Rationale/Lesson Abstract: Students will graph exponential functions, identify key features and learn how the variables a, h and k in f x a b

Transcription

1 Grade Level/Course: Algebra Lesson/Unit Plan Name: Graphing Eponential Functions Rationale/Lesson Abstract: Students will graph eponential functions, identif ke features h and learn how the variables a, h and k in f a b k affect the parent graph f b. Timeframe: class periods Da focus: Graphing functions of the form f b and f Da focus: Graphing functions of the form f a b. for b,,. b h Da focus: Graphing functions of the form f a b k. Common Core Standard(s): F-IF.e Graph eponential and logarithmic functions, showing intercepts, end behavior, and trigonometric functions, showing period, midline, and amplitude. F-BF. Identif the effect on the graph of replacing f b f k, kf, f k k, and f for specific values of k (both positive and negative); find the value of k given the graphs. Eperiment with cases and illustrate an eplanation of the effects on the graph using technolog. Include recognizing even and odd functions from their graphs and algebraic epressions for them. Notes: A Warm-Up is on page. For Da, use the blank grid handout for eample and tr problem. Instructional Resources/Materials: Warm-Up Dail Student Graphing Handout ( options) Dail Student Eit Ticket (strip) Graphing Tool (optional) Page of MCC@WCCUSD 0//

2 Lesson Da Think-Pair-Share: List all that ou know about the function h. Then compare f and predict what the graph f of is going to look like. Possible Descriptions of h : h is a quadratic function. The graph is a parabola that has a verte and an ais of smmetr. Eample : Graph the function f. h to Compare h to f : is the base in h and the eponent in f, is the eponent of h and the base in f Discuss the graph of f : Show the graph of f on a graphing tool. Discuss the ke features (domain, range, etc). Use the table of values to duplicate the graph onto handout. If no technolog is available, proceed to Eample and create the graph using a table. f, f f f,, 0 0 f 0 0, f, f, f, Input-Output Table: Create a column table: Choose non-negative inputs to start. Choose a few non-positive input values and review the properties a 0 and a n n. a Notice that as decreases, f approaches zero. The line 0, or the -ais, is called a horizontal asmptote. 0 Page of MCC@WCCUSD 0//

3 An eponential function is a function of the form a b where 0 a, b 0 and b. *For this lesson a and b Tr: Partner A: f Partner B: f Think-Pair: What kind of functions are the? Eplain. Predict what each graph is going to look like. Each partner graphs their function. Partners echange papers, graph their function on partner s paper. Compare and contrast. 0 0 Eample : Graph the function g. 0 f, f f f 0 0, 0, 0 f, Page of MCC@WCCUSD 0//

4 How are the functions g and f related? The are eponential functions of the form b f. Their graphs have the same shape, -intercept and asmptote at 0. Their bases are reciprocals. Their graphs are reflections about the -ais. Identif other points on f :,,,,,, and,. Tr: Partner A: g Partner B: g Think-Pair: What kind of functions are the? Eplain. Predict what each graph is going to look like. Each partner graphs their function. Hint: Use the previous graphs for reference. Identif the ke features of each graph. Partners echange papers, graph their function on partner s paper. Compare and contrast. 0 0 Note: Students might identif the pattern (geometric sequence) created in the tables of the function b f. The common ratio could be used to generate the outputs quickl. Page of MCC@WCCUSD 0//

5 Generalize: The graph of f b goes through the points,, b the shape of the graph, these three points are sufficient to graph the function. Ke Features of b f : Domain: All Real Numbers, 0, and, b or Range: All Positive Real Numbers 0 Intercept(s): No -intercept, -intercept is 0, Asmptote: 0 End Behavior when End Behavior when 0 b : As increases, As decreases, b : As increases, As decreases, f increases. f or 0 f approaches zero. f approaches zero. f increases.. So knowing Recommended: Revisit graphs. Write/discuss the ke features of each. Eample : Think-Pair: Write the function whose graph is given. What kind of function does the graph represent? Make a conjecture about the b value in f b. Identif a point or points on the graph Solution: This is going to be an eponential function like those in eample, where b. Two points on the graph are, 0 and, 00. Notice that we can write the points as, 0 and, 0. I predict that an point on the graph is, 0. Therefore, the graph could be modeled b the function f 0. EXIT TICKET: Identif the ke features of the graph of f. Page of MCC@WCCUSD 0//

