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Toby and Uma are playing a game. Toby shows Uma a graph and transforms it, or changes it according to some rule. If Uma can figure out the rule, she wins; otherwise, Toby wins.
For example, Toby first draws the dashed line f(x) below. Then, he draws a transformation of the original curve in solid line called g(x).
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Uma notices that the original curve was a sine, and the new curve is the same sine curve just shifted up. She explains to Toby that g(x)=f(x)+2. Uma wins the first round!
Vertical translation of graphs y=f(x)+b
The graph of y=f(x)+b can be obtained from the graph of y=f(x) by a vertical translation b units upwards.
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Checkpoint
It is given that f(x)=x2+4x−5 and g(x)=x2+4x+4. The graph of f translated up b units is the graph of g.
Find b.
Select the correct option
Discussion
Toby draws an original function f(x) and its transformation g(x).
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Help Uma describe the transformation.
The three zeros of f are at x=−5, −2, 1, while those of g are at x=−3, 0, 3. Each root has increased by 2, and the local extrema shift similarly. We see that f(−5)=g(−3), f(−2)=g(0), and f(1)=g(3).
Hence g is f moved 2 units to the right, or, equivalently, g(x)=f(x−2). Either a qualitative or algebraic description is sufficient.
Horizontal translation
The graph of y=f(x−a) can be obtained from the graph of y=f(x) by a horizontal translation a units to the right.
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Checkpoint
It is given that f(x)=2(x−2) and g(x)=2(x+7). The graph of f translated right a units is the graph of g.
Find a.
Select the correct option
Discussion
Toby draws an original function f(x) and its transformation g(x).
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Help Uma describe the transformation.
Both curves have their peaks, zeros and troughs at the same x-values, so there is no horizontal shift or change of period. The only difference is that f oscillates between y=±1 while g oscillates between y=±5.
In other words, for any x the output of g is five times the output of f. Equivalently, every point (x,f(x)) on the graph of f is taken to (x,5f(x))—its distance from the x-axis is multiplied by 5. Hence, g(x)=5f(x).
Vertical scaling of graphs y=af(x)
The graph of y=pf(x) can be obtained from the graph of y=f(x) by a vertical stretch with scale factor p.
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Checkpoint
It is given that f(x)=ex+2 and g(x)=5ex+10. The graph of f scaled vertically by a factor of p is the graph of g.
Find p.
Select the correct option
Discussion
Toby draws an original function f(x) and its transformation g(x).
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Help Uma describe the transformation.
Both curves reach the same maximum y-value, so there is no vertical stretch. We do see that the zeros of f are at (±2,0), but the zeros of g are at (±8,0). Thus, every zero of f at x=a becomes a zero of g at x=4a—all the roots lie four times as far from the y-axis.
We see that other points follow a similar pattern, the points (±1,6) lie on f, while the points (±4,6) lie on g. Thus, we can conclude that the points of g are 4 times further from the y-axis than the points of f. Equivalently, each point (a,f(a)) on f is carried to (4a,f(a)). In other words, to reproduce f’s output you must divide the input by 4. Hence, g(x)=f(x/4).
Horizontal scaling
The graph of y=f(qx) can be obtained from the graph of y=f(x) by a horizontal stretch with scale factor q1.
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Checkpoint
It is given that f(x)=sin(x)+4 and g(x)=sin(4x)+4. The graph of f scaled horizontally by a factor of q is the graph of g.
Find q.
Select the correct option
Discussion
Toby draws an original function f(x) and its transformation g(x).
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Help Uma describe the transformation.
The graph of g is exactly the “mirror image” of the graph of f. In other words, every point (a,f(a)) on the graph of f corresponds to the point (−a,f(a)) on the graph of g.
Algebraically this means
For example, since f(1)=0 then
Reflection in the y-axis
The graph of y=f(−x) can be obtained from the graph of y=f(x) by a reflection in the y-axis.
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Discussion
Toby draws an original function f(x) and its transformation g(x).
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Help Uma describe the transformation.
The graph of g is exactly the vertical “mirror image” of the graph of f. In other words, every point (a,f(a)) on f corresponds to the point (a,−f(a)) on g.
Algebraically this means
For example, since f(0)=1 then
Reflection in the x-axis
The graph of y=−f(x) can be obtained from the graph of y=f(x) by a reflection in the x-axis.
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Discussion
In the last round, Toby graphs f and its transformation g below.
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Uma claims the transformation is a horizontal shift 16 units to the right follow by a horizontal stretch by a factor of
41, while Toby argues that the transformation is a horizontal stretch by a factor of 41 followed by a horizontal shift 4 units to the right.
Who is right?
We compare the two orders of operations on a general function f.
Toby’s order:
– Horizontal stretch by factor 1/4 means replace x with 4x,
– Shift right by 4 means replace x by x−4 in h1,
Hence Toby’s net result is
Uma’s order:
– Shift right by 16 gives
– Horizontal stretch by factor 1/4 then replaces x with 4x in h2,
Thus Uma also obtains
Since both sequences of transformations yield the same formula g(x)=f(4x−16), Toby and Uma are technically both correct. However, that means Uma wins the game since she correctly described Toby's transformation.
Combined horizontal scale and translation f(ax+b)
The graph of y=f(ax+b)=f[a(x+ab)] can be obtained from the graph of y=f(x) by a horizontal translation b units to the left, followed by a horizontal stretch with scale factor a1.
OR a horizontal stretch with scale factor a1 followed by a horizontal translation ab units to the left.
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