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AI HL · 12.5

Functions

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12.5.7

Composite transformations

12.5.7

Composite transformations

Target: I can say whether the order of two transformations matters, and apply them in the right order. (+6 more)

By the end of this section

Key points

Order matters

  • Two vertical changes: the order matters. \(y=pf(x)+b\) is a stretch (or reflection) with scale factor p first, then a translation by b.
  • Two horizontal changes: the order matters. \(y=f(qx+c)\) is a translation by \(-c\) first, then a stretch with scale factor \(1/q\).
  • One vertical and one horizontal change: either order gives the same graph.
  • To sketch, move each key point: \((x,y)\) goes to \(\left(\dfrac{x-c}{q},\ py+b\right)\) for \(y=pf(qx+c)+b\).
  • A translation in both directions can be written as a column vector.

Stretches and reflections

  • \(y=pf(x)\) is a vertical stretch with scale factor p. \(y=f(qx)\) is a horizontal stretch with scale factor \(1/q\).
  • A negative p reflects in the x-axis. A negative q reflects in the y-axis. Both together give \(y=-f(-x)\).
  • For a power function, \(a(qx)^n=aq^nx^n\). A vertical stretch with scale factor \(q^n\) equals a horizontal stretch with scale factor \(1/q\).
  • To stretch away from the line \(y=k\), keep that line fixed: \(y=k+p\big(f(x)-k\big)\).

Describing from an equation

  • Write the new equation in terms of f, then read the order from it. Complete the square first for a quadratic.
  • \(y=pf(x)+b\) can also be made by translating by \(b/p\) first, then stretching with scale factor p.
  • \(f(qx+c)=f\big(q(x+c/q)\big)\): the stretch can come first if the translation is \(-c/q\).
  • To find unknown coefficients, apply the transformations to \(ax^2+bx+c\) and compare coefficients.

Transformations and asymptotes

  • A translation moves an asymptote with the graph. A translation up or down changes a horizontal asymptote; a translation left or right changes a vertical one.
  • A stretch from the x-axis leaves the line \(y=0\) where it is. A stretch applied after a translation also stretches the shifted asymptote: do the transformations in order.
  • A reflection in the x-axis turns \(y=c\) into \(y=-c\).

Explore: transformations of a graph

The dashed graph is y = f(x). The orange graph is y = p f(qx + c) + b. Drag the first orange point to translate the graph, or choose a kind of change and move the sliders. Say what happens to the labelled points before you read the answer.

Move a slider to change the graph.

  1. Choose Translation and drag A′ 3 units to the left. What is c now? Why is that the opposite of what the plus sign suggests?
  2. Choose Stretch and set q = 2. Is the graph twice as wide or half as wide? Then choose Reflection and tick the y-axis box. Which coordinate of each labelled point changes sign?
  3. Choose All and set c = 2 and q = 2. Read the order in the sentence. Does the graph move 2 units, or a different amount? Why?
  4. Choose 1/x or eˣ and translate the graph. Where does each asymptote go? Which numbers move the vertical one, and which the horizontal one?

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
A translation then a stretch

The graph of \(y=f(x)\) is a cubic with a maximum at (−1, 3) and a minimum at (1, −1). Sketch the graph of \(y=2f(x-3)\), and mark its turning points.

−3−2−1123−4−2246xyy = f(x)(−1, 3)(1, −1)
Working
  1. Horizontal

    \(f(x-3)\): translate 3 to the right: (−1, 3)→(2, 3), (1, −1)→(4, −1)

    One vertical and one horizontal change: the order does not matter. Do the translation first.

  2. −4−22468−5510xyy = f(x)y = 2f(x − 3)(2, 6)(4, −2)
    Vertical

    \(2f(x-3)\): multiply every y-coordinate by 2: maximum (2, 6), minimum (4, −2)

    A stretch with scale factor 2 keeps the x-coordinates and doubles the heights.

  3. Sketch

    Maximum (2, 6), minimum (4, −2)

    Draw the same cubic shape through the two new turning points.

Video walkthrough coming
A reflection then a translation

The graph of \(y=f(x)\) is a cubic with a maximum at (−1, 1) and a minimum at (1, −3). Sketch the graph of \(y=5-f(x)\), and mark its turning points.

−3−2−1123−6−4−224xyy = f(x)(−1, 1)(1, −3)
Working
  1. Order

    \(y=5-f(x)=-f(x)+5\)

    Two vertical changes: the order matters. Reflect in the x-axis first, then translate up 5.

  2. Reflect

    \(-f(x)\): (−1, 1)→(−1, −1), (1, −3)→(1, 3)

    Change the sign of each y-coordinate. A maximum becomes a minimum.

  3. −4−3−2−11234−6−4−2246810xyy = f(x)y = 5 − f(x)(−1, 4)(1, 8)
    Translate

    \(-f(x)+5\): add 5 to each y-coordinate: (−1, 4) and (1, 8)

    The new graph has a minimum at the first point and a maximum at the second.

Video walkthrough coming
Translate, then stretch, inside the bracket

The graph of \(y=f(x)\) is a cubic with a maximum at (0, 1) and a minimum at (2, −3). Sketch the graph of \(y=f(2x+6)\), and mark its turning points.

−2−11234−6−4−224xyy = f(x)(0, 1)(2, −3)
Working
  1. Two horizontal changes

    \(f(2x+6)\): translate by −6, then stretch with scale factor \(\tfrac12\)

    The order matters. The translation is the one in the bracket as it stands: \(f(x+6)\), then replace x by 2x.

