F. Functions
- As the Wheel Turns
- Average Cost
- Containers
- F-IF From the flight deck
- F-IF Graphing Stories
- Hoisting the Flag 1
- Hoisting the Flag 2
- How is the Weather?
- Influenza epidemic
- Lake Sonoma
- Logistic Growth Model, Abstract Version
- Logistic Growth Model, Explicit Version
- Model air plane acrobatics
- Modeling London's Population
- Playing Catch
- Solar Radiation Model
- Telling a Story With Graphs
- The Aquarium
- The Canoe Trip, Variation 1
- The Canoe Trip, Variation 2
- The story of a flight
- Throwing Baseballs
- Warming and Cooling
- Words - Tables - Graphs
F-IF.A. Understand the concept of a function and use function notation.
F-IF.A.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.A.2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
F-IF.A.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 $n \ge 1$.
F-IF.B. Interpret functions that arise in applications in terms of the context.
F-IF.B.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.
F-IF.B.5. 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.B.6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
F-IF.C. Analyze functions using different representations.
F-IF.C.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
F-IF.C.7.a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
F-IF.C.7.b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
F-IF.C.7.c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
F-IF.C.7.d. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
F-IF.C.7.e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
F-IF.C.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
F-IF.C.8.a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
F-IF.C.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.C.9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
- Building a General Quadratic Function
- Building an Explicit Quadratic Function by Composition
- Building a quadratic function from $f(x) = x^2$
- Exploring Sinusoidal Functions
- Identifying Even and Odd Functions
- Identifying Quadratic Functions (Standard Form)
- Identifying Quadratic Functions (Vertex Form)
- Medieval Archer
- Transforming the graph of a function
- No tasks yet illustrate this standard.
F-BF.A. Build a function that models a relationship between two quantities.
F-BF.A.1. Write a function that describes a relationship between two quantities.
F-BF.A.1.a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
F-BF.A.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.A.1.c. Compose functions. For example, if $T(y)$ is the temperature in the atmosphere as a function of height, and $h(t)$ is the height of a weather balloon as a function of time, then $T(h(t))$ is the temperature at the location of the weather balloon as a function of time.
F-BF.A.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.B. Build new functions from existing functions.
F-BF.B.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.
F-BF.B.4. Find inverse functions.
F-BF.B.4.a. Solve an equation of the form $f(x) = c$ for a simple function $f$ that has an inverse and write an expression for the inverse. For example, $f(x) =2 x^3$ or $f(x) = (x+1)/(x-1)$ for $x \neq 1$.
F-BF.B.4.b. Verify by composition that one function is the inverse of another.
F-BF.B.4.c. Read values of an inverse function from a graph or a table, given that the function has an inverse.
F-BF.B.4.d. Produce an invertible function from a non-invertible function by restricting the domain.
F-BF.B.5. Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
- Comparing Exponentials
- Comparing Graphs of Functions
- Dido and the Foundation of Carthage
- Extending the Definitions of Exponents, Variation 2
- Rising Gas Prices – Compounding and Inflation
- Triangular Numbers
- What functions do two graph points determine?
- Basketball Bounces, Assessment Variation 1
- Basketball Bounces, Assessment Variation 2
- Boiling Water
- Choosing an appropriate growth model
- Exponential Functions
- Interesting Interest Rates
- Linear or exponential?
- Algae Blooms
- Allergy medication
- A valuable quarter
- Basketball Bounces, Assessment Variation 1
- Basketball Bounces, Assessment Variation 2
- Basketball Rebounds
- Boiling Water
- Boom Town
- Carbon 14 dating in practice II
- Choosing an appropriate growth model
- Decaying Dice
- Do two points always determine a linear function?
- Do two points always determine a linear function II?
- Do two points always determine an exponential function?
- Exponential Parameters
- Finding Parabolas through Two Points
- F-LE, A-CED Paper Folding
- Identifying Exponential Functions
- Moore's Law and Computers
- Population and Food Supply
- Predicting the Past
- Rumors
- Sandia Aerial Tram
- Snail Invasion
- Temperatures in degrees Fahrenheit and Celsius
- Two Points Determine an Exponential Function I
- Two Points Determine an Exponential Function II
F-LE.A. Construct and compare linear, quadratic, and exponential models and solve problems.
F-LE.A.1. Distinguish between situations that can be modeled with linear functions and with exponential functions.
F-LE.A.1.a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
F-LE.A.1.b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
F-LE.A.1.c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
F-LE.A.2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
F-LE.A.3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
F-LE.A.4. For exponential models, express as a logarithm the solution to $ab^{ct} = d$ where $a$, $c$, and $d$ are numbers and the base $b$ is 2, 10, or $e$; evaluate the logarithm using technology.
F-LE.B. Interpret expressions for functions in terms of the situation they model.
F-LE.B.5. Interpret the parameters in a linear or exponential function in terms of a context.
- No tasks yet illustrate this standard.
- No tasks yet illustrate this standard.