Sign in to jmaths to print this page or save it as a PDF: jmaths-8ce638778fd6.herokuapp.com/maths/ai-hl/functions/12-5-7/problems/

jmaths

IB Mathematics: Applications and Interpretation HL · 12.5: Functions · 12.5.7

Composite transformations Problem set A Worked solutions

Skills

  1. Write down the equation of the graph after each change.

    1. (a) The graph of \(y=5x^4\) is stretched vertically with scale factor 6.

      • Multiply the whole function by 6: \(y=6\times5x^4\)
      • \(y=30 x^{4}\)

    2. (b) The graph of \(y=4 x^{3} - x^{2}\) is reflected in the x-axis and then reflected in the y-axis.

      • Reflect in the x-axis: \(y=-(4 x^{3} - x^{2})=- 4 x^{3} + x^{2}\)
      • Reflect in the y-axis: replace x by −x
      • \(y=4 x^{3} + x^{2}\)

    3. (c) The graph of \(y=3^x\) is translated 2 units in the positive x direction and then stretched vertically with scale factor 5.

      • Translate: replace x by \(x-2\): \(y=3^{x-2}\)
      • Stretch: multiply by 5
      • \(y=5\times 3^{x-2}\)

    4. (d) The graph of \(y=x^{2} - 4 x\) is translated 1 unit in the negative x direction and then stretched vertically with scale factor 2. Give your answer in the form \(y=ax^2+bx+c\).

      • Translate: \(y=(x+1)^2-4(x+1)\)
      • Stretch: multiply everything by 2
      • \(y=2 x^{2} - 4 x - 6\)

  2. A vertical stretch can sometimes be replaced by a horizontal stretch.

    1. (a) The graph of \(y=x^2\) is stretched vertically with scale factor 16. Write down the equation of the new graph.

      • Multiply the function by 16: \(y=16x^2\)

    2. (b) Find the scale factor of the horizontal stretch that has the same effect on \(y=x^2\).

      • \((qx)^2=q^2x^2\), so \(q^2=16\) and \(q=4\)
      • Horizontal stretch with scale factor \(1/q=0.25\)

    3. (c) The graph of \(y=2x^3\) is stretched vertically with scale factor 125. Find the scale factor of the horizontal stretch that has the same effect.

      • \(2(qx)^3=2q^3x^3\), so \(q^3=125\) and \(q=5\)
      • Horizontal stretch with scale factor \(1/q=0.2\)

  3. The diagram shows the graph of \(y=f(x)\), a cubic. On separate diagrams sketch each graph, marking the turning points.

    −112345−4−224xyy = f(x)(1, 2)(3, −2)

    1. (a) \(y=3f(x+1)\)

      • Turning points of f: (1, 2) and (3, −2)
      • Translate 1 left: subtract 1 from each x-coordinate
      • Stretch vertically with scale factor 3: multiply each y-coordinate by 3
      • Image points: (0, 6) and (2, −6)

    2. (b) \(y=\tfrac12f(x-4)\)

      • Turning points of f: (1, 2) and (3, −2)
      • Translate 4 right: add 4 to each x-coordinate
      • Stretch vertically with scale factor 0.5: multiply each y-coordinate by 0.5
      • Image points: (5, 1) and (7, −1)

  4. The diagram shows the graph of \(y=f(x)\), a cubic. On separate diagrams sketch each graph, marking the turning points.

    −112345−4−224xyy = f(x)(1, 2)(3, −2)

    1. (a) \(y=-f(x)+2\)

      • Turning points of f: (1, 2) and (3, −2)
      • Stretch/reflect first: multiply each y-coordinate by −1
      • Then translate: add 2 to each y-coordinate
      • Image points: (1, 0) and (3, 4)

    2. (b) \(y=1-2f(x)\)

      • Turning points of f: (1, 2) and (3, −2)
      • Stretch/reflect first: multiply each y-coordinate by −2
      • Then translate: add 1 to each y-coordinate
      • Image points: (1, −3) and (3, 5)

  5. The diagram shows the graph of \(y=f(x)\), a cubic. On separate diagrams sketch each graph, marking the turning points.

