Functions

This math unit develops the understanding and skills related to slopes and equations of lines, with a specific focus on parallelism. Initially, students learn to recognize and convert line equations between different forms, starting from understanding simple forms such as slope-zero intercept and slope-intercept forms, to more complex transformations involving standard forms and decimal representations of slope. As the unit progresses, the emphasis shifts to applying these foundational skills to understand parallel lines. Students practice identifying parallel slopes by converting equations between various formats including zero-intercept, slope-y-intercept, fraction form, and graph representation to standard forms. Through these exercises, students enhance their ability to interpret and manipulate different algebraic expressions of linear equations, deepening their grasp of how slopes indicate parallelism and how lines can be graphically and algebraically analyzed and compared for this property.Skills you will learn include:

• Parallels
• Y = mx + b
• Ax + Cy = C
• Parallels of line equations
• Graphs from parallels

This math unit begins by developing students' abilities to interpret and identify different types of function graphs based on their corresponding algebraic equations. Initially, students match specific graphs to their equations and recognize the type of function each equation represents. The unit progresses to enhance understanding of function domains and ranges through visual representations, starting from identifying domains and ranges, to determining whether given relations qualify as functions based on their visual mappings. Further along, the unit delves deeper into defining and describing domains and ranges more formally. Students learn to translate number line visuals into verbal descriptions and vice versa, convert verbal descriptions into mathematical inequalities without unions, and then express these inequalities using set builder notation. Finally, the unit teaches students to transform set builder notations into interval notations and accurately represent these intervals on a number line, further solidifying their comprehension of domain and range concepts in the context of functions.Skills you will learn include:

• Functions as maps
• Domain and range
• Definition of a function
• Inequalities
• Set builder
• Number line

This math unit focuses on understanding and applying the concept of perpendicular slopes within different contexts of linear equations. It begins with basic calculations to find the negative reciprocal of given integer slopes and progresses to handling fractional and decimal slopes to identify perpendicular lines. The unit further develops by having learners convert and determine perpendicular slopes between different forms of linear equations, such as slope-intercept form, zero-intercept form, and standard form. Additionally, learners practice converting these equations for graphical representation, aiding in visual understanding and verification of perpendicular relationships. The depth of the unit increases as students move from initially identifying perpendicular slopes in simpler formats to manipulating complex algebraic forms and graphing them, thus building a comprehensive skill set in analyzing and constructing perpendicular lines within coordinate geometry. Throughout the unit, the primary emphasis remains on mastering the concept that the product of the slopes of perpendicular lines is -1, and applying this understanding in various mathematical scenarios.Skills you will learn include:

• Perpendiculars
• Y = mx + b
• Ax + Cy = C
• Perpendiculars of line equations
• Graphs from perpendiculars

This math unit begins with the foundational concept of identifying y-intercepts from linear equations in slope-intercept and standard forms using integer coefficients. As the unit progresses, it introduces the concept of x-intercepts, requiring students to manipulate equations set to zero in either variable while still using integer values. The complexity increases as the unit shifts to equations involving decimal coefficients. This additional challenge tests the students' ability to work with more precise values and enhances their algebraic manipulation skills. Towards the end of the unit, the focus shifts to finding intersection points between different types of lines including horizontal, vertical, and other linear equations demonstrating both integer and decimal solutions. This progression from basic intercept identification to solving for intersections between various lines helps students understand the graphical behavior of linear equations and their points of intersection.Skills you will learn include:

• Intersections
• Y = mx + b
• Horizontal line intersections
• Vertical line intersections
• Two linear equation intersections

This math unit delves into the fundamentals of graphing circles on a coordinate plane, beginning with basic identification and placement of circle centers. Students first learn to graph circles given the center coordinates and then advance to calculating circle graphs based on specified radii. Progressing further, the unit teaches conversion of a circle's equation into its graphical representation by interpreting the meaning of value coefficients in the standard circle equation \( (x-h)^2 + (y-k)^2 = r^2 \). Students continue to refine their skills by extracting center coordinates and radii directly from the circle equations, enhancing their ability to visualize and plot these geometric forms accurately. The subsequent lessons expand these concepts by guiding students to derive a circle's standard equation using both center coordinates and radius. Towards the end of the unit, learners are equipped to transform a complete circle graph back into its algebraic equation, solidifying their understanding of the relationship between a circle’s graphical and algebraic representations. This step-by-step progression effectively builds foundational knowledge crucial for solving geometry and coordinate graphing problems involving circles.Skills you will learn include:

• Graphing circles
• Circle equation format
• Finding radius
• Finding center coordinate

This math unit begins by teaching students how to visually interpret and define functions through relation maps and progresses to detailed explanations of domain and range using various mathematical notations and representations. Starting with basic function definitions, students learn how to find domain and range using number lines and inequalities, gradually moving to translating these into set builder and interval notations without unions. The complexity increases as students learn to navigate between different representations, including verbal descriptions and symbolic expressions. Towards the end of the unit, the scope expands to include unions in domains and ranges, enhancing their ability to handle more complex scenarios by interpreting inequalities, set builder notations, and intervals that involve combining separate mathematical intervals. This progression builds a foundational understanding of functions, crucial for further study in calculus and algebra.Skills you will learn include:

