We need an m x n matrix A to allow a linear transformation from Rn to Rm through Ax = b In the example, T R2 > R2 Hence, a 2 x 2 matrix is needed If we just used a 1 x 2 matrix A = 1 2, the transformation Ax would give us vectors in R1Q If the triangle is transformed following the rule (x,y)> (x5,y3), what transformation is modeled?Plotting the points from the table and continuing along the xaxis gives the shape of the sine functionSee Figure \(\PageIndex{2}\) Figure \(\PageIndex{2}\) The sine function Notice how the sine values are positive between \(0\) and \(\pi\), which correspond to the values of the sine function in quadrants I and II on the unit circle, and the sine values are negative between \(\pi\) and \(2
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Transformations of y=x^2 parent parabola-Q ∆QRS contains the points Q (4, 2) R (5, 1) S (3,7) If the triangle is reflected across the yaxis, what will S' be?Here is an example of a transformation question Describe, using appropriate terminology, the transformations that occurred to y=x^2 in order to become y=2(x3)^25 First there is a vertical reflection because of the in front of the 2 Second there is a
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1¾2) {U = X=Y » C(0;1){V = X ¡Y » N(0;2¾2)† What is the joint distribution of U = X Y and V = X=Y if X » Gamma(fi;‚) and Y » Gamma(fl;‚) and X and Y are independent Approaches 1 CDF approach fZ(z) = d dzFZ(z) 2We can use this graph that we know and the chart above to draw f(x)2, f(x) 2, 2f(x), 1 2f(x), and f(x) Or to write the previous five functions without the name of the function f, these are the five functions x22,x22, 2x2, x2 2,andx2 These graphs are drawn on the next page 68 Answers ( 1 ) Step 1 a) Stert from the graph of the parent function f ( x) = log 3 x As we can see the given function y = log 3 ( x 2) can be expressed in terms of the parent function f as y = f ( x 2) This indicates that the graph of the function y = log 3 ( x 2) will be the same as the graph of the parent function f ( x
Question 2 SURVEY 900 seconds Report an issue Q Identify the transformation from the graph of f (x)=x 2 to the graph g (x)= (x5) 2 answer choices shifted up five units reflection in xaxisGraphic designers and 3D modellers use transformations of graphs to design objects and images Part of Maths (y=x^2 a\) represents a translation of the graph of \(y = x^2\) by the vector \VCE Maths Methods Unit 3 Transformation of functions y=x 32 Re!ections (in the x axis) 5 • Re!ections !ip the graph around the x or y axis • Re!ections keep the shape of the graph the same • A re!ection in the x axis signs are changed for y values (1,3)
X y y = − 2 x ← D i l a t i o n a n d r e f l e c t i o n − 1 − 2 y = − 2 − 1 = − 2 ⋅ 1 = − 2 0 0 y = − 2 0 = − 2 ⋅ 0 = 0 1 − 2 y = − 2 1 = − 2 ⋅ 1 = − 2 Use the points {(−1, −2), (0, 0), (1, −2)} to graph the reflected and dilated function y = − 2 x Then translate thisY = f ( x ), we can write this formula as ( x, f ( x )) → ( x, f (x) ) Translations of Functions f (x) k and f (x k) Translation vertically (upward or downward) f (x) k translates f (x) up or down Changes occur "outside" the function (affecting the yvalues)And we loop through those points, making new points using the 2×2 matrix "a,b,c,d" for (let i = 0;
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Y y= x y 2 4 6 8 10 x 224 4 0 step 3 vertical translation step 2 vertical reflection = 2x 3 Method 2 Use Mapping Notation Apply each transformation to the point (xy, ) to determine a general mapping notation for the transformed function Transformation of y= √ __ x Mapping Horizontal stretch by a factor of 1 _ (2 x, y) → (1 _x 2, y7 23ATypicalApplication Let Xand Ybe independent,positive random variables with densitiesf X and f Y,and let Z= XYWe find the density of Zby introducing a new random variable W,as follows Z= XY, W= Y (W= Xwould be equally good)The transformation is onetoone because we can solve for X,Yin terms of Z,Wby X= Z/W,Y= WIn a problem of this type,we must alwaysIn two dimensions, linear transformations can be represented using a 2×2 transformation matrix Stretching A stretch in the xyplane is a linear transformation which enlarges all distances in a particular direction by a constant factor but does not affect distances in the perpendicular direction We only consider stretches along the xaxis
