UNIT #1: Transformation of Functions; Exponential and Log. Goals: Review core function families and mathematical transformations.

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1 UNIT #1: Transformation of Functions; Exponential and Log Goals: Review core function families and mathematical transformations. Textbook reading for Unit #1: Read Sections

2 2 Example: The graphs of e x, ln(x), x 2 and x 1 2 are shown below. Identify each function s graph. y1 x

3 Unit 1 Transformation of Functions; Exponential and Log 3 Comment on the properties of the graphs of inverse functions - exponentials - logarithms - powers of x -

4 4 Knowing the graphs and properties of essential families of functions is crucial for effective mathematical modeling. Name other families of functions.

5 Unit 1 Transformation of Functions; Exponential and Log 5 Give examples of members of each family, and state some of their common properties.

6 6 The core families of functions can be made even more versatile by being transformed. Example: Sketch the graph of y = x 2, over the interval x [ 4,4]. On the same axes, sketch the graph of y = (x+1)2.

7 Unit 1 Transformation of Functions; Exponential and Log 7 Review the four common types of function transformations. Type Form Example

8 8 Type Form Example

9 Unit 1 Transformation of Functions; Exponential and Log 9 Example: Consider the data shown below, showing the concentration of a chemical produced in a reaction vessel, over time. Concentration (ppm) Time (hours) What family of functions would best describe this graph? Point out specific features of the graph that make the choice a reasonable one.

10 10 Give a general mathematical form for the function, based on the shape of the graph. e.g. C(t) =... Concentration (ppm) Time (hours)

11 Unit 1 Transformation of Functions; Exponential and Log 11 Determine as many of the numerical values in the formula C(t) =... as you can, given the graph. Sketching related graphs along the way might be helpful. Concentration (ppm) Time (hours)

12 12 Looking closely at the graph, you see that after 30 hours, the concentration has reached almost exactly 12 ppm. Determine the value for the final missing parameter in your concentration function. Concentration (ppm) Time (hours)

13 Unit 1 Transformation of Functions; Exponential and Log 13 Logarithm Review Most students are quite comfortable with exponential functions, but many find logarithms less familiar. To address this we will do a more comprehensive review of the logarithmic function and its use in transforming equations. Log/Exponential Equivalency Simplify log a (a 7 ). a c = x means log a x = c Simplify a log a(25).

14 14 Without using a calculator, find log 10 (1/100) and log 10 (10,000).

15 Unit 1 Transformation of Functions; Exponential and Log 15 These problems suggest the following equations, which also follow from the fact that a x and log a (x) are inverse functions. log a (a x ) = x and a log ax = x Rules for Computing with Logarithms 1.log a (AB) = log a A+log a B 2.log a (A/B) = log a A log a B 3.log a (A P ) = P log a A

16 16 Changing logarithmic bases The functions a x and log a are not provided on calculators unless a = 10 or a = e (seenextsectionofthesenotes). Forothervaluesofa,a x andlog a canbeexpressed in terms of 10 x and log 10. To calculate log a x, we use the following formula: Conversion of Log Bases log a x = logx loga Prove the above formula, using the Rules for Computing Logarithms and the fact that log a x = c means x = a c.

17 Unit 1 Transformation of Functions; Exponential and Log 17 Graphs of Logarithmic Functions Since the logarithm in base 10 is commonly used in science, we define logx (no subscript) to mean log 10 x, for brevity. The graph of logx may be obtained from the graph of 10 x by reversing the axes (that is, by reflecting the graph in the line y = x). (If drawing the graph of inverse functions is unfamiliar, please read Section 1.3 in the text.)

18 x log 10 (x) 10 What is the domain of logx? What is the range of logx? Sketch the logarithm function for the bases e and 2.

19 Unit 1 Transformation of Functions; Exponential and Log 19 Classic Applications of Exponentials and Logarithms Example: Based on H-H, Section 1.4 #48: A cup of coffee contains 100 mg of caffeine, which leaves the body at a continuous rate of 17% per hour. Sketch the graph of caffeine level over time, after drinking one cup of coffee.

20 20 There are two natural interpretations of the question statement which lead to two different formulae for A(t). Write down both formulae. Compare the predicted caffeine level after 10 hours, using each model. Based on those values, how similar are these two models in practice?

21 Unit 1 Transformation of Functions; Exponential and Log 21 The key phrase continuous rate has a special meaning in mathematics and science, and it associated with the natural exponential form e rt. It is typically associated with processes like chemical reactions, population growth, and continuously compounded interest. Common alternative statements about percentage growth or decay, where the rate is assumed to be measured at the end of one time period (hour, day year), are usually of the form (1±r) t.

22 22 Write out an appropriate mathematical model for the following scenarios: Infant mortality is being reduced at a rate of 10% per year. My $10,000 investment is growing at 5% per year. A savings account offers daily compound interest, at a 4% annual rate. Bacteria are reproducing at a continuous rate of 125% per hour.

23 Unit 1 Transformation of Functions; Exponential and Log 23 We now return to our earlier modeling problem. Example: A cup of coffee contains 100 mg of caffeine, which leaves the body at a continuous rate of 17% per hour. Write the formula for A(t). What is the caffeine level at t = 4 hours? At what time does the caffeine level reach A = 10 mg? Find the half-life of caffeine in the body.

Example: The graphs of e x, ln(x), x 2 and x 1 2 are shown below. Identify each function s graph.

Familiar Functions - 1 Transformation of Functions, Exponentials and Loga- Unit #1 : rithms Example: The graphs of e x, ln(x), x 2 and x 1 2 are shown below. Identify each function s graph. Goals: Review