• Domain and range of functions
• Unions in domain and range
• Inequalities
• Set builder
• Number line

This math unit progresses through a series of skills, focusing on understanding and defining the domain and range of functions through various representations and conditions. Initially, the unit addresses basic visual and symbolic representations involving the translation of inequalities and verbal descriptions into number lines and set builder notation, utilizing unions to depict composite intervals. As students progress, they learn to convert these representations into interval notation, further solidifying their understanding of function behavior over specified domains and ranges. The unit advances into more complex scenarios where students must determine function domains involving higher polynomials, linear and quadratic expressions under square roots, and functions defined by fractional expressions over linear, quadratic, and radical denominators. The tasks require careful analysis to avoid undefined expressions and adhere to the conditions provided by each mathematical model, enhancing proficiency in translating between different mathematical representations of domains, such as number lines and algebraic conditions. The increasing complexity aids in mastering the algebraic and graphical understanding necessary to analyze functions critically within defined parameters.Skills you will learn include:

• Domain and range of functions
• Unions in domain and range
• Domain of Linear Functions
• Domain of Fractions
• Domain of Radicals

This math unit focuses on the end behavior of polynomial functions, progressing from basic understanding to more complex application and analysis. Initially, students learn to interpret the end behavior of polynomials through graphical analysis, identifying trends as \(x\) approaches positive or negative infinity. They then progress to matching these behaviors with appropriate graphical representations and deducing rules based on graph trends. As the unit advances, students deal with the correlation between the highest power and the leading coefficient of polynomials and how these influence the end behavior. They learn to determine these elements from graphs, rules, and directly from polynomial functions. Finally, the unit covers predicting polynomial graphs based on a given set of highest power and leading coefficient values, wrapping up with an ability to match polynomial functions to their respective end behaviors and rules, essentially preparing them for practical applications and deeper studies in polynomial functions.Skills you will learn include:

• End behaviour of functions
• Highest power and leading coefficient rules
• Identify functions from end behavour

This math unit progresses through a sequence of increasing complexity in understanding and identifying the domain of various functions. It begins with basic skills to determine the domain of simple linear functions, recognizing which number line illustrates these domains correctly. The unit introduces polynomial functions of higher degrees, including understanding domains for fractional functions with linear and quadratic components, and progresses to involve more complex function forms like square roots and those with complex roots. Learners start with linear and simple polynomial functions, extending their skills to scenarios involving square and cube roots, which require understanding conditions for real number outputs. The focus then shifts to more complex algebraic structures, such as rational functions where the numerator and/or the denominator may consist of linear or quadratic expressions, including those with complex roots. Towards the advanced stages, the emphasis is on mastering domains involving the interplay of roots and quadratic functions, culminating with fractions that impose multiple domain constraints. This comprehensively builds the learners’ ability to analyze, calculate, and graphically represent function domains across various algebraic configurations.Skills you will learn include:

• Domain and range of functions
• Unions in domain and range
• Domain of Linear Functions
• Domain of Fractions
• Domain of Radicals

This math unit starts with understanding the basics of function composition, focusing on identifying the input, output, inner, outer, first, and second functions in various composite configurations. Skills progress to calculating the outputs of these compositions, reinforcing comprehension through practical application. The unit then advances to evaluating the domains of these compositions, particularly when involving root, linear, rational, or quadratic functions, underscoring the importance of determining the set of input values that make the function compositions well-defined and meaningful. The culmination of the unit involves mastering the concept of function inverses, both for straightforward functions and more complex exponential and logarithmic forms. Students learn to discern whether pairs of functions are inverses and calculate the inverse of given functions to deepen their understanding of functional relationships, preparing them for more advanced studies in function operations and algebra.Skills you will learn include:

• Composition of Functions
• Domain of Composed Functions
• Inversion of Functions

This math unit ushers students through a progressive exploration of the domain of functions represented by various types of fractional expressions. It begins with simpler concepts like determining the domain of functions formed by fractions where the denominator is a linear or quadratic expression, emphasizing understanding whether denominators could hit zero or if roots are complex. The unit then escalates to handle square roots in the denominator, moving from linear roots to more complex quadratic roots, immediately instructing students to address non-negative conditions that ensure valid domains. Equipped with these foundational skills, students advance to scenarios where both numerators and denominators incorporate linear functions, quadratic functions, or square roots. Each type of fractional function challenges students to consider multiple domain constraints such as avoiding undefined expressions and non-negative square roots. By the culmination of the unit, students are proficient in graphically representing these domains on number lines. The skill set includes algebraic manipulation to identify domain boundaries and use of number lines to clearly illustrate valid ranges of inputs for diverse and increasingly complex rational functions.Skills you will learn include:

• Domain and range of functions
• Unions in domain and range
• Domain of Linear Functions
• Domain of Fractions
• Domain of Radicals