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Linear Algebra for MATH 308
Lecture 1: Vectors, Linear Independence, and Spanning Sets
Lecture 2: Operations with Matrices and Vectors
Lecture 3: Systems of Equations
Lecture 4: Determinant
Lecture 5: Eigenvectors and Eigenvalues
Lecture 6: Matrix Inverses and Diagonalization
Lecture 7: Systems of Differential Equations
Lecture 8: Systems of Differential Equations
Quantitative Finance
Several Variables Calculus
Section 1: Functions of Several Variables
Section 2: Limits and Continuity
Section 3: Partial Derivatives
Section 4: Tangent Planes and Linear Approximations
Section 5: The Chain Rule
Section 6: Directional Derivatives and the Gradient Vector
Section 7: Maximum and Minimum Values
Section 8: Lagrange Multipliers
Differential Equations
Section 1: Integrating Factor
Section 2: Separable Equations
Section 3: Compound Interest
Section 4: Variation of Parameters
Section 5: Systems of Ordinary Differential Equations
Section 6: Matrices
Section 7: Systems of Equations, Linear Independence, and Eigenvalues & Eigenvectors
Section 8: Homogeneous Linear Systems with Constant Coefficients
Section 9: Complex Eigenvalues
Section 10: Fundamental Matrices
Section 11: Repeated Eigenvalues
Section 12: Nonhomogeneous Linear Systems
Mathematical Probability
Section 1: Probabilistic Models and Probability Laws
Section 2: Conditional Probability, Bayes’ Rule, and Independence
Section 3: Discrete Random Variable, Probability Mass Function, and Cumulative Distribution Function
Section 4: Expectation, Variance, and Continuous Random Variables
Section 5: Discrete Distributions
Section 6: Continuous Distributions
Section 7: Joint Distribution Function, Marginal Probability Mass Function, and Uniform Distribution
Section 8: Independence of Two Random Variables, Covariance, and Correlation
Section 9: Conditional Distribution and Conditional Expectation
Section 10: Moment Generating Function
Section 11: Markov’s Inequality, Chebyshev’s Inequality, and Weak Law of Large Numbers
Section 12: Convergence and the Central Limit Theorem
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Section 1.6&7 – Parent Functions and Transformations
Section 1.6&7 – Parent Functions and Transformations
Section Details.
Learning the properties and graphs for a catalog of standard functions
Vertical and horizontal shifts of functions
Reflections of functions about the \(x\)-axis and \(y\)-axis
Vertical stretches and shrinks of functions
Horizontal stretches and shrinks of functions
Learing the order to apply the transformations
Using the parent function and transformations to graph a function
Writing the function for graph by identifying the parent function and all transformations
Practice Problems
Directions.
The following are review problems for the section. We recommend you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it.
The graph of a function \(g\) is given below.
Identify the parent function \(f\).
Describe the sequence of transformations from \(f\) to \(g\).
Find the function \(g\).
Use function notation to write \(g\) in terms of \(f\).
Answer
Parent function: \(f(x)=x^2\)
Transformations: Vertical shrink by 1/4 (or horizontal stretch by 2), Reflect over x-axis, Left 3, Up 4
\(g(x)=-\dfrac{1}{4}(x+3)^2+4\)
\(g(x)=-\dfrac{1}{4}f(x+3)+4\)
Video
To see the full video page and find related videos, click the following link.
WIR2 20B M150 V13
Consider the function \(g(x)=2\sqrt{-x+3}-4\).
Identify the parent function \(f\).
Describe the sequence of transformations from \(f\) to \(g\).
Use function notation to write \(g\) in terms of \(f\).
Answer
Parent function: \(f(x)=\sqrt{x}\)
Transformations: Reflect over \(y\)-axis, Right 3, Vertical stretch by 2, Down 4 OR Left 3, Reflect over \(y\)-axis, Vertical stretch by 2, Down 4
\(g(x)=2f(-x+3)-4\)
Video
To see the full video page and find related videos, click the following link.
WIR2 20B M150 V14