Post on 28-Sep-2020
Logarithmic Functions Lesson 2-‐ Graphing and Transforming Logarithmic Functions We know that: The inverse of an exponential function is a logarithmic function Consider the exponential equation: 𝑦 = 𝑏! , where b >0 and b≠ 1. The inverse of this function is: It is written as a logarithmic function: Determine the inverse of the function Ex1: 𝑓 𝑥 = 3! Ex2: 𝑓 𝑥 = log!(5𝑥) Ex3: Given 𝑓 (𝑥)=10!!!" − 7, what is the value of 𝑓!! 4 to the nearest hundredth? Graphing a Logarithmic Function In the logarithmic function 𝑦 = log! 𝑥, where b > 0 and b ≠ 1, the value of b changes the shape of the graph. To understand the effect of parameter b, consider the following:
a) State the inverse of 𝑓 𝑥 = 3! and sketch a graph of the inverse. Identify: Domain, Range, Intercepts, and
asymptotes Inverse:
x y x y -‐3 -‐2 -‐1 0 1 2 3
Negative values of b are excluded because a logarithmic function with a negative b-‐value yields an exponential function with a negative base
𝑓(𝑥) = 3! 𝑓!!(𝑥) = log! 𝑥
Domain: Range: x-‐intercept: y-‐intercept: Asymptotes:
To calculate/graph logarithmic functions on calculators:
MATH à A: LogBASE à ENTER
Plug in values for the base and your input x
Homework: Page380#1,8,9,16 Page389#1,2,5,6,8,10
Ex: Sketch the graph of the function 𝑦 = log!!𝑥. Determine the vertical asymptote, and state the domain and range.
Inverse: Step 1: Build a table of values
Applying Transformations to the Graph of y = log! x , Where b > 0 You can graph a logarithmic function by performing a series of transformations on the graph y = log! 𝑥, where b>0. When a series of transformations are performed on the graph y = log! 𝑥, the result is the graph:
𝑦 = 𝑎 log! 𝑐 𝑥 − ℎ + 𝑘 Where:
• A vertical stretch about the x-‐axis by a factor of 𝑎
• A horizontal stretch about the y-‐axis by a factor of !!
• If h > 0, a horizontal translation h units right. If h < 0, a horizontal translation h units left
• If k > 0, a vertical translation k units up. If k < 0, a vertical translation k units down
x y x y -‐2 -‐1.5 -‐1 -‐0.5 0 1
𝑦 = log!!𝑥
Domain: Range: x-‐intercept: y-‐intercept: Asymptotes:
Translations of a logarithmic function
a. Use transformations to sketch the graph of the function 𝑦 = log!(𝑥 + 9) + 2 b. Identify the following characteristics of the graph: equation of asymptotes, the domain and range, &
intercepts
Reflections, Stretches and Translations of a Logarithmic Function
a. Use transformations to sketch the graph of the function 𝑦 = − log!(2𝑥 + 6) b. Identify the following characteristics of the graph: equation of asymptotes, the domain and range, &
intercepts