5: Quadratic functions
- Page ID
- 238570
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 5.1: Introduction to quadratic functions
- This page covers methods to find zeros of quadratic functions, including factoring and the quadratic formula, alongside graphing techniques based on intercepts and properties. It provides exercises on locating zeros, sketching graphs, and deriving formulas. Additionally, it features a real-world application related to the height of a thrown ball, which emphasizes the importance of interpreting mathematical results contextually.
- 5.2: Vertex of a parabola
- This page explores quadratic functions, focusing on finding vertices, intercepts, and zeros. It discusses transformations and graphical representations, including the axis of symmetry and standard versus vertex forms. The page highlights characteristics of absolute maxima and minima and applies these concepts to problem-solving, like maximizing areas under constraints. Overall, it emphasizes mastering quadratic equations for effective analysis and graphing.
- 5.3: Power functions
- This page examines power functions with various exponents, including positive, negative, and fractional, focusing on graph characteristics and rewriting methods. It discusses how even and odd integers affect graphs and translates functions with negative or fractional powers into positive forms. Additionally, the long-run behavior as \(x\) increases or nears zero is analyzed, resulting in general rules about the behavior of power functions based on the exponent value \(p\).
- 5.4: Power functions and proportionality
- This page explains direct and inverse proportionality in mathematical functions, illustrating with equations and real-world applications such as oil spills and microwave cooking. It offers guidance on finding constants and forming related equations, alongside tasks for practice and verification of learned concepts.
- 5.5: Graphs of Polynomial Functions
- This page covers mathematical functions, particularly polynomials, along with their properties and graphical representations. It addresses finding zeros for functions, analyzing polynomial behavior, and examining a population model and water volume function to interpret maximum values and density implications. The emphasis is on understanding polynomial behavior and its applications in statistical modeling.
- 5.6: Rational functions
- This page discusses mathematical concepts in cost analysis and rational functions, using copper concentration in bronze as an example. It emphasizes the evaluation of functions and details total and average production costs. The trends of average costs with scaling production and the long-run behavior of rational functions are examined. Additionally, the page highlights the use of Desmos for graphing and interpreting these economic principles, enhancing understanding of the concepts presented.
- 5.7: Quadratic Equation Vertex Form
- This page focuses on quadratic functions, detailing the analysis and transformation processes. It teaches how to identify the vertex and axis of symmetry from equations, convert standard forms to vertex form, and adjust constants to ensure the function passes through specific points. The page also covers creating quadratic equations based on properties such as vertex, intercepts, and symmetry, and encourages using graphing tools like Desmos for a better visual comprehension.
- 5.8: Dividing Polynomials
- This page offers practice problems on polynomial division, beginning with long division for the first six problems featuring various degrees divided by linear expressions. It then shifts to synthetic division for problems 7 to 11, also involving specific polynomials and linear factors. Overall, the exercises aim to enhance comprehension of both long and synthetic division methods in algebra.
- 5.9: Dividing Polynomials
- This page features problems on polynomial division, specifically long division and synthetic division, with six long division and five synthetic division exercises. It challenges readers to divide polynomials by linear divisors. Additionally, it includes a Desmos exploration task for students to evaluate a synthetic division problem at a specific value, encouraging them to analyze the connection between the function's value and the remainder from the division.
- 5.10: Zeros of Polynomial Functions
- This page outlines exercises on polynomial functions, including graphing, analyzing behavior in various viewing windows, and finding zeros. It covers factoring polynomials and the effects of even and odd powers on graph shapes. The page also involves practical applications, such as designing an open-top box, which requires sketching, deriving volume formulas, and determining maximum volume conditions.
- 5.11: Rational functions
- This page discusses cost and concentration in alloys, using bronze as a case study for copper-tin compositions. It explores how changes in copper concentration affect production costs and average costs, illustrating this relationship with examples. The page evaluates trends in average cost calculations and the long-run behavior of rational functions, supported by graphical representations.