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Mastering Exponential Functions: Graphs and Transformations

Learn to graph exponential growth and decay functions, understand domain and range, transformations, asymptotes, and factors. Explore examples and practical applications in real-world scenarios.

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Mastering Exponential Functions: Graphs and Transformations

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  1. LESSON 7–1 Graphing Exponential Functions

  2. TEKS

  3. You graphed polynomial functions. • Graph exponential growth functions. • Graph exponential decay functions. Then/Now

  4. exponential function • exponential growth • asymptote • growth factor • exponential decay • decay factor Vocabulary

  5. Concept

  6. Graph Exponential Growth Functions Graph y = 4x. State the domain and range. Make a table of values. Connect the points to sketch a smooth curve. Example 1

  7. Graph Exponential Growth Functions Answer: The domain is all real numbers, and the range is all positive real numbers. Example 1

  8. A.B. C.D. Which is the graph of y = 3x? Example 1

  9. Concept

  10. Graph Transformations A. Graph the function y = 3x – 2. State the domain and range. The equation represents a translation of the graphy = 3x down 2 units. Example 2A

  11. Graph Transformations Answer: Domain = {all real numbers} Range = {y│y > –2} Example 2A

  12. Graph Transformations B. Graph the function y = 2x – 1. State the domain and range. The equation represents a translation of the graphy = 2x right 1 unit. Example 2B

  13. Graph Transformations Answer: Domain = {all real numbers} Range = {y │y ≥ 0} Example 2B

  14. A.B. C.D. A. Graph the function y = 2x – 4. Example 2A

  15. A.B. C.D. B. Graph the function y = 4x – 2+ 3. Example 2B

  16. Graph Exponential Growth Functions INTERNETIn 2006, there were 1,020,000,000 people worldwide using the Internet. At that time, the number of users was growing by 19.5% annually. Draw a graph showing how the number of users would grow from 2006 to 2016 if that rate continued. First, write an equation using a = 1.020 (in billions), and r = 0.195. y = 1.020(1.195)t Then graph the equation. Example 3

  17. Graph Exponential Growth Functions Answer: Example 3

  18. A.B. C.D. CELLULAR PHONESIn 2006, there were about 2,000,000,000 people worldwide using cellular phones. At that time, the number of users was growing by 11% annually. Which graph shows how the number of users would grow from 2006 to 2014 if that rate continued? Example 3

  19. Concept

  20. A. Graph the function State the domain and range. Graph Exponential Decay Functions Example 4A

  21. Graph Exponential Decay Functions Answer: Domain = {all real numbers} Range = {y│y > 0} Example 4A

  22. B. Graph the function State the domain and range. The equation represents a transformation of the graph of Graph Exponential Decay Functions Examine each parameter. ● There is a negative sign in front of the function: The graph is reflected in the x-axis. ● a = 4: The graph is stretched vertically. Example 4B

  23. Graph Exponential Decay Functions ● h = 1: The graph is translated 1 unit right. ● k = 2: The graph is translated 2 units up. Answer: Domain = {all real numbers} Range = {y│y < 2} Example 4B

  24. A. Graph the function A.B. C.D. Example 4A

  25. B. Graph the function A.B. C.D. Example 4B

  26. Graph Exponential Decay Functions A. AIR PRESSUREThe pressure of the atmosphere is 14.7 lb/in2 at Earth’s surface. It decreases by about 20% for each mile of altitude up to about 50 miles. Draw a graph to represent atmospheric pressure for altitude from 0 to 50 miles. y = a(1 – r)t = 14.7(1 – 0.20)t = 14.7(0.80)t Example 5A

  27. Graph Exponential Decay Functions Graph the equation. Answer: Example 5A

  28. Graph Exponential Decay Functions B. AIR PRESSUREThe pressure of the atmosphere is 14.7 lb/in2 at Earth’s surface. It decreases by about 20% for each mile of altitude up to about 50 miles. Estimate the atmospheric pressure at an altitude of 10 miles. y = 14.7(0.80)t Equation from part a. = 14.7(0.80)10 Replace t with 10. ≈ 1.58 lb/in2 Use a calculator. Answer: The atmospheric pressure at an altitude of about 10 miles will be approximately 1.6 lb/in2. Example 5B

  29. A.B. C.D. A. AIR PRESSUREThe pressure of a car tire with a bent rim is 34.7 lb/in2 at the start of a road trip. It decreases by about 3% for each mile driven due to a leaky seal. Draw a graph to represent the air pressure for a trip from 0 to 40 miles. Example 5A

  30. B. AIR PRESSUREThe pressure of a car tire with a bent rim is 34.7 lb/in2 at the start of a road trip. It decreases by about 3% for each mile driven due to a leaky seal. Estimate the air pressure of the tire after 20 miles. A. 15.71 lb/in2 B. 16.37 lb/in2 C. 17.43 lb/in2 D. 18.87 lb/in2 Example 5B

  31. LESSON 7–3 Logarithms and Logarithmic Functions

  32. Targeted TEKS A2.2(C) Describe and analyze the relationship between a function and its inverse (quadratic and square root, logarithmic and exponential), includingthe restriction(s) on domain, which will restrict its range. A2.5(C) Rewrite exponential equations as their corresponding logarithmic equations and logarithmic equations as their corresponding exponential equations. Also addresses A2.2(A) and A2.5(A). Mathematical Processes A2.1(B), Also addresses A2.1(G). TEKS

  33. You found the inverse of a function. • Evaluate logarithmic expressions. • Graph logarithmic functions. Then/Now

  34. logarithm • logarithmic function Vocabulary

  35. Concept

  36. Concept

  37. Because 3 > 1, use the points (1, 0), and (b, 1). Graph Logarithmic Functions A.Graph the function f(x) = log3x. Step 1 Identify the base. b = 3 Step 2 Determine points on the graph. Step 3 Plot the points and sketch the graph. Example 4

  38. Graph Logarithmic Functions (1, 0) (b, 1) → (3, 1) Answer: Example 4

  39. B.Graph the function Graph Logarithmic Functions Step 1 Identify the base. Step 2 Determine points on the graph. Example 4

  40. Graph Logarithmic Functions Step 3 Sketch the graph. Answer: Example 4

  41. A. B. C.D. A. Graph the function f(x) = log5x. Example 4

  42. B. Graph the function . A. B. C.D. Example 4

  43. Concept

  44. ● : The graph is compressed vertically. Graph Logarithmic Functions This represents a transformation of the graph f(x) = log6 x. ● h = 0: There is no horizontal shift. ● k = –1: The graph is translated 1 unit down. Example 5

  45. Graph Logarithmic Functions Answer: Example 5

  46. Graph Logarithmic Functions ● |a| = 4: The graph is stretched vertically. ● h = –2: The graph is translated 2 units to the left. ● k = 0: There is no vertical shift. Example 5

  47. Graph Logarithmic Functions Answer: Example 5

  48. A. B. C.D. Example 5

  49. A. B. C.D. Example 5

  50. LESSON 7–1 Graphing Exponential Functions

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