Making Graphs from Tables and Graphing Horizontal and Vertical Lines - Black Level Problems

Transcription

1 Making Graphs from Tables and Graphing Horizontal and Vertical Lines - Black Level Problems Black Level Hperbola. Give the graph and find the range and domain for. EXPONENTIAL Functions - The following problem illustrates an eponential relation. Double or Quits: Your friend owes ou $. Instead of paing ou, our friend challenges ou to double or quits on whether the Warriors will win their net game. Your friend will either pa ou $ if the lose or nothing if the win. You win the challenge, and instead of paing ou $, our friend sas double or quits again. What happens to the amount our friend owes ou if ou win each bet in a row? Draw a graph to show the pattern.. Complete the table below: Number of bets Doubling 4 Amount owed = 4 = 8. The formula for the amount owed is =. a. What does represent? b. What does represent? 4. Draw a smooth curve using the values from the table. 5. Use a calculator to investigate: a. How man bets take place before our friend owes ou more than $5. b. How man bets take place before our friend owes ou more than $,,. Amount owed ($) Number of bets

2 6. Use a calculator to work out (.5), (.5) and (.5) 4, and then use these values to draw the graph of = Use our calculator to complete this table, and then draw the graph of =. Use a scale of - to for the -ais and to for the -ais = What is the equation of this eponential graph? It passes through (, ), (, 4) and (, 6). 9. The graphs of = 5 and = 6 are drawn here. Which one is the graph of = 5

3 Deca curves Eponential graphs (equation = a ) do not alwas show growth. If the base a is less than, the graph slopes downwards. Then the graph is called a deca curve.. Eample : Draw the graph of = ( ) Answer : A calculator helps complete this table of values work out powers of.5 rather than : - (.5) 4 Plot these points and join them up with a smooth curve Use our calculator to complete this table, and then fraw the graph of = (.4). Note that (.4) is the same as. Use a scale of - to fot the -ais, and 5 to for the -ais = (.4) Use a calculator to work out (.8) -4, (.8) -, (.8) and (.8) 4, then use these values to draw the graph of =.8.. Draw these eponential curves b plotting points. a. =. 4 b. = (.7). The graphs of = (.55) and = (.6) are drawn below. Which one is the graph of = (.55)?

4 Features of eponential graphs The general form of a simple eponential graph is = a. If the base is greater than, we get a growth curve (a > ), the graph slopes upwards everwhere. If the base is less than, we get a deca curve ( < a < ). a > < a < The -intercept is alwas. That is, eponential graphs alwas pass through the point (, ). This is because an number to power of gives a result of. The -ais is an asmptote. On a growth curve for negative values of the graph becomes ver close to the -ais but never actuall touches it. The base number a affects the steepness of the graph. For eample, the graph of = is steeper than the graph of =. An eponential function, of the form = a, can take an -value, but the -value can onl be a positive number. Technicall, we sa the domain of the function = a is the real numbers, and the range of the function = a is {: >, є R}. 4. Eplain wh the graph of = a alwas passes through on the -ais. 5. The point (, ) is on the graph of = a. What is the value of a? 6. The point (.5, k) is on the graph of =. What is the value of k? 7. Eplore what happens to the graph of = a when a is a number ver close to either just below or just above.

5 Changes of scale the form = b a. Multipling each -value on the curve given b = a b the same constant alters the vertical scale. To transform the graph of = a to the graph of = b a scale the height of each point from the -ais b a factor of b. This is shown most clearl b the position of the -intercept. On the graph of = a the intercept on the vertical ais is (, ). On the graph of = b a the vertical intercept is (, b). For eample, each point on the graph of = 4 (.) is four times the distance from the -ais as the corresponding point on the graph of = (.). Two points on the graph of = (.) are (, ) and (,.) = 4 (.) = (.) The points on the graph of = 4 (.) with the corresponding -values are (, 4) and (, 5.). Eample Draw the graph of 8 6 =. 5 = Note: the graph of = is also shown for comparison. 4 = Draw these graphs. a. = b. = 5 c. = 4(.7) b. = (.5)

6 9. Draw these graphs. a. = b. = 4(.9). B drawing a graph or otherwise, find the co-ordinates of the point(s) where the graph of = intersects the graph of =.. Write down the -intercept for each of these graphs. a. = 5 4 b. = c. = 8(.6) d. = (.75) Match each equation (a-e) with one of the graphs (A-E). a. = 4 b. = 4 c. = 4 d. = 4 e. = 4. Match each equation (a-d) with one of the graphs (A-D). a. = b. = c. = d. =

