This unit extends students’ previous work with linear and quadratic functions as they investigate polynomials of higher degree. Students rewrite polynomials in different forms, recognizing the benefits of the various forms for their ability to reveal the structure of key features of their graphs.

The unit begins with an introduction to two situations that can be modeled by a polynomial function. Students build their understanding of what polynomials are and what their graphs can look like. Certain aspects, such as end behavior, will be important in a later unit when students explore the end behavior of rational functions.

Graph of polynomial function y = x squared, xy-plane, origin O.  Horizontal axis, scale -75 to 75, by 25’s. Vertical axis, scale -2,000 to 4,000, by 1,000’s. Polynomial graph comes from Quadrant 2, passes down through (-50 comma 2,500), descending in a smooth curve through the origin, up into Quadrant 1 ascending in a smooth curve to the upper right through (50 comma 2,500).

Graph of polynomial function y = x cubed, xy-plane, origin O.  Horizontal axis, scale -75 to 75, by 25’s. Vertical axis, scale -8,000 to 8,000, by 2,000’s. Polynomial graph comes from Quadrant 3, passes up through (-20 comma 8,000), ascending in a smooth curve through the origin, up into Quadrant 1 ascending in a smooth curve to the upper right through (20 comma 8,000).

Graph of polynomial function, xy-plane, origin O.  Horizontal axis, scale -75 to 75, by 25’s. Vertical axis, scale -2,000 to 8,000, by 2,000’s. Polynomial graph comes from Quadrant 3, passes through (-10 comma 10,000), descending in a smooth curve through the origin, up into Quadrant 1 and ascends in a smooth curve to the upper right through (10 comma 10,000).

Focusing on functions expressed in factored form and their graphs, students connect that a factor of \((x-a)\) means \(a\) is a zero of the function and \((a, 0)\) is a horizontal intercept. The effect of the degree and leading coefficient on end behavior is established along with the effect of multiplicity on the shape of the graph near zeros of the function. Taking in all of these features, students learn to sketch polynomial functions expressed as a product of linear factors.

In a previous course, students used the distributive property to multiply factors, and also factored quadratics. Opportunities to review these skills and apply them to polynomials of higher degree are embedded throughout the unit and in practice problems. This prepares students for the final section where students divide a polynomial written in standard form by a suspected factor. From there, the connection between division and multiplication equations is used to establish the Remainder Theorem. This allows the conclusion that if a polynomial has a zero at \(x=a\), then it must also have \((x-a)\) as a factor.

Building on the “Polynomial Functions” unit, in this unit, students transition to working with rational functions, rational equations, and identities.

The unit begins with an introduction to rational functions as students consider situations they can model, such as when minimizing surface area or determining average costs. Students examine the asymptotic behavior of their graphs, which relates to the structure of the equations. Students continue to build on structure as they use polynomial division to rewrite rational expressions for the purpose of identifying the end behavior of the function. Students then focus on solving rational equations and making sense of how some methods can lead to possible solutions that are in fact not solutions (so-called “extraneous solutions”).

In the final section, students study identities. Students sharpen their skills manipulating expressions while proving, or disproving, that two expressions are equivalent. The unit concludes with a return to geometric sequences first examined in the previous unit and, using a polynomial identity proved at the start of the section, students derive the formula for the sum of a finite geometric series before using the formula to solve problems.

In this unit, students use what they know about exponents and radicals to extend exponent rules to include rational exponents, solve equations involving squares and square roots, develop an understanding of complex numbers, and solve quadratic equations that include complex roots.

The unit opens with an optional review of exponent rules before using those rules to justify why \(x^{\frac{a}{b}} = \sqrt[b]{x^a}\) when \(a\) and \(b\) are integers and \(x\) is positive. The new rule leads to an exploration of square and cube roots as solutions to equations of the form \(x^2 = c\) or \(x^3 = c\). In particular, students learn that positive numbers have two square roots and that \(\sqrt{c}\) represents only the positive root.

Students explore equations involving square and cube roots to recognize that squaring each side of an equation can introduce solutions that are not solutions to the original equation.

Note that there are claims in the unit like, “When a square is equal to a negative number, there are no solutions,” which would be more precisely stated as “When a square is equal to a negative number, there are no real number solutions.” However, until students learn about imaginary numbers in the next section, it doesn't make sense to make this distinction.

The third section introduces imaginary and complex numbers by proposing \(i\) as a solution to the equation \(x^2 = \text{-}1\). Students explore the implications of this new type of number by representing it on an imaginary axis off of the real number line. This exploration includes adding imaginary and real numbers together to get complex numbers, and then adding and multiplying complex numbers.

Next, students use their new understanding of complex numbers to solve more quadratic equations. Finally, there is an optional exploration of data representation on a map using circles with area proportional to the data, in which students must use their understanding of square roots to find the radii of the appropriate circles.

In this unit, students explore exponential and logarithmic functions. The unit begins with students recalling geometric sequences and drawing a connection to the growth or decay of values by a constant growth factor. This leads to expressing exponential relationships using functions of the form \(f(x) = a\boldcdot b^x\), where \(a\) is the value of the function when \(x = 0\) and \(b\) is the growth factor. 