6 Graphing Eponential Functions (Da ) f g f g f g Page of MCC@WCCUSD 0//

7 Graphing Eponential Functions (Da ) Page of MCC@WCCUSD 0//

8 Lesson Da Think-Pair-Share: Review the graphs from Da. Compare the functions How would ou classif (or name) the functions f? How would ou classif (or name) the functions g? f and g. Similarities between the functions f and g : The are all eponential functions. Their graphs have the same shape. The have the same domain, range, -intercept and asmptotes. Major Difference between the functions f and g f increases from left to right. Specificall, as g decreases from left to right. Specificall, as : End behavior, f., 0 g. An eponential growth function is a function of the form f a b An eponential deca function is a function of the form f a b where a 0 and b. where a 0 and 0 b. Eample : Graph the function h Think-Pair:. What kind of function is h? Eplain. Predict what the graph is going to look like. How is the going to affect the graph? h, h f 0 0,, 0,,, Input-Output Table: Refer to the table of f. Create a similar table for h. Discuss h f : the outputs of h are times the outputs of f. Find the outputs of h b multipling the outputs of f b. 0 Page of MCC@WCCUSD 0//

9 h Tr: Partner A: h Partner B: Think-Pair: What kind of functions are the? Eplain. Predict what each graph is going to look like. Each partner graphs their function. Hint: Use previous graphs for reference. Partners echange papers, graph their function on partner s paper. Compare and contrast. 0 0 Discuss: How does the value of a affect the graph of an eponential function? [The value of a changes the -intercept and stretches or compresses the graph.] Input-Output Table: Eample : Graph the function g. Think-Pair: What kind of function is g? Eplain. Predict what the graph is going to look like. How is the going to affect the graph? How is g related to h g, g,, 0 0,,,? Refer to the table of h 0. Create a table for g. Discuss g h : the outputs of g are the opposite of the outputs of h. Find the outputs of of h b. g b multipling the outputs Page of MCC@WCCUSD 0//

10 g Tr: Partner A: g Partner B: Think-Pair: What kind of functions are the? Eplain. Predict what each graph is going to look like. How will ou graph each function? Each partner graphs their function. Hint: Use the previous graphs for reference. Partners echange papers, graph their function on partner s paper. Compare and contrast. 0 0 Discuss: How do the graphs compare when a is positive to when a is negative? Identif some points on the graph of f a b. When a 0, the function values are positive and when a 0 the function values are negative. f a b a contains the points, b Domain: All Real Numbers Range: If 0, 0, a, and, ab. Ke Features of f a b : or a : All Positive Real Numbers 0 If 0 a : All Negative Real Numbers 0 Intercept(s): No -intercept and -intercept at 0, a f or 0 f or 0 End Behavior: The end behavior depends on a and b; discuss the cases (see Eit Ticket). For eample, where does the function approach zero? If f approaches zero. b : as decreases, b : as increases, If 0 f approaches zero. Asmptote: 0 Page 0 of MCC@WCCUSD 0//

11 : a) Identif and find function values of the parent function. b) Identif the points of the transformed function. c) Determine how to draw and label aes. Eample : Graph the function f 0, f,, 0, 0,, 0 0 Tr: Partner A: f Partner B: f Think-Pair: What kind of functions are the? Eplain. Predict what each graph is going to look like. How will ou graph the function? How will ou draw and label the aes? Each partner graphs their function. Partners echange papers, graph their function on partner s paper. Compare and contrast Page of MCC@WCCUSD 0//

12 EXIT TICKET: Match the functions to their graphs.. f 0. f. f.. f Graph A Graph B Graph C Graph D Page of MCC@WCCUSD 0//

13 Graphing Eponential Functions (Da ) h g 0 h g 0 h g 0 Page of MCC@WCCUSD 0//

14 Graphing Eponential Functions (Da ) Page of MCC@WCCUSD 0//

15 Lesson Da Think-Pair-Share: Each graph below can be written as a function of the form f a b. What can ou determine about the values of a and b for each graph? Graph A Graph B Graph C Graph D Graph A Graph B Graph C Graph D a 0 b a 0 b a 0 0 b a 0 0 b Eample : Graph the function Think-Pair: What kind of function is h h.? Eplain. Identif the parent function. How does the affect the graph?, h,, Input-Output Table: Review the table of f. Create a similar table for h. Discuss h f : the outputs of h are less than the outputs of f. Find the outputs of h b subtracting from the outputs of f. Another option: graph f and translate the points down units to obtain h. 0 0,,,, Page of MCC@WCCUSD 0//

16 Eample continued: Think-Pair-Share: How do the graphs of the parent function f and h compare? Plot the points of the parent function f is on the same coordinate plane and compare them to h of h can be obtained b translating the points of down. Some of the ke features of h are:. The points f units Tr: Domain: Range: Intercept(s): Asmptote: All real numbers 0,,., 0 Choose a method and graph the functions. Method #- Use an input-output table Method #- Translate the graph of the parent function. Partner A: h Partner B: h Think-Pair: What kind of function is h? Eplain. Identif the parent function. Predict what the graph is going to look like. Each partner graphs their function. Partners echange papers, graph their function on partner s paper. Compare and contrast. Think-Pair-Share: In general, for f a b k, which ke features are affected when k 0? Domain: [No] Range: [Yes] Intercept(s): [Yes] Asmptote: [Yes] Page of MCC@WCCUSD 0//

