = 464=1296
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Evaluating
• You need to be able to find the missing values in equations involving logarithms.
• Log4x = 3 what is x?
• Log2 1/16
= y what is y?
• Logx18 = 4 what is x?
• Log√xx3=a what is a?