# Rectangle $ABCD$ABCD​ is translated and then reflected to create rectangle $A'B'C'D'$A′B′C′D′​ . Do rectangle $ABCD$ABCD​ and rectangle $A'B'C'D'$A′B′C′D′​ have the same area? Justify your answer.

Question
Performing transformations
Rectangle $ABCD$ABCD​ is translated and then reflected to create rectangle $A'B'C'D'$A′B′C′D′​ . Do rectangle $ABCD$ABCD​ and rectangle $A'B'C'D'$A′B′C′D′​ have the same area? Justify your answer.

2021-01-16
When we translate the rectangle ABCD we do this by sliding the figure based on the transformation vector or directions of translations. When performing a translation we are sliding a given figure up, down, left or right. The orientation and size of the figure does not change. When applying the transformations we will transform each point and then redraw the figure.
when we reflect the rectangle we are just shifting the rectangle to the other side of either x or y axis (according to the reflection) at a distance equidistant from the axis. Hence each vertex shifts to a new position but remains at the same distance from the other vertex as previously before the reflection.
Hence there would be no change in the area of the rectangle if we translate and then reflect it.

### Relevant Questions

Your friend attempted to describe the transformations applied to the graph of $$y=sinx$$ to give the equation $$f(x)=1/2sin(-1/3(x+30))+1$$.
They think the following transformations have been applied. Which transformations have been identified correctly, and which have not? Justify your answer.
a) f(x) has been reflected vertically.
b) f(x) has been stretched vertically by a factor of 2.
c) f(x) has been stretched horizontally by a factor of 3.
d) f(x) has a phase shift left 30 degrees.
e) f(x) has been translated up 1 unit.
Starting with the function $$f(x) = e^x$$, create a new function by performing the following transformations.
i. First shift the graph of f (x) to the left by two units.
$$f_1 (x) =$$?
ii. Then compress your result by a factor of $$1/7$$
$$f_2 (x) =$$ ?
iii. Next, reflect it across the x-axis.
$$f_3 (x) =$$ ?
iv. And finally shift it up by eight units to create g (x).
g (x) =?
Create a new function in the form $$y = a(x-h)^2 + k$$ by performing the following transformations on $$f (x) = x^2$$.
Give the coordinates of the vertex for the new parabola.
g(x) is f (x) shifted right 7 units, stretched by a factor of 9, and then shifted down by 3 units. g(x) = ?
Coordinates of the vertex for the new parabola are:
x=?
y=?
Sketch the graph of the function $$f(x) = -2^x+1 +3$$ using transformations. Do not create a table of values and plot points
Answer true or false to each of the following statements and explain your answers.
a. In using the method of transformations, we should only transform the predictor variable to straighten a scatterplot.
b. In using the method of transformations, a transformation of the predictor variable will change the conditional distribution of the response variable.
c. It is not always possible to fnd a power transformation of the response variable or the predictor variable (or both) that will straighten the scatterplot.
Which of the following is the correct equation used to solve for the measure of each angle? OA.
$$\displaystyle{m}\angle{A}−{m}\angle{B}−{m}\angle{C}={180}∘$$ O B. $$\displaystyle{m}\angle{A}+{m}\angle{B}−{m}\angle{C}={180}∘$$ O c. $$\displaystyle{m}\angle{A}−{m}\angle{B}+{m}\angle{C}={180}∘$$ O D. $$\displaystyle{m}\angle{A}+{m}\angle{B}+{m}\angle{C}={180}∘$$
The measure of analeAis Click to select your answer(s). Save for Later 0.
(i) 170
(1) 7
(1) 5
(1) 5
(1) 5
(1) 5
Create a new function in the form $$y = a(x- h)^2 + k$$ by performing the following transformations on $$f (x) = x^2$$
Give the coordinates of the vertex for the new parabola.
h(x) is f (x) shifted right 3 units, stretched by a factor of 9, and shifted up by 7 units. h(x) = ?
Edit Coordinates of the vertex for the new parabola are:
x=?
y =?
Let T $$\displaystyle{P}_{{2}}\rightarrow\mathbb{R}^{{3}}$$ be a transformation given by
$$\displaystyle{T}{\left({f{{\left({x}\right)}}}\right)}={\left[\begin{array}{c} {f{{\left({0}\right)}}}\\{f{{\left({1}\right)}}}\\{2}{f{{\left({1}\right)}}}\end{array}\right]}$$
(a)Then show that T is a linear transformation.
(b)Find and describe the kernel(null space) of T i.e Ker(T) and range of T.
(c)Show that T is one-to-one.