Exponential Functions and Equations

Unit 8

Common Core Says...

Although the notion of number changes, the four operations stay the same in important ways. The commutative, associative, and distributive properties extend the properties of operations to the integers, rational numbers, real numbers, and complex numbers. Extending the properties of exponents leads to new and productive notation; for example, since the properties of exponents suggest that (5^1/3)^3= 5^(1/3)^3= 5^1= 5, we define 51/3?to be the cube root of 5.

Mathematics - High School: Number and Quantity - Introduction

Big Picture Lesson Planning for the Common Core

At this point in the year, I am pushing my students to connect math concepts, like order of operations, properties of equalities and patterns to non-linear concepts.

Week #20 Introduction to Exponential Functions

• Identify parts of an expression (A.SSE.1)
• Identify a function and it's domain and range (F.IF.1)
• Find an appropriate domain given a context (F.IF.5)
• Evaluate exponential functions (F.IF.2)

Week #21 Radical Functions and Rational Exponents

• Change expressions using the rules of exponents (N.RN.2)
• Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values (N.RN.1)
• Use the properties of exponents to interpret expressions for exponential functions. (F.IF.8.b)
• Write a function that describes a relationship between two quantities. (F.BF.1.a)

Week #22 Modeling Exponential Functions

• Graph exponential functions (F.IF.7.e)
• Identify intercepts and end behavior (F.IF.7.e)
• Translate exponential functions (F.BF.3)
• Compare the key components of a linear function graph to an exponential function graph. (F.IF.4)
• Fit a function to the data; use functions fitted to data to solve problems in the context of the data. (S.ID.6.a)

Lesson Ideas

All of these resources and more can be found in the livebinder below.

Common Core Standards

A.SSE.1.a Interpret parts of an expression, such as terms, factors, and coefficients.

A.SSE.3.c Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15^t can be rewritten as (1.15^1/12)^12t ? 1.01212^t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

N.RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (5^1/3)^3 = 5^(1/3)^3 to hold, so (5^1/3)^3 must equal 5.

N.RN.2 (Rewrite expressions involving radicals and rational exponents using the properties of exponents.)

F.IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

F.IF.2 Understand the concept of a function and use function notation. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

F.IF.3 (Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n - 1) for n = 1.)

F.IF.4?For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.* (Emphasize quadratic, linear, and exponential functions and comparisons among them)

F.IF.5?Interpret functions that arise in applications in terms of the context. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*

F.IF.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

F.IF.8.b Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.

F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, give a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

F.BF.1.a Write a function that describes a relationship between two quantities. (Emphasize linear, quadratic, and exponential functions). Determine an explicit expression, a recursive process, or steps for calculation from a context. Video explaination

F.BF.1.b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.?

F.BF.3 Identify the effect on the graph of 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); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Video explanation

S.ID.6.a Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear and exponential models.

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