17 Discuss: Draw and label the asmptote on each graph. h : and h : How do ou determine the asmptote given the function (without graphing)? The asmptote of the function f a b k is k How does the value of k affect the graph of f a b k? The value of k is the vertical shift up or down of the parent function k units. If k 0, then the graph shifts up k units. If k 0, then the graph shifts down k units. Eample : Graph the function g. Think-Pair: What kind of function is (Start) 0 g? Identif the parent function., g,,,,,, Input-Output Table: Start with the -value that makes the eponent zero, or another nonnegative number. Then find at least two integer values less than and greater than our starting -value. The parent function of g is f coordinate plane and compare them to g. The points of of f units right. Discuss the ke features of g : domain is all real numbers range is all positive real numbers asmptote is 0 (there is no vertical shift). Plot the points of the parent function on the same g can be obtained b translating the points g is an eponential growth function whose graph can be obtained b translating the graph of f to the right units. Page of MCC@WCCUSD 0//

18 Tr: Choose a method and graph the functions. Method #- Use an input-output table. Method #- Translate the graph of the parent function. Partner A: g Partner B: g Think-Pair: What kind of functions are the? Eplain. Identif the parent function. How will ou graph the functions? Each partner graphs their function. Partners echange papers, graph their function on partner s paper. Compare and contrast. 0 0 h Think-Pair-Share: In general, for f a b, which of the following are affected when h 0? Domain: [No] Range: [No] Intercept(s): [Yes] Asmptotes: [No] Discuss: h How does the value of h affect the graph of f a b? The value of h is the horizontal shift left or right of the parent function h units. If h 0, then the graph shifts right h units. If h 0, then the graph shifts left h units. h Generalize: To graph the eponential function f b k, graph the parent function f b and translate the graph horizontall h units and verticall k units. Page of MCC@WCCUSD 0//

19 Page of 0// Eample : Graph the function f : Method : Make an input-output table. Start with the -value that makes the eponent zero, then find at least two integer values less than and greater than our starting -value. Asmptote: f,,, (Start) 0,, 0 0, 0 0

20 Method : Translate the graph of the parent function. a) Identif and find points on the parent function. The parent function is g. b) Identif the vertical and horizontal shift of the parent function. h f. h, the horizontal shift is units left and k, the vertical shift is units up. Find h and k b comparing f a b k and Since c) Graph f b translating the points of f units left and units up.,,, f,,,,, 0 0, 0,,,,,,, 0, f f Tr: Partner A: f Partner B: f Think-Pair: What kind of functions are the? Eplain. Identif the parent functions. How will ou graph the functions? Choose a method: Method #- Use an input-output table. Method #- Translate the graph of the parent function. Each partner graphs their function. 0 Page 0 of MCC@WCCUSD 0//

21 Eit Ticket : Identif the ke features of the graph of f Domain: Range: Intercept(s): End Behavior: Asmptote:. Eit Ticket : Match the functions to their graphs. f 0 f f. f Graph A Graph B Graph C Graph D Eit Ticket : Determine if the function is an eponential growth or deca function. Identif the parent function, the horizontal shift and vertical shift of each function. Function Eponential Growth or Deca? Parent Function Horizontal Shift Vertical Shift f 0 g 0 Page of MCC@WCCUSD 0//

22 Graphing Eponential Functions (Da ) Page of MCC@WCCUSD 0//

23 Warm-Up CCSS: N-RN. Indicate if each epression is: Between 0 and Equal to Greater than CCSS: F-IF.a: Graph the function f. A) B) C) D) CCSS A-SSE.: CCSS F-IF.: Solve the equations b rewriting the right side of the equation as a power with the same base as the left side of the equation. Use the graph of the function f above to identif the ke features. a) Domain A) B) C) b) Range c) Intercept(s) d) Maimum(s)/Minimum(s) e) Intervals where the function is increasing and/or decreasing c) Intervals where the function is positive and/or negative Page of MCC@WCCUSD 0//

24 CCSS: N-RN. Indicate if each epression is: Between 0 and Equal to Greater than Warm-Up Solutions CCSS: F-IF.a: Graph the function f. A) B) ❸ ❶ C) ❸ D) ❷ CCSS A-SSE.: CCSS F-IF.: Solve the equations b rewriting the right side of the equation as a power with the same base as the left side of the equation. Use the graph of the function to identif the ke features. f above A) C) B) 0 a) Domain: All real numbers b) Range: All real numbers f c) Intercepts: 0,,, 0,, 0 d) The minimum occurs at, minimum value is. 0 where the e) The function is increasing for 0. The function is decreasing for 0. c) The function is positive for and. The function is negative for. Page of MCC@WCCUSD 0//