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Five-Minute Check (over Chapter 6) CCSS Then/Now New Vocabulary Key Concept: Parent Function of Exponential Growth Functions Example 1: Graph Exponential Growth Functions Key Concept: Transformations of Exponential Functions Example 2: Graph Transformations Example 3: Real-World Example: Graph Exponential Growth Functions Key Concept: Parent Function of Exponential Decay Functions Example 4: Graph Exponential Decay Functions Example 5: Real-World Example: Graph Exponential Decay Functions Lesson Menu
Solve 4a2 – 9 = 0. A. ± 1 B. ± C. ± 2 D. 2, __ 3 2 5-Minute Check 1
Solve 4a2 – 9 = 0. A. ± 1 B. ± C. ± 2 D. 2, __ 3 2 5-Minute Check 1
Solve 6y3 + 13y2 + 5y = 0. A. 3, 2, 1 B. C. D. 5-Minute Check 2
Solve 6y3 + 13y2 + 5y = 0. A. 3, 2, 1 B. C. D. 5-Minute Check 2
Find (f + g)(x) if f(x) = 3x + 7 and g(x) = x2 – 10. A. (f + g)(x) = 3x2 – 70 B. (f + g)(x) = x2 + 3x – 3 C. (f + g)(x) = x2 – 3x + 17 D. (f + g)(x) = 3x – 3 5-Minute Check 3
Find (f + g)(x) if f(x) = 3x + 7 and g(x) = x2 – 10. A. (f + g)(x) = 3x2 – 70 B. (f + g)(x) = x2 + 3x – 3 C. (f + g)(x) = x2 – 3x + 17 D. (f + g)(x) = 3x – 3 5-Minute Check 3
Determine whether f(x) = 4x – 9 and g(x) = are inverse functions. A. yes B. no 5-Minute Check 4
Determine whether f(x) = 4x – 9 and g(x) = are inverse functions. A. yes B. no 5-Minute Check 4
A. –9xy2 B. –9x2y4 C. –3xy2 D. 3xy2 5-Minute Check 5
A. –9xy2 B. –9x2y4 C. –3xy2 D. 3xy2 5-Minute Check 5
A. –7 B. C. –2 D. 4 5-Minute Check 6
A. –7 B. C. –2 D. 4 5-Minute Check 6
Mathematical Practices Content Standards F.IF.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. F.IF.8.b Use the properties of exponents to interpret expressions for exponential functions. Mathematical Practices 3 Construct viable arguments and critique the reasoning of others. CCSS
You graphed polynomial functions. Graph exponential growth functions. Graph exponential decay functions. Then/Now
exponential function exponential growth asymptote growth factor exponential decay decay factor Vocabulary
Concept
Graph y = 4x. State the domain and range. 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
Graph Exponential Growth Functions Answer: Example 1
Graph Exponential Growth Functions Answer: The domain is all real numbers, and the range is all positive real numbers. Example 1
Which is the graph of y = 3x? A. B. C. D. Example 1
Which is the graph of y = 3x? A. B. C. D. Example 1
Concept
A. Graph the function y = 3x – 2. State the domain and range. Graph Transformations A. Graph the function y = 3x – 2. State the domain and range. The equation represents a translation of the graph y = 3x down 2 units. Example 2A
Graph Transformations Answer: Example 2A
Domain = {all real numbers} Range = {y│y > –2} Graph Transformations Answer: Domain = {all real numbers} Range = {y│y > –2} Example 2A
B. Graph the function y = 2x – 1. State the domain and range. Graph Transformations B. Graph the function y = 2x – 1. State the domain and range. The equation represents a translation of the graph y = 2x right 1 unit. Example 2B
Graph Transformations Answer: Example 2B
Domain = {all real numbers} Range = {y │y ≥ 0} Graph Transformations Answer: Domain = {all real numbers} Range = {y │y ≥ 0} Example 2B
A. Graph the function y = 2x – 4. A. B. C. D. Example 2A
A. Graph the function y = 2x – 4. A. B. C. D. Example 2A
B. Graph the function y = 4x – 2 + 3. A. B. C. D. Example 2B
B. Graph the function y = 4x – 2 + 3. A. B. C. D. Example 2B
First, write an equation using a = 1.020 (in billions), and r = 0.195. Graph Exponential Growth Functions INTERNET In 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
Graph Exponential Growth Functions Answer: Example 3
Graph Exponential Growth Functions Answer: Example 3
CELLULAR PHONES In 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? A. B. C. D. Example 3
CELLULAR PHONES In 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? A. B. C. D. Example 3
Concept
A. Graph the function State the domain and range. Graph Exponential Decay Functions A. Graph the function State the domain and range. Example 4A
Graph Exponential Decay Functions Answer: Example 4A
Domain = {all real numbers} Range = {y│y > 0} Graph Exponential Decay Functions Answer: Domain = {all real numbers} Range = {y│y > 0} Example 4A
B. Graph the function State the domain and range. Graph Exponential Decay Functions B. Graph the function State the domain and range. The equation represents a transformation of the graph of 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
● h = 1: The graph is translated 1 unit right. Graph Exponential Decay Functions ● h = 1: The graph is translated 1 unit right. ● k = 2: The graph is translated 2 units up. Answer: Example 4B
● h = 1: The graph is translated 1 unit right. 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
A. Graph the function A. B. C. D. Example 4A
A. Graph the function A. B. C. D. Example 4A
B. Graph the function A. B. C. D. Example 4B
B. Graph the function A. B. C. D. Example 4B
Graph Exponential Decay Functions A. AIR PRESSURE The 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 20 miles. y = a(1 – r)t = 14.7(1 – 0.20)t = 14.7(0.80)t Example 5A
Graph the equation. Answer: Graph Exponential Decay Functions Example 5A
Graph the equation. Answer: Graph Exponential Decay Functions Example 5A
y = 14.7(0.80)t Equation from part a. Graph Exponential Decay Functions B. AIR PRESSURE The 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: Example 5B
y = 14.7(0.80)t Equation from part a. Graph Exponential Decay Functions B. AIR PRESSURE The 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
A. AIR PRESSURE The pressure of a car tire with a bent rim is 34 A. AIR PRESSURE The 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. A. B. C. D. Example 5A
A. AIR PRESSURE The pressure of a car tire with a bent rim is 34 A. AIR PRESSURE The 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. A. B. C. D. Example 5A
B. AIR PRESSURE The pressure of a car tire with a bent rim is 34 B. AIR PRESSURE The 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
B. AIR PRESSURE The pressure of a car tire with a bent rim is 34 B. AIR PRESSURE The 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
End of the Lesson