Transformations & asymptotes

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Linear Transformations

Discussion

What do the two curves have in common?

Both curves are the same “S-shaped” cubic—they have identical turning-point behaviour and inflection pattern.

Visually, one is just shifted up-and-to-the-left and the other down-and-to-the-right, but their shape (and the relative heights of their peaks and depths of their troughs) is exactly the same.

Therefore, we say that the curves are transformations of each other, which means that either curve could be produced by applying some consistent rule to the other curve. In this case, the rule is a diagonal shift. Notice that each point is shifted in the exact same direction with the exact same distance from one curve to another.

Toby and Uma are playing a game. Toby shows Uma a graph and transforms it. If Uma can figure out the "rule" that produced the transformation, 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).

Discussion

Help Uma win the first round by describing graphically or algebraically what transformation can be applied to f to produce g.

The two curves have the same shape, amplitude and period, and at each x-value the value of g exceeds f by 2. In particular,

g(0)=2=f(0)+2

Hence g is obtained from f by a vertical shift up 2 units. Algebraically, this means that the ouput of g is always 2 greater than the output of f, so

g(x)=f(x)+2.

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 (if b<0, the transformation may also be called a translation ∣b∣ units down).

Discussion

Toby draws an original function f(x) and its transformation g(x).

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 graphical or algebraic description is sufficient.

In the discussion, either describing the transformation as a shift 2 units to the right or g(x)=f(x−2) was a sufficient answer. Indeed, a shift h units to the right is equivalent to the transformation f(x)→f(x−h). This is not as natural as a vertical shift k units up, which is simply f(x)→f(x)+k.

Can we make sense of why f(x−h) increases the x-value of each point by h?

Let's interpret what the algebraic form of the transformation, g(x)=f(x−2), actually means:

We get g(1) by taking f(−1), g(2) by taking f(0), g(3) by taking f(1), and so on.
We see each output of g at a is the output of f at a−2.
Thus, the reason a horizontal shift h units to the right is represented by x−h is because the value of g at every x is equivalent to the value that f ouputs h units to the left of x.

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 (if a<0, the transormation may also be called a translation ∣a∣ units to the left).

Checkpoint

The graph of g is obtained by translating the graph of f 1.3 units to the right.

Find g(3.3).

Select the correct option

Translation by a vector

If a point P is translated by a vector (ab), apply a translation a units to the right and b units up:

P(x,y)P′(x−a,y+b).

Checkpoint

The image of P=(x,y) after a translation by (52) is the point (9,6). Find the coordinates of P.

Select the correct option

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.

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

Again, we notice something peculiar about a horizontal transformation. Specifically, to describe f getting 4 times wider algebraically, we have f(x)→f(4x), which is less natural than if we have a function that becomes 4 times taller: f(x)→4f(x).

Can we make sense of why f(qx) increases the x-value of each point by a factor of q?

Let's interpret what the algebraic form of the transformation, g(x)=f(4x), actually means:

We get g(1) by taking f(41), g(4) by taking f(1), g(8) by taking f(2), and so on.
We see each output of g at a is the output of f at 4a.
Thus, the reason a horizontal stretch by a factor q is represented by qx is because the value of g at every x is equivalent to the value that f ouputs q times closer to the x-axis.

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.

Exercise

It is given that f(x)=ex+sinx.

Find an expression for g, the image of f after a translation 2 units to the right and 3 units down.
Find an expression for h, the image of f after a horizontal stretch by a factor of 5.

Select the correct option

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.

Checkpoint

Find the reflection of f(x)=x2+6x+5 in the y-axis.

Select the correct option

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.

Checkpoint

Find the reflection of x2+4ex in the x-axis.

Select the correct option

Exercise

g is the image of f under a horizontal stretch by a factor of 43 and reflection in the y-axis. Write an expression for g in terms of f.

Select the correct option

Discussion

In the last round, Toby graphs f and its transformation g below.

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 correct?

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,
h1(x)=f(4x)
– Shift right by 4 means replace x by x−4 in h1,
g(x)=h1(x−4)=f(4(x−4))=f(4x−16)
Hence Toby’s net result is
g(x)=f(4x−16)
Uma’s order:

– Shift right by 16 gives
h2(x)=f(x−16)
– Horizontal stretch by factor 1/4 then replaces x with 4x in h2,
g(x)=h2(4x)=f((4x)−16)=f(4x−16)
Thus Uma also obtains
g(x)=f(4x−16)

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.

Why is this true? (animation: Parabola translates out then stretches vs stretches then translates out -- two desmos? or one long animation? ***james question)

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Separate note or same -- why doesn't 4x+4 work for the example above? "order of transformations" point -- enough for a skill?

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.

Checkpoint

A curve is translated 8 units to the left and then horizontally stretched by a factor of 161. Equivalently, this transformation can be described as a horizontal stretch by a factor of c and a horizontal translation to the right by k units.

Find c and k.

Select the correct option

Exercise

It is given that g(x)=f(ax+b). g is the image of f after a horizontal stretch by a factor of 3 followed by a translation 2 units left. Find a and b.

Select the correct option

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