1: Relating Changing Quantities
- Page ID
- 233731
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Functions are tools for describing and understanding relationships and change. In this unit, students learn what a function is and how to read, notate, and interpret its graph; how linear and quadratic functions model real-world situations; and how combining, composing, inverting, and transforming a function changes its behavior and meaning. The focus is on interpreting meaning — not just calculating answers — as a foundation for calculus and STEM problem-solving.
- 1.1: Introduction to Functions (FN1)
- This page introduces the concept of functions, defining them as relations where each input corresponds to one output. It presents the vertical line test as a method to visually determine if a graph qualifies as a function. Readers learn to assess different mathematical representations using these criteria.
- 1.2: Introduction to Function Notation (FN2) and Characteristics of a Function's Graph (FN3)
- This page introduces function notation, explaining how to succinctly represent relationships between variables. It emphasizes the significance of understanding functions in mathematics and covers graphical representations of functions, detailing characteristics like intercepts, slopes, and symmetry. These features help in understanding the behavior and applications of functions, making this section a foundational aspect of analyzing mathematical concepts.
- 1.3: Introduction to Slope and Average Rate of Change (LF1)
- This page introduces slope and average rate of change, essential for linear relationships. Slope is defined as the vertical change over horizontal change between two points on a line. Average rate of change, a specific use of slope, measures the average change of a quantity over an interval. The importance of these concepts in various mathematical contexts is highlighted, providing foundational knowledge for analyzing functions and their behavior.
- 1.4: Introduction to Equations of Lines (LF2), Graphs of Linear Equations (LF3), and Linear Models and Meanings (LF5)
- This page provides an overview of linear equations, focusing on their mathematical representation, graphical interpretation, and practical applications. It highlights key concepts such as slope and intercepts, and emphasizes the significance of linking algebraic forms with graphical structures to effectively analyze data and understand linear models in real-world contexts.
- 1.5: Introduction to Graphing Quadratic Functions (PR1) and Quadratic Models and Meaning (PR2)
- This page presents the fundamentals of graphing quadratic functions, focusing on key features such as parabolas' vertex, axis of symmetry, and opening direction. It highlights the practical applications of quadratic models across disciplines like physics and economics, underlining their significance in mathematical modeling and interpretation.
- 1.6: Introduction to Combining and Composing Functions (FN5)
- This page covers the mathematical concepts of combining and composing functions, including addition, subtraction, multiplication, and division. It emphasizes the characteristics of resultant functions and explains function composition, where the output of one function serves as the input to another. Notation and evaluation methods for composed functions are highlighted, along with their significance in various applications, underscoring their importance in function analysis.
- 1.7: Introduction to Finding the Inverse Function (FN6)
- This page explains methods for finding the inverse of a function, detailing how to determine invertibility, swap variables, and solve for the dependent variable. It emphasizes the horizontal line test for assessing if a function is one-to-one, which is essential for the existence of an inverse. Examples are included to enhance understanding of the inverse function process.
- 1.8: Introduction to Transformations of Functions (FN4)
- This page covers function transformations, detailing how translations, reflections, stretches, and compressions alter a function's graph. It emphasizes horizontal and vertical shifts and differentiates between positive and negative transformations. Mastery of these concepts is essential for analyzing and predicting function behavior, laying the groundwork for advanced studies in function manipulation and graph interpretation.