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TRANSFORMATIONS OF EXPONENTIAL FUNCTIONS

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## TRANSFORMATIONS OF EXPONENTIAL FUNCTIONS

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**Chapter 2 Exponents and Logarithms**TRANSFORMATIONS OF EXPONENTIAL FUNCTIONS 2.2 2.2.1 MATHPOWERTM 12, WESTERN EDITION**Learning Outcomes:**• Learn to apply translations, stretches, and reflections to the graphs of exponential functions • To represent the transformations in the equations of exponential functions • To solve problems that involve exponential growth or decay**Consider an exponential function of the form:**• 1. Graph each set of functions on one set of coordinate axes. 2. Compare the graphs in set A. For any constant k, describe the relationship between the graphs of and .**continued**3. Compare the graphs in set B. For any constant h, describe the relationship between the graphs of and .**Investigate Further:**4. Graph each set of functions on one set of coordinate axes.**5. Compare the graphs in set C. For any real value a,**describe the relationship between the graphs of and . 6. Compare the graphs in set D. For any real value b, describe the relationship between the graphs and .**The graph of a function of the form is obtained by applying**transformations to the graph of the base function , where c > 0. (see text p. 348)**Example:**Transform the graph to Describe the effects on the domain, range, equation of the horizontal asymptote, and intercepts. Solution Vertical: no vertical stretch but the graph is translated 3 units down This affects the asymptote: y = 0 becomes y = -3 so the range becomes y > -3 Horizontal: Graph is reflected in the y - axis and is stretched by a factor of ½ , then translated 5 units left. The domain is not affected.**Solution continued: Intercepts**• To determine the intercepts after multiple transformations, solve algebraically by setting x = 0 to solve for y and y = 0 to solve for x.**Example:**• A cup of water is heated to 100˚C and then allowed to cool in a room with an air temperature of 20˚C. The temperature, T, in degrees Celsius, is measured every minute as a function of time, m, in minutes, and these points are plotted on a coordinate grid. The temperature of the water is found to decrease exponentially at a rate of 25% every 5 min. A smooth curve is drawn through the points, resulting in the graph to the right.**a. What is the transformed exponential function in the form**that can be used to represent the situation? (decreases by 25%, which means 75% of previous value) The exponent m represents any time. The exponent needs to be because the five represents 5 minute intervals (decreases by 25% every 5 minutes) There is a horizontal asymptote at T = 20, therefore the function has been vertically translated upward 20 units.**Solution continued**• The y-intercept occurs at (0, 100). So, there must be a vertical stretch factor. Solve for a:**Assignment:**• P. 355 #3a,c, 4, 5, 7b,c, 9, 11, C1