  2. Translate

    (0, 1)→(−6, 1), (2, −3)→(−4, −3)

    Subtract 6 from each x-coordinate.

  3. −6−4−224−6−4−2246xyy = f(x)y = f(2x + 6)(−3, 1)(−2, −3)
    Stretch

    Halve each x-coordinate: (−3, 1) and (−2, −3)

    A horizontal stretch with scale factor \(\tfrac12\) squeezes the graph towards the y-axis.

Video walkthrough coming
Combining changes on a piecewise graph

The graph of \(y=f(x)\) is made of straight lines joining (−4, 3), (−2, −1), (2, −1), (4, 5). Sketch the graph of \(y=2f(x+1)-3\).

−6−4−2246−4−22468xy
Working
  1. Horizontal

    \(f(x+1)\): subtract 1 from each x-coordinate

    The horizontal change can be done at any point.

  2. Vertical, in order

    \(2f(\,)-3\): multiply each y-coordinate by 2, then subtract 3

    Two vertical changes: stretch first, then translate.

  3. −6−4−2246−5510xy
    Image points

    (−4, 3)→(−5, 3), (−2, −1)→(−3, −5), (2, −1)→(1, −5), (4, 5)→(3, 7)

    Join the four image points with straight lines.

Video walkthrough coming
Completing the square to read the transformations

Write \(2 x^{2} - 12 x + 19\) in the form \(a(x-h)^2+k\). Hence describe two transformations that map the graph of \(y=x^2\) to the graph of \(y=2 x^{2} - 12 x + 19\).

Working
  1. Take out the factor

    \(2 x^{2} - 12 x + 19=2(x^2-6x)+19\)

    Take 2 out of the x² and x terms only.

  2. Complete the square

    \(=2\big[(x-3)^2-9\big]+19=2(x-3)^2+1\)

    Half of −6 is −3. The bracket is \(2(x-3)^2-18+19\).

  3. Read it as f

    With \(f(x)=x^2\): \(y=2f(x-3)+1\)

    The stretch factor is 2, the horizontal translation is +3 and the vertical translation is +1.

  4. Describe

    Vertical stretch with scale factor 2, then translation by the vector \(\begin{pmatrix}3\\1\end{pmatrix}\)

    The stretch comes before the vertical translation. The horizontal translation can go with either.

Video walkthrough coming
A vertical stretch as a horizontal stretch

The graph of \(y=3x^3\) is stretched vertically with scale factor 64. (a) Write down the equation of the new graph. (b) Find the scale factor of the horizontal stretch that has the same effect.

Working
  1. Vertical stretch

    \(y=64\times 3x^3=192x^3\)

    Multiply the whole function by 64.

  2. Horizontal form

    \(3(qx)^3=3q^3x^3\), so \(q^3=64\) and \(q=4\)

    Replace x by qx and match the multiplier.

  3. Scale factor

    \(y=3(4x)^3\): horizontal stretch with scale factor \(\tfrac14=0.25\)

    Replacing x by 4x squeezes the graph: the scale factor is 1/4, not 4.

Video walkthrough coming
Two sequences for one graph

Find two sequences of two transformations that map the graph of \(y=f(x)\) to the graph of \(y=f(3x-6)\).

Working
  1. As it stands

    \(f(3x-6)\): translate by +6, then stretch horizontally with scale factor \(\tfrac13\)

    Translate first in the bracket as written: \(f(x-6)\), then replace x by 3x.

  2. Factor the bracket

    \(f(3x-6)=f\big(3(x-2)\big)\)

    Take out the 3. The stretch is now inside, the translation outside it.

  3. The other order

    Stretch horizontally with scale factor \(\tfrac13\), then translate by +2

    Both sequences give the same graph.

Video walkthrough coming
A stretch away from a line

The graph of \(y=f(x)\) is stretched vertically with scale factor 3 away from the line \(y=2\), so the line \(y=2\) does not move. The new graph is \(y=3f(x)+b\). Find b.

Working
  1. Move the line to the x-axis

    \(y=f(x)-2\)

    Translate down 2. The line \(y=2\) is now the x-axis.

  2. Stretch

    \(y=3\big(f(x)-2\big)\)

    A stretch with scale factor 3 keeps the x-axis fixed.

  3. Move back

    \(y=3\big(f(x)-2\big)+2=3f(x)-4\)

    Translate up 2. Check: where \(f(x)=2\), \(3\times2-4=2\). The line is fixed.

Video walkthrough coming
Where does the asymptote go?

The graph of \(y=2^x\) has the horizontal asymptote \(y=0\). The graph is stretched vertically with scale factor 3, then translated 2 units to the right and 4 units down. Find the equation of the image and the equation of its horizontal asymptote.

Working
  1. Stretch first

    \(y=3\times2^x\). The asymptote \(y=0\) stays at \(y=0\).

    Multiplying 0 by 3 gives 0: a stretch from the x-axis leaves the line \(y=0\) where it is.

  2. Translate 2 right

    \(y=3\times2^{x-2}\). The asymptote is still \(y=0\).

    A horizontal shift does not move a horizontal line.

  3. Translate 4 down

    \(y=3\times2^{x-2}-4\). The asymptote moves to \(y=-4\).

    Every point, and the asymptote, goes down 4.

  4. State

    Image: \(y=3\times2^{x-2}-4\); horizontal asymptote \(y=-4\)

    Check with the calculator: the graph levels off at \(-4\).

Video walkthrough coming