Students use different rational inputs, including negative values and values between integers, to better understand exponential functions and their meaning in various contexts. This includes an exploration of growth factors over intervals of different lengths. For example, the same population growth can be described using a growth factor per decade or a different growth factor per year.

The exploration of variables used as an exponent leads to a need to solve equations for the variable and to the introduction of logarithms. Students are presented with traditional logarithm patterns and asked to find patterns and relationships, which leads them to discover that logarithms represent a way to rewrite exponential equations. That is, \(a^y = x\) can be rewritten as \(\log_a (x) = y\). 

The constant \(e\) is introduced and students compare functions of the form \(f(t) = P \boldcdot e^{rt}\) with functions of the form \(g(t) = P \boldcdot (1 + r)^t\).

Finally, students solve exponential and logarithmic equations using graphical and algebraic methods and then interpret the solutions in context. The logarithms in this unit are primarily focused on the bases 10, 2, and \(e\), although other positive bases are mentioned.

Note that, throughout the unit, the bases for both exponential expressions and logarithms are assumed to be positive.

Coordinate plane, horizontal, time since study began, months, 0 to 25 by 5, vertical, ant populations, thousands, 0 to 16 by 4. Dotted lines connect 20 comma 15 million to corresponding spots on the x and y axis. Line c begins at 0 comma 8 point 1. Line r begins at 0 comma 5 point 4.  The lines meet at approximately 20 comma 15 million.

In this unit, students consider functions as a whole and understand how they can be transformed to fit the needs of a situation, which is an aspect of modeling with mathematics. Prior to this unit, students have worked with a variety of function types, such as polynomial, radical, and exponential. Students will build on this work in a later unit to transform periodic functions. By saving the introduction of trigonometric functions until after a study of transformations, students have the opportunity to revisit transformations from a new perspective, which reinforces the idea that all functions, even periodic ones, behave the same way with respect to translations, reflections, and scale factors.

The unit begins with students informally describing transformations of graphs, eliciting their prior knowledge, and establishing language that will be refined throughout the unit. Students begin their investigation with vertical and horizontal translations and reflections across the axes. This leads to both algebraic and geometric descriptions of a function as even, odd, or neither.

Students continue to deepen their understanding by exploring the effects of multiplying the input and output of a function by a scale factor. Students then apply their understanding of translations, reflections, and scaling to graphs and equations of many function types. The unit ends with students applying transformations to different functions to model a real-world data set.

Graph of function g on a coordinate plane. X axis from negative 3 to 5. Y axis from negative 4 to 4. From left to right, the function begins around negative 3 comma 2, moves downward and to the right to about negative 1 comma negative 2, stays level until about 2 comma negative 2, moves upward and to the right to about 3 comma 2, moves downward and to the right to 4 comma negative 1, then stays level until 5 comma negative 1.

In this unit, students are introduced to trigonometric functions. This unit builds directly on the work of a previous unit by having students apply their knowledge of transformations to trigonometric functions and use these functions to model periodic situations.

The unit begins with a deep study of the unit circle, as students study circular motion within familiar contexts. Students then use the Pythagorean Theorem and right-triangle trigonometry to determine the coordinates of points on a circle. They use the similarity of circles and right triangles to generalize to the unit circle, focusing on the important fact that when the hypotenuse has unit length, the length of the legs can be expressed with cosine and sine.

Then students transition to thinking about cosine and sine as functions. They use the unit circle to graph cosine and sine, then expand the domain to all real numbers as they learn the meaning of radian angles greater than \(2\pi\) and less than 0. Students also reason about the input values where the tangent function does not exist and how the output values repeat at regular intervals, leading to the tangent function’s periodic nature.

Next, students apply their previous work with transformations of graphs to trigonometric functions as they identify important features of periodic functions—including midline, amplitude, and period. They apply the work of this unit by modeling periodic or approximately periodic situations both algebraically and graphically.

Students create their own unit circle display in the unit. This display is meant to be a reference tool for students throughout the unit as they transition from a right-triangle-focused understanding of trigonometry to seeing cosine, sine, and tangent as functions with their own inputs and outputs. Students should have access to a unit circle display throughout the unit unless otherwise noted.

This unit focuses on some important uses of randomization in statistics. Initially, students consider three types of study (experimental, observational, and survey) as ways of collecting data. In each form of study, random selection of participants and assignment into any subgroups is important for being able to generalize findings to a larger population. Students notice that an estimation of a proportion or mean of a population from sample data comes with some variability because different samples can result in different estimates. This can be expressed by including a margin of error along with any point estimates given.

Next, students examine the normal distribution as a common model for bell-shaped distributions. With technology, students can use a normal distribution model to approximate the proportion of data within certain intervals. This provides a way to quantify the confidence students should have in their estimates of population proportions or means. It also provides a way to check whether data from an experimental study has enough evidence to conclude that a difference in means between a control and treatment group is due to the treatment.

The unit concludes with an experimental study that students can do together, from study design to data collection and analysis. Then students draw a conclusion about the experiment based on their analysis.

Histogram from 16.5 to 25.5 by 0 point 5’s. Handspan, centimeters. Beginning at 16.5 up to but not including 17, height of bar at each interval is .001, .007, .013, .016, .052, .059, .087, .132, .159, .132, .104, .098, .069, .032, .023, .010, .004, .002, 0. A bell-curve line is drawn at the top of each of the bars across the graph.