In these activities, students model the height of a projectile using a quadratic function. The formula forthis function is derived using scaling and shifting, starting with the function 𝑓(𝑡) = 𝑡2. They alsoinvestigate different algebraic representations for the same function by composing sequences oftransformations and inverting these sequences. Students practice using standard form, vertex form, androot form for the formula of a quadratic function in the context of a model. The activities reinforcestudents’ algebraic skills in solving equations and using the quadratic formula, and introduces them to theskill of factoring quadratic polynomials. The concept of total change as the area under a rate of changecurve is explored. Since the total change is quadratic when the rate of change varies as a (non-constant)linear function, many of the techniques of previous lessons for interpreting quadratic functions arereviewed.

In these activities, students solve exponential equations using the compound interest formula. Theyexplore the basics of savings bonds to see that the interest rate has the greatest influence on theinvestment. Students also apply logarithmic rules learned in previous activities to solve for the variable inan exponential equation. Students are introduced to the ideas of guaranteed savings bonds and how theactual value differs from the federally guaranteed amount. They are then introduced to the “Rule of 72” toapproximate doubling times of exponential functions quickly. Students also develop their skills at solvingequations that contain a single logarithmic expression or two or more logarithmic expressions and checkfor extraneous solutions.

In these activities, students extend their knowledge of function composition to rational functions. Studentssee that the domain of a composite function 𝑔(𝑓(𝑥)) is a subset of the domain of 𝑓(𝑥) and may be furtherrestricted by the domain of 𝑔(𝑥). Similarly, they see how asymptotes of the composite function 𝑔(𝑓(𝑥))are related to asymptotes of the functions 𝑔(x) and 𝑓(x). Students also continue to explore how change istransmitted through function composition. By exploring a distance-rate-time scenario, students realize theimportance of being able to add two rational expressions. Direct instruction leads students through thealgebra of finding a common denominator and helps them compute the final sum. Adding rationalexpressions that require a first step of finding a common denominator is focused on, as well. The need toperform the addition is emphasized as it allows one to solve certain contextual problems. Studentscontinue adding and subtracting rational functions to create new rational functions. Students focus onadding or subtracting rational functions in which the denominators share common factors. 

In these activities, students use tables and graphs to examine behavior near discontinuities of a rationalfunction. They also explore the end behavior of rational functions and relate the end behavior to STEMproblems.

In these activities, students work with functions that feature fractional (or rational) exponents. Theycontinue using the rules of exponents to simplify expressions involving fractional exponents. Studentsalso work with functions that feature fractional exponents, determine the inverse of such a function, andcontinue using the rules of exponents to simplify complicated expressions that involve exponents. Theyalso learn to identify whether a function has an inverse. Students study functions with fractionalexponents and their graphs. They explore the shapes of graphs of power functions 𝑦 = 𝑥𝑝/𝑞, where 𝑝 and 𝑞are integers, and describe properties of the graph depending on whether 𝑝 and 𝑞 are even or odd. 

In these activities, students begin learning the terminology of dependent and independent variables anddifferentiating between these two types of variables given algebraic or graphical representations offunctions. Students also start to connect the independent variable to input values of functions and thedependent variable to output values. Students discover that a function is more than a formula, table, orgraph, and learn that a function is a process that takes input values and assigns output values. Givendefinitions for the domain and range of a function, students also explore how arithmetic as well as contextcan affect domain and range. Finally, students will begin evaluating, graphing, and interpretingexpressions such as f(x + 1) and f(x) + 1.

In these activities, students are introduced to polynomial functions in the context of modeling the heightof an object and the motion of an object along a straight line. Students investigate these models usinggraphing calculators or apps and reflect on the advantages and limitations of modeling with polynomialfunctions. A method for finding a formula for a polynomial given its graph is also presented, and studentswill practice with multiplying polynomials. Students are also given additional exposure to modeling withpolynomials, investigating their graphs, and manipulating them algebraically. The problem of factoringpolynomials as an example of reverse engineering is also presented.