# Recent questions in Transformation properties

Transformation properties

### For large value of n, and moderate values of the probabiliy of success p (roughly, $$0.05\ \Leftarrow\ p\ \Leftarrow\ 0.95$$), the binomial distribution can be approximated as a normal distribution with expectation mu = np and standard deviation $$\sigma = \sqrt{np(1\ -\ p)}$$. Explain this approximation making use the Central Limit Theorem.

Transformation properties

### Complete the tasks to determine: a)To graf: The function $$g{{\left({x}\right)}}={2}^{{{x}-{4}}}.$$ b)The domain of the function $$g{{\left({x}\right)}}={2}^{{{x}-{4}}}$$ in the interval notation. c)The range of the function $$g{{\left({x}\right)}}={2}^{{{x}-{4}}}$$ in the interval notation. d)The equation of the asymptote of the function $$g{{\left({x}\right)}}={2}^{{{x}-{4}}}.$$

Transformation properties

### Describe how to obtain the graph of g and f if $$g(x)=2(x\ +\ 2)^{2}\ -\ 3,\ h(x)=2x^{2},\ and\ h(x)=2(x\ -\ 3)^{2}\ +\ 2$$

Transformation properties

### To determine the solution of the initial value problem $${y}{''}+{4}{y}= \sin{{t}}+{u}_{\pi}{\left({t}\right)} \sin{{\left({t}-\pi\right)}}:$$ $$y(0) = 0,$$ $$y'(0) = 0.$$ Also, draw the graphs of the solution and of the forcing function and explain the relation between the solution and the forcing function..

Transformation properties

### a) To graph: the $${k}{e}{r}{\left({A}\right)},{\left({k}{e}{r}{A}\right)}^{\bot}{\quad\text{and}\quad}{i}{m}{\left({A}^{T}\right)}$$ b) To find: the relationship between im $$(A^{T})$$ and ker (A). c) To find: the relationship between ker(A) and solution set S d) To find vecx_0 at the intersection of $${k}{e}{r}{\left({A}\right)}{\quad\text{and}\quad}{\left({k}{e}{r}{A}\right)}^{\bot}$$ e) To find: the lengths of $$\vec{{x}}_{{0}}$$ compared to the other vectors in S

Transformation properties

### Provide answers to all tasks using the information provided. a) Find the parent function f. Given Information: $$g{{\left({x}\right)}}=-{2}{\left|{x}-{1}\right|}-{4}$$ b) Find the sequence of transformation from f to g. Given information: $$f{{\left({x}\right)}}={\left[{x}\right]}$$ c) To sketch the graph of g. Given information: $$g{{\left({x}\right)}}=-{2}{\left|{x}-{1}\right|}-{4}$$ d) To write g in terms of f. Given information: $$g{{\left({x}\right)}}=-{2}{\left|{x}-{1}\right|}-{4}{\quad\text{and}\quad} f{{\left({x}\right)}}={\left[{x}\right]}$$

Transformation properties

### To calculate: The reduced form of the provided matrix, $${\left[\begin{matrix}{1}&{0}&-{3}&{1}\\{0}&{1}&{2}&{0}\\{0}&{0}&{3}&-{6}\end{matrix}\right]}$$ with the use of row operations.

Transformation properties

### Prove these examples are correct: a) What is the area of the largest rectangle that fits inside of the ellipse $$x^{2}\ +\ 2y^{2} = 1?$$ b) Prove the following: Let c in (a, b). If f is continuous on $$[a,\ b],$$ differentiable on (a, b)?

