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.⋆
F.IF.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.⋆
FIF.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.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. ⋆

Graph linear and quadratic functions and show intercepts, maxima, and minima

Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

F.IF.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

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.

Use the properties of exponents to interpret expressions for exponential functions.

F.IF. 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

F.BF.1. Write a function that describes a relationship between two quantities.⋆

Determine an explicit expression, a recursive process, or steps for calculation from a context.

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.

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.3. Identify the effect on the graph of replacing f(x) by f(x)+k, kf(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.4. Find inverse functions.

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)=2x3 or f(x)=(x+1)/(x−1) for x≠1.

Verify by composition that one function is the inverse of another.

Read values of an inverse function from a graph or a table, given that the function has an inverse.

Produce an invertible function from a non-invertible function by restricting the domain.

F.LE.3. Distinguish between situations that can be modeled with linear functions and with exponential functions. ⋆

Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

## Standards

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.⋆

F.IF.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.⋆

FIF.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.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. ⋆

- Graph linear and quadratic functions and show intercepts, maxima, and minima
- Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
- Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
- Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
- Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

F.IF.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.- 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.
- Use the properties of exponents to interpret expressions for exponential functions.

F.IF. 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).F.BF.1. Write a function that describes a relationship between two quantities.⋆

- Determine an explicit expression, a recursive process, or steps for calculation from a context.
- 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.
- 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.3. Identify the effect on the graph of replacing f(x) by f(x)+k, kf(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.4. Find inverse functions.

F.LE.3. Distinguish between situations that can be modeled with linear functions and with exponential functions. ⋆

## Examples of Learning Cycles and Tasks

Writing & Graphing Quadratics with real world activities (St. George CA, Ramona F.IF.7 and F.IF.8)

Graphs of non-linear function (North Davis CA, refers to F.IF.7, ... )

Graphs of Quadratics (Murray CA, refers to F.IF.7,...)

Modelling statistical data using quadratic functions

Transformations (North Davis CA, refers to F.BF.3)

Solving Quadratic Equations (? CA, refers to ? )

Completing the Square (Orem CA, refers to A.REI.4)

Completing the Square (to be used after the Orem LC) (Wastach CA refers to A.SSE.3)

Graphing linear, quadratic, absolute value equations (Snow Canyon CA, refers to F.IF.7)

Intro Quadratics (Wasatch CA, refers to F.IF.4)

Piecewise Functions (Salem UT, refers to F.IF.7)

Maximums and Minimums

Recursive equations for quadratic functions (West High CA 2012, refers to F.BF.1)

Functions and Modeling (CopperHills, refers to F.BF.3)

Transformations of Quadratic and Absolute Value Graphs (Cache, UT, refers to F.BF.3)

Quadratic Transformations (Unit Plan, Task 1, Task 2, Task 3) (Copper Hills refers to F.BF.3)

More Quadratic Transformations (West High CA, refers to F.BF.3)

Average Rate of Change (Cache, UT, F.IF.6)

Getting Ready for Quadratic Functions (Build a Function that models a relationship between two quantities)

Diving into Math at Lake Powell ( Richfield CA, refers to F.1F.4, F.1F7a, F.1F.8)

Quadratic Transformations Exploration (Ogden CA F.BF.3)