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We can flip it leftright by multiplying the xvalue by −1 g(x) = (−x) 2 It really does flip it left and right!Start studying Transformation Rules (x,y)> Learn vocabulary, terms, and more with flashcards, games, and other study tools2 Let a and ß be the following transformations a (x, Cheggcom Math Advanced Math Advanced Math questions and answers 2 Let a and ß be the following transformations a (x, y) = (x y, y), B (x, y) = (x – 2y, y) Find formulas for a o B and Boa, and show that a o B # Boa Question 2
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2 For the following linear transformations T Rn!Rn, nd a matrix A such that T(~x) = A~x for all ~x 2Rn (a) T R2!R3, T x y = 2 4 x y 3y 4x 5y 3 5 Solution To gure out the matrix for a linear transformation from Rn, we nd the matrix A whose rst column is T(~e 1), whose second column(You should be able to tell without graphing) Reflect in the axis Solution (a) 2 Reflect in the axis Left 2 f x x g x hx x x y o o Note In part (a), hx can also be written as h x x 2A X,Y B X,Y,Z C X,Y,W D Xw,Yw,w ANSWER D Two consecutive transformation t1 and t2 are _____ A Additive B Substractive C Multiplicative D None of these ANSWER A Reflection about the line Y=X is equivalent to _____,followed by a anticlockwise rotation 90 ̊ A Reflection about yaxis B Reflection about xaxisC Reflection about origin D
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Geometry – Transformations ~10~ NJCTLorg Draw the line of reflection you can use to map one figure onto the other 8 9 10 Find the image of K(4, 3) after two reflections, first across line ℓ 1, and then across line ℓ 2I assume the intent of this question is What transformations are performed to change the graph of mathy=f(x)/math to mathy=f(2x)/math?Y = 2x y = 2 x The transformation from the first equation to the second one can be found by finding a a, h h, and k k for each equation y = abx−h k y = a b x h k Find a a, h h, and k k for f (x) = 2x f ( x) = 2 x a = 1 a = 1 h = 0 h = 0 k = 0 k = 0 The horizontal shift depends on
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F ( x) = x2 A function transformation takes whatever is the basic function f (x) and then "transforms" it (or "translates" it), which is a fancy way of saying that you change the formula a bit and thereby move the graph around For instance, the graph for y = x2 3 looks like this This is three units higher than the basic quadratic, f (x) = x2I) { let pt = shapeptsi let x = a * pt0 b * pt1 let y = c * pt0 d * pt1 newPtspush({ x x, y y }) } We are given the quadratic function y = f (x) = (x 2)2 For the family of quadratic functions, the parent function is of the form y = f (x) = x2 When graphing quadratic functions, there is a useful form called the vertex form y = f (x) = a(x −h)2 k, where (h,k) is the vertex Data table is given below (both for the parent function and
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1) y = (x 3)2 Transformations reflected in the xaxis (flipped upside down) translated 3 units to the left 2) y = 1/2x2 4 Transformations vertically compressed by a factor of 1/2 translated 4 units up 3) y = (x 2)2 3Have students predict what they think the graphs of y = sin(x 2) and y = sin(x 2) will look like y = sin(x 2) y = sin(x 2) Notice that in the graph of y = sin(x 2) the sine curve has been translated to the left two units In the graph of y = sin(x 2)We can use matrices to translate our figure, if we want to translate the figure x3 and y2 we simply add 3 to each xcoordinate and 2 to each ycoordinate If we want to dilate a figure we simply multiply each x and ycoordinate with the scale factor we want to dilate with
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The answer is either 1 Translation 2 units to theLecture 16 General Transformations of Random Variables 163 Differentiating, we get f Y(y) = f X(g−1(y)) 1 g0(g−1(y)) The second term on the right hand side of the above equation is referred to as the Jacobian of the transfor mation g(·) It can be shown easily that a similar argument holds for a monotonically decreasing function gas well and we obtain Vertex form y=a(xh)^2k All parabolas are the result of various transformations being applied to a base or "mother" parabola This base parabola has the formula y=x^2, and represents what a parabola looks like without any transformations being applied to it The table of values for a base parabola look like this