Transformation properties

### Which of the following are linear transformations from $$RR^{2} \rightarrow RR^{2} ?$$ (d) Rotation: if $$x = r \cos \theta, y = r \sin \theta,$$ then $$\overrightarrow{T}(x,y)=(r \cos(\theta+ \varphi), r \sin (\theta+ \varphi))$$ for some constants $$\angle \varphi$$ (f) Reflection: given a fixed vector $$\overrightarrow{r} = (a, b), \overrightarrow{T}$$ maps each point to its reflection with respect to $$\overrightarrow{r} \overrightarrow{T}(\overrightarrow{x})=\overrightarrow{x}-2\overrightarrow{x}_{r \perp}$$ $$=2 \overrightarrow{x}_{r}-\overrightarrow{x}$$

Transformation properties

### (a) The new coordinates geometrically if X represents the point $$(0,\ \sqrt{2})$$ m and this point is rotated about the origin $$45^{\circ}$$ clockwise and then translated 2 units to the right and 3 units upward. (b) The value of $$Y = ABX,$$ and explain the result. (c) If ABX equal to BAX. Interpret the resul. (d) A matrix that translate Y back to X

Transformation properties

### To find: The Laplace transform of the function $${L}{\left\lbrace{t}^{4}-{t}^{2}-{t}+ \sin{\sqrt{{{2}{t}}}}\right\rbrace}$$

Transformation properties

### Let $$T : U \rightarrow U$$ be a linear transformation and let beta be a basis of U Define the determinant det(T) of T as det$$(T) = det([T]_{\beta}).$$ Show ta det (T) is well-defined, i. e. that it does not depend on the choice of the basis beta Prove that T is invertible if and only if det $$(T) \neq 0.$$ If T is invertible, show that det $$(T^{-1}) = \frac{1}{det(T)}$$

Transformation properties

### a) Find the sequence of transformation from f to g Given information: $$g{{\left({x}\right)}}=\frac{1}{{2}}{\left|{x}-{2}\right|}-{3}{\quad\text{and}\quad} f{{\left({x}\right)}}={x}^{3}$$ b) To sketch the graph of g. Given information: $$f{{\left({x}\right)}}={\left|{x}\right|}$$ c) To write g in terms of f. Given information: $$g{{\left({x}\right)}}=\frac{1}{{2}}{\left|{x}-{2}\right|}-{3}{\quad\text{and}\quad} f{{\left({x}\right)}}={\left|{x}\right|}$$

Transformation properties

### To sketch: (i) The properties, (ii) Linear transformation. Let $${T}:\mathbb{R}^{2}\to\mathbb{R}^{2}$$ be the linear transformation that reflects each point through the $$x_{1} axis.$$ Let $${A}={\left[\begin{matrix}{1}&{0}\\{0}&-{1}\end{matrix}\right]}$$

Transformation properties

### The table shows some values of the derivative of an unknown function f.Complete the table by finding the derivative of each transformation of f, it possible a) $$g(x) = f(x)\ -\ 2$$ b) $$h(x) = 2 f(x)$$ c) $$r(x) = f(-3x)$$

Transformation properties

### Show that mapping $${y}\mapsto{r}{e}{f}{l}_{{L}}{y}$$ is a linear transformation.

Transformation properties

### Guided Proof Let $${v_{1}, v_{2}, .... V_{n}}$$ be a basis for a vector space V. Prove that if a linear transformation $$T : V \rightarrow V$$ satisfies $$T (v_{i}) = 0\ for\ i = 1, 2,..., n,$$ then T is the zero transformation. To prove that T is the zero transformation, you need to show that $$T(v) = 0$$ for every vector v in V. (i) Let v be the arbitrary vector in V such that $$v = c_{1} v_{1} + c_{2} v_{2} +\cdots + c_{n} V_{n}$$ (ii) Use the definition and properties of linear transformations to rewrite T(v) as a linear combination of $$T(v_{j})$$ . (iii) Use the fact that $$T (v_{j}) = 0$$ to conclude that $$T (v) = 0,$$ making T the zero transformation.

Transformation properties

### To determine the solution of the initial value problem $${y}{''}+{4}{y}=\delta{\left({t}-\pi\right)}-\delta{\left({t}-{2}\pi\right)},{y}{\left({0}\right)}={0},{y}'{\left({0}\right)}={0}$$ and draw the graph of the solution.

Transformation properties