Lesson 31 Graphs of Logarithmic Functions Example 1: Complete the input/output table for the function 푓(푥) = log2(푥), and use the ordered pairs to sketch the graph of the function. After graphing, list the domain, range, zeros, positive/negative intervals, increasing/decreasing intervals, and the intercepts. 푓(푥) Outputs Inputs Outputs (these are (these are exponents powers of the that produce the powers base 2) of the base 2) 푥 푓(푥) = log2(푥) 푥 → 0 푓(푥) → −∞ 1 1 log ( ) = 2 16 −ퟒ Notice that the graph of 16 푓(푥) = log2(푥) is 1 1 increasing throughout its log2 ( ) = −ퟑ domain. When a function 8 is always increasing or 8 always decreasing that function is one-to-one, and 1 1 it will have an inverse log ( ) = 2 4 −ퟐ function. The inverse of a 4 logarithmic function is an exponential function. 1 1 log2 ( ) = −ퟏ Domain: (ퟎ, ∞) 2 2 Range: (−∞, ∞) 1 log2(1) = ퟎ Zeros: 푓(푥) = 0 when 푥 = ퟏ Positive intervals: 2 log2(2) = ퟏ 푓(푥) > 0 when 푥 is (ퟏ, ∞) Negative intervals: 4 log ( 4) = ퟐ 2 푓(푥) < 0 when 푥 is (ퟎ, ퟏ) 퐈퐧퐜퐫퐞퐚퐬퐢퐧퐠 퐢퐧퐭퐞퐫퐯퐚퐥퐬: 8 log (8) = ퟑ 2 풇(풙) 퐢퐬 퐫퐢퐬퐢퐧퐠 퐰퐡퐞퐧 풙 퐢퐬 (ퟎ, ∞) Decreasing intervals: 16 log (16) = ퟒ 2 푓(푥) is falling when 푥 is 퐍퐎퐍퐄 Intercepts: 푥 → ∞ 푓(푥) → ∞ 푥 − intercept: (ퟏ, ퟎ) 푦 − intercept: 퐍퐎퐍퐄 1 Lesson 31 Graphs of Logarithmic Functions Vertical Asymptote: - a vertical line (푥 = #) that the graph of a function approaches, but never touches or crosses, when the inputs approach an undefined value (푥 → #, where # is a value that is not part of the domain) o in the case of 푓(푥) = log2(푥), as the inputs get closer and closer to zero (푥 → 0), the outputs get smaller and smaller (푓(푥) → −∞), so the graph has a vertical asymptote at 푥 = 0 o the graph of every logarithmic function will have a vertical asymptote (푥 = #) - in a later example (Example 2) I will denoted the vertical asymptote with a dotted line to make it easier to identify Example 2: Re-write the function 푔(푥) = log2(푥 + 2) in terms of 푓(푥) = log2(푥). Then find the 푥-intercept of 푔 and find its graph by transforming the graph of the original function 푓. Enter exact answers only (no approximations) for the 푥-intercept. 푓(푥) Outputs 푓(푥) = log2(푥) Re − write 푔(푥) in terms of 푓(푥): 품(풙) = 푥 Inputs When solving problems like this on the homework, you can use the transformation and the 푥-intercept 풙 − 퐢퐧퐭퐞퐫퐜퐞퐩퐭: to get the graph, or you can simply use transformations only to get the 0 = log2(푥 + 2) graph, and then identify the 푥-intercept from the graph. 0 2 = 푥 + 2 To find the 푥-intercept algebraically, remember that a logarithm 1 = 푥 + 2 is an exponent, so anything equal to a logarithm is also an exponent. In this example, the logarithm log2(푥 + 2) represents −1 = 푥 the exponent that makes the base 2 equal to the argument 푥 + 2. Since 0 is equal to log2(푥 + 2), 0 is the exponent that makes 2 (−ퟏ, ퟎ) equal to 푥 + 2. 2 Lesson 31 Graphs of Logarithmic Functions Example 3: Re-write each of the following functions in terms of 푓(푥) = log2(푥), then match the transformation with the appropriate graph. Also, find the 푥-intercepts of each function. 푓(푥) Outputs a. ℎ(푥) = − log2(푥) 푓(푥) = log (푥) 2 풉(풙) = 풙 − 퐢퐧퐭퐞퐫퐜퐞퐩퐭: 푥 0 = − log2(푥) → Inputs ( ,0) 푓(푥) Outputs b. 푗(푥) = log2(−푥) 푓(푥) = log2(푥) 풋(풙) = 풙 − 퐢퐧퐭퐞퐫퐜퐞퐩퐭: 푥 0 = log2(−푥) → Inputs ( ,0) 풙 − 퐢퐧퐭퐞퐫퐜퐞퐩퐭: 푙(푥) = (log2(푥)) − 2 3 Lesson 31 Graphs of Logarithmic Functions LON-CAPA Problem : Given the function 푓(푥) = , along with it’s graph below, complete the following: a. Express the new function ℎ(푥) = 2 푥+2 in terms of the original function 푓. 풉(풙) = 풚 − 퐢퐧퐭퐞퐫퐜퐞퐩퐭: b. Find the 푥-intercept of the function ℎ(푥) = 2 푥+2 and enter your answer as an ordered pair (푥, 푦). Enter exact answers only, no approximations. 풙 − 퐢퐧퐭퐞퐫퐜퐞퐩퐭 (풙, 풚) = c. Transform the graph of 푓 to get the graph of ℎ(푥) = 2푥+2 . Use the 푥-intercept of ℎ to verify that your transformation is correct. 푓(푥) Outputs Keep in mind that even though the point you’re given on the graph maybe the 푥-intercept, after transforming the point it 푥 Inputs may no longer be an intercept. Use the answer from part a. to determine how to transform the point that you are given on the graph. After transforming the graph, use your answer from part b. to verify that the 푥-intercept on the new graph is correct. 4 Lesson 31 Graphs of Logarithmic Functions Example 4: Re-write each of the following functions in terms of 푓(푥) = log2(푥), then match the transformation with the appropriate graph. Also, find the 푥-intercepts of each function. 푓(푥) a. 푘(푥) = (log2(푥)) − 2 Outputs 푓(푥) = log2(푥) 풌(풙) = 풙 − 퐢퐧퐭퐞퐫퐜퐞퐩퐭: 푥 Inputs 0 = (log2(푥)) − 2 → ( ,0) 풙 − 퐢퐧퐭퐞퐫퐜퐞퐩퐭: 푓(푥) = log2(푥) b. 푛(푥) = log2(2푥) 풏(풙) = 푥 풙 − 퐢퐧퐭퐞퐫퐜퐞퐩퐭: Inputs 0 = log2(2푥) → ( ,0) 5 Lesson 31 Graphs of Logarithmic Functions Answers to Examples: 2. 푔(푥) = 푓(푥 + 2), 푥 − intercept: (−1, 0) ; 3a. ℎ(푥) = −푓(푥), 푥 − intercept: (1, 0) ; 3b. 푗(푥) = 푓(−푥), 푥 − intercept: (−1, 0) ; 4a. 푘(푥) = 푓(푥) − 2 , 푥 − intercept: (4, 0) ; 1 4b. 푛(푥) = 푓(2푥), 푥 − intercept: ( , 0) ; 2 6 .