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Hence, when we add 2 then the parent function will get shifted upward by 2 units The graph of is attached below (fig 2) Finally, x is replaced by x hence the graph is reflected about y axis In figure 3 the graph of the given function is shownDescribe the transformation compared to y=x2?That is, the rule for this transformation is –f (x) To see how this works, take a look at the graph of h(x) = x 2 2x – 3
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Y =x2 y = x 2 Translating the function up the y y axis by two produces the equation y =x2 2 y = x 2 2 And translating the function down the y y axis by two produces the equation y =x2 −2 y = x 2 − 2 Vertical translations The function f(x)=x2 f ( x) = x 2> How do you graph the transformation of f(2x)?Transformations Involving Joint Distributions Want to look at problems like † If X and Y are iid N(0;¾2), what is the distribution of {Z = X2 Y2 » Gamma(1;
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Math Algebra 2 Transformations of functions Graphs of exponential functions Graphs of exponential functions happen at x equals negative 3 y is equal to 1 so notice we shifted to the left by 3 likewise in our original graph when X is 2 y is 4 well how do we get y equals 4 in our in this thing right over here well for y to be equal to 4 thisTransformation is given by w 1 = x w 2 = y with standard matrix A= 1 0 0 1 Re ection about the line y= x The schematic of re ection about the line y= xis given below The transformation is given by w 1 = y w 2 = x with standard matrix A= 0 1 1 0 { Projection Operators Projected onto xaxis The schematic of projection onto the xaxis is given(a) Reflect in the yaxis, then shift left 2 units (b) Shift left 2 units, then reflect in the yaxis (c) Do parts (a) and (b) yield the same function?
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Describe the Transformation y=x^2 y = x2 y = x 2 The parent function is the simplest form of the type of function given y = x2 y = x 2 For a better explanation, assume that y = x2 y = x 2 is f (x) = x2 f ( x) = x 2 and y = x2 y = x 2 is g(x) = x2 g ( x) = x 2 f (x) = x2 f ( x) = x 2 g(x) = x2 g ( x) = x 2Now consider a transformation of X in the form Y = 2X2 X There are five possible outcomes for Y, ie, 0, 3, 10, 21, 36 Given that the function is onetoone, we can make up a table describing the probability distribution for Y TABLE 3 ProbabilityofaFunction oftheNumberofHeadsfromTossing aCoin Four Times Y = 2 * (# heads)2 # of headsCos(x) = cos (x), so discussing reflection in the yaxis using this as a first example might not be helpful (the same is true for x 2 and (x) 2) Trig functions can be helpful for the multiplication transformations, however, given that 2sin(x) looks very different from sin(2x), but might not be as useful for the leftright translation because
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Y = sin(x) in blue y = sin(x 1) in purple y = sin(x 2) in red It appears that the function shifted to the left c units The negative characteristic of the shift is consistent with other functions who have a similar parameter (ie y = (x1)^2 is shifted one unit to the left of y = x^2) In fact, one may be tempted to determine that thisUse transformations to graph y = 3x^2Y=sqrt(x) Transformations Loading Y=sqrt(x) Transformations Y=sqrt(x) Transformations Log InorSign Up y = a bx − h k 1 h = 0 2 k = 0 3 a = 1 4 b = 1 5 y = x 6 7 powered by powered by $$ x $$ y $$ a 2
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Q Is the picture being reflected in the yaxis or xaxis?Purplemath The last two easy transformations involve flipping functions upside down (flipping them around the xaxis), and mirroring them in the yaxis The first, flipping upside down, is found by taking the negative of the original function;But you can't see it, because x 2 is symmetrical about the yaxis So here is another example using √(x) g(x) = √(−x) This is also called reflection about the yaxis (the axis where x
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