7 Step Functions Consider the step function [] =. Define it b = the value of the greatest integer less than or equal to. The step function defines a broken line. Some tets also call this function the floor function, denoted. 4. Graph the following step function: =. 5. Find the range and domain for = and graph it. 6. Find the range and domain for = and graph it. 7. Give the graph for = Give the graph and find the range and domain for =. Cubic Functions 9. Use a table of values to graphs =, the basic form of a cubic function. Use a table of values to graph = ( ). Use a table a velues to graph = +. Draw the graphs for each of these cubics, labelling the intercepts on the - and -aes. a. = b. = c. = ( ) d. = ( + ) e. = ( ) f. = ( + )

8 . Draw the graphs of these cubics. a. = + 4 b. = 4. Match each graph a to d with the correct equation (A) to (D). a. b. c. d. (A) = ( ) (B) = + (C) = ( + ) (D) = NOTE if the equation starts with when epanded, the general trend is for the graph to go up from left to right, if the equation starts with, the graph will trend down from left to right Upward trend Up-Down-Up Downward trend Down-Up-Down

9 Circle: Simple circles, with their center at the origin (, ), have an equation in the form: + = number Points lie on a graph if their coordinates satisf the equation. So, for eample, (, 4) will be a point on the graph that has the equation + = 5 because + 4 = 5. The table shows si points, each of which lie on the graph of + = Write down the coordinates of si more points which fit the equation. (, 4) (-, -4) (, 5) (4, ) (-4, ) (-5, ) 6. Use a compass to draw a circle through the points. 7. What is the radius of this circle? Tip: The radius of the circle + = number is number 8. Consider that the relation + = 9. Making the subject: = ± 9. a. Cop and complete the table below. =± ±.6 - b. Plot the points represented b the pairs in the table. Join up the points with a smooth curve. 9. Draw these circles b using a table of values a. + = 4 b. + = 6 c. + = d. = 4. What is the radius of each of these circles?

10 a. + = 49 b. + = c. + = 64 d. + = The diagram shows the circle + = 6. What are the co-ordinates of A and B? A B 4. The point (6, 8) lies on th circle + = k. Calculate the value of k. 4. The point (-, 5) lies on the circle + = k. Calculate the value of k.

11 Making Graphs from Tables and Graphing Horizontal and Vertical Lines - Black Level Solutions Number of bets Doubling Amount owed = 4 4 = = = 6 6 = = 8. a. represents the number of bets. b. represents the amount owed Amount owed ($) Number of bets

12 5. a. Nine bets b. bets 6..5,.75, = = 4 9. B = (.4) ,.56,.64,.4

13 . a. = 4 b.. A 4. To calculate the -intercept substitute = ; this gives because a = (an number to the power of is equal to ). 5. a = 6. k = =.7 (4 sf) 7. The graph is almost linear, passing through on the -ais. 8. a. b. c. d. 9. a. b.. (, 6)

14 . a. (, 5) b. (, ) c. (, 8) d. (, -). a. D b. A c. B d. E e. C. a. D b. A 4. c. C d. B 5. Solution: The domain of the function consists of. The range is. 6. Solution: Notice ; while takes on onl two values:, -.

15 The graph is the same as the basic cubic but moved unit to the right.. The graph is the same as the basic cubic but moved units upwards.

16 . a. b. c. d. e. f.. a. b. 4. a. (B) b. (C) c. (D) d. (A) 5. (, -5), (-4, -), (, -4), (-, 4), (4, -), (5, ) units

17 8. a. b. =± ±.6 - ±.88 ± ±.88 ± a. b c. d a. 7 b. c. 8 d A = (, 4), B = (-4, ) 4. k = 4. k = 69

18 Bibliograph Information Teachers attempted to cite the sources for the problems included in this problem set. In some cases, sources were not known. Problems Bibliograph Information, 4- Cook, Allen, and Natalia Romalis. Content Area Mathematics for Secondar Teachers The Problem Solver. New York: Christopher-Gordon, Inc., 6. -, -4 Barton, David. Theta Mathematics: NCEA Level, Pearson, 5.