Chapter 6 Functions Checklist

Chapter 6 Standards Checklist

• Calculate and interpret the average rate of change of a function, whether presented algebraically or as a table, over a specified interval and estimate and interpret the rate of change from a graph.
• Graph functions expressed algebraically, showing key features by hand and with technology.
• Compare properties of functions represented algebraically, graphically, numerically in tables, or verbally, which may be of different types and/or represented in different ways.
• Identify the effect on a graph when replacing f(x) by f(x) + k, k * f(x), f(kx), and f(x + k) for specific values of k (both positive and negative), and find the value of k given the graphs.
• Interpret expressions representing a quantity in terms of its context.
• Interpret parts of an expression, such as terms, factors, and coefficients.
• Interpret complicated expressions by viewing parts as single entities.
• Write a function defined by an expression in different but equivalent forms to reveal and explain different properties.
• Use the properties of exponents to interpret expressions for exponential functions in a real-world context.
• Represent data from two quantitative variables on a scatter plot and describe the relationship between the variables then fit a function to the data and use it to solve problems, along with using units to understand and choose appropriate levels of accuracy.
• Find the inverse of a function on an appropriate domain.
• Understand the relationship between exponential functions and logarithmic functions.
• Solve exponential equations using various strategies, including logarithms.
• Understand that a logarithm is the solution to a^bc = d, where a, b, c, and d are numbers.
• Evaluate logarithms using technology.
• Create equations and inequalities in one variable and use them to solve problems in a real-world context.
• Understand solving equations as a process of reasoning and construct a viable argument to justify a solution method.
• Build a function that describes a relationship between two quantities.
• Define sequences as functions, including recursive definitions, whose domain is a subset of the integers.
• Express explicit and recursive formulas for arithmetic and geometric sequences in context and relate them to linear and exponential functions.

Topic 6-1: Key Features of Exponential Functions

• Exponential functions can be expressed as graphs, tables, and equations.
• Transformations of exponential functions display intercepts and end behavior.
• Exponential functions are used to model increasing or decreasing quantities by a fixed percentage over consistent time periods.
• An exponential function takes the form y = a * bx where a and b are constants, a ≠ 0, b > 0, and b ≠ 1.
• Key features include domain, range, intercepts, asymptotes, and end behavior.
• Transformations include shifts and reflections.
• "a" determines the shift (up/down), and the reflection
• If 'a' is negative, the function is reflected
• |a| > 1 is a vertical stretch, 0 < |a| < 1 is a vertical compression.

Topic 6-2: Exponential Models

• Exponential functions can be rewritten to identify rates and the parameters within an exponential function can be interpreted within the context of compound interest problems.
• Dividing the annual growth rate by 12 will not provide the accurate monthly growth rate.
• Compound interest involves earning interest on previously earned interest.
• The compound interest formula is: A=P(1+r/n)^(nt)"
  • P = the initial principal invested
  • r = annual interest rate, written as a decimal
  • n = number of compounding periods per year
  • t = number of years
  • A = the value of the account after t years
• The value "e" is defined as the value that (1 + 1/x)^x approaches as x approaches +∞.
• The number "e" is an irrational number and "e" is the base in A = Pert
  • P = the initial principal invested
  • e = the natural base
  • r = annual interest rate, written as a decimal
  • t = number of years
  • A = the value of the account after t years

Topic 6-3: Logarithms

• Logarithms are the inverse of exponents.
• Logarithms are used to solve exponential models.
• Logarithms can be evaluated with technology.
• ab = c is the same as loga c = b, where log of the result with the given base equals the exponent.
• To convert from exponential form to logarithmic form and vice versa, use B.O.B.
• The base 10 logarithm is called the common logarithm and is written as log x, the base of 10 is implied.
• The base "e" logarithm is called the natural logarithm and is written as "ln x".

Topic 6-4: Logarithmic Functions

• Logarithmic functions are graphed and key features are interpreted and are the inverse of exponential and logarithmic functions.

Topic 6-5: Properties of Logarithms

• Properties of logarithms are used to rewrite logarithmic expressions.
• The Change of Base Formula is used to evaluate logarithmic expressions and solve equations.
• Product Property: log(m*n) = log(m) + log(n)
• Quotient Property: log(m/n) = log(m) - log(n)
• Power Property: logm^n = n * log(m)
• Change of Base Formula: logam = logb m / logb a

Topic 6-6: Exponential and Logarithmic Equations

• Logarithms are used to express the solutions to exponential models, besides solving exponential and logarithmic equations.
• If b > 0 and b ≠ 1, then bx = by if and only if x = y.
• logb x = logb y if and only if x = y.

Topic 6-7: Geometric Sequences and Series

• Geometric sequences can be constructed using a graph, table, or description of a relationship.
• Geometric sequences can be translated between recursive and explicit forms and the formula for the sum of a finite geometric series can be used to solve problems.
• A geometric sequence is a sequence with a constant ratio between consecutive terms and is called the common ratio "r".
• The general definition for any geometric sequence is an = a1 * rn-1.
• Geometric series: Sn=a1(1−rn)/1−r,r = 1
• Sn = (a1 - anr) / (1-r), r = 1