7.1 Exponential Functions...Exponential functions of the form !=!!, where 0
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Chapter 7- Exponential Functions 7.1 Characteristics of Exponential Functions Exponential Functions are of the form = ! , where c is a constant (c > 0) and x is a variable. Consider the functions: = 2 ! and = ( 1 2 ) Increasing Decreasing The graph of an exponential function such as = ! , is increasing when > 1, and decreasing when 0 < < 0, and neither increasing nor decreasing when = 1. From the graph you can determine characteristics such as domain and range, intercepts, and any asymptotes. When: = 1 !
Transcript of 7.1 Exponential Functions...Exponential functions of the form !=!!, where 0
Chapter 7- Exponential Functions 7.1 Characteristics of Exponential Functions Exponential Functions are of the form 𝑦 = 𝑐!, where c is a constant (c > 0) and x is a variable.
Consider the functions: 𝑦 = 2! and 𝑦 = ( 12 )𝑥
Increasing Decreasing
The graph of an exponential function such as 𝑦 = 𝑐! , is increasing when 𝑐 > 1, and decreasing when 0 < 𝑐 < 0, and neither increasing nor decreasing when 𝑐 = 1. From the graph you can determine characteristics such as domain and range, intercepts, and any asymptotes.
When: 𝑦 = 1!
Ex 1: Graph each exponential function. Identify the following: -Domain/Range -x-intercept, y-intercept -Increasing or decreasing function? -equation of the horizontal asymptote a) 𝑦 = 4𝑥 b) 𝑦 = ( 13 )
𝑥
Solution a) b)
Domain
Range
x-intercept
y-intercept
Increasing/decreasing
Horizontal Asymptote
Domain
Range
x-intercept
y-intercept
Increasing/decreasing
Horizontal Asymptote
Ex 2: Writing an Exponential Function Given Its Graph
Ex 3: A radioactive sample of radium has a half-life of 15 days. The mass, m, in grams, of radium remaining over time, t, in 15-day intervals, can be modeled using the exponential graph shown a) What is the initial mass of radium in the sample? What value does the mass of radium approach as time passes? b) What are the domain and range of this function? c) Write the exponential decay model that relates to the mass of radium
d) Estimate how many days it would take for radium to decay to !!"
of its original mass
Exponential Functions of the form 𝑦 = 𝑐! , where 𝑐 > 1, can be used to model exponential growth. This signifies the increasing (or growing) pattern of values. Exponential functions of the form 𝑦 = 𝑐! , where 0 < 𝑐 < 1 can be used to model exponential decay. This signifies the decreasing (or decaying) pattern of values Half Life: The length of time for an unstable element to spontaneously decay to one-half it’s original mass.
Time (15-‐day intervals)
Mass of radium (g)
Homework: page 342#1,2,3,4(a), 5(b,d), 7, 8, 10, 11