    −112345−4−224xyy = f(x)(1, 2)(3, −2)

    1. (a) \(y=f(\tfrac12x-1)\)

      • Turning points of f: (1, 2) and (3, −2)
      • Translate by 1 in the x direction: add 1 to each x-coordinate
      • Then stretch horizontally with scale factor 2: multiply each x-coordinate by 2
      • Image points: (4, 2) and (8, −2)

    2. (b) \(y=f(2x-3)\)

      • Turning points of f: (1, 2) and (3, −2)
      • Translate by 3 in the x direction: add 3 to each x-coordinate
      • Then stretch horizontally with scale factor 0.5: multiply each x-coordinate by 0.5
      • Image points: (2, 2) and (3, −2)

  6. The graph of \(y=f(x)\) is made of straight lines joining (−4, 1), (−2, −3), (0, −3), (2, 2), (4, 2). Sketch each graph.

    −6−4−2246−6−4−224xy

    1. (a) \(y=2f(x-1)+1\)

      • Horizontal: add 1 to each x-coordinate
      • Vertical: multiply each y-coordinate by 2, then add 1 to each y-coordinate
      • Image points: (−4, 1)→(−3, 3), (−2, −3)→(−1, −5), (0, −3)→(1, −5), (2, 2)→(3, 5), (4, 2)→(5, 5)
      • Join the points with straight lines

    2. (b) \(y=f(2x+4)\)

      • Horizontal: subtract 4 from each x-coordinate, then multiply each x-coordinate by 0.5
      • No vertical change
      • Image points: (−4, 1)→(−4, 1), (−2, −3)→(−3, −3), (0, −3)→(−2, −3), (2, 2)→(−1, 2), (4, 2)→(0, 2)
      • Join the points with straight lines

  7. Describe a sequence of transformations that maps the graph of \(y=2 x^{2} - x\) to the graph of the given equation.

    1. (a) \(y=6 x^{2} - 3 x + 4\)

      • Let \(f(x)=2 x^{2} - x\)
      • \(6 x^{2} - 3 x + 4=3(2 x^{2} - x)+4=3f(x)+4\)
      • The stretch comes before the translation

    2. (b) \(y=2 x^{2} - 17 x + 37\)

      • \(y=f(x-4)+1\)
      • Both changes are translations, so the order does not matter

  8. The graph of \(f(x)=3^x\) has the horizontal asymptote \(y=0\). For each change write the equation of the image and the equation of its horizontal asymptote.

    1. (a) A translation 2 units up.

      • Translate up 2: \(y=3^x+2\)
      • The asymptote moves up 2: \(y=2\)

    2. (b) A reflection in the x-axis, then a translation 5 units up.

      • Reflect in the x-axis: \(y=-3^x\), asymptote \(y=0\)
      • Translate up 5: \(y=-3^x+5\), asymptote \(y=5\)

    3. (c) A vertical stretch with scale factor 4, then a translation 1 unit down.

      • Stretch: \(y=4\times3^x\), asymptote \(y=0\)
      • Translate down 1: \(y=4\times3^x-1\), asymptote \(y=-1\)

  9. The graph of \(y=\dfrac{1}{x}\) has the asymptotes \(x=0\) and \(y=0\). The graph is translated 3 units to the left and 2 units down. Write the equation of the image and the equations of its asymptotes.

    • Translate 3 left: replace x by \(x+3\): \(y=\dfrac{1}{x+3}\)
    • Translate 2 down: \(y=\dfrac{1}{x+3}-2\)
    • The vertical asymptote moves 3 left: \(x=-3\). The horizontal asymptote moves 2 down: \(y=-2\).

In context

  1. The curve of a suspended cable between two towers is modelled by \(y=\frac{x^{2}}{2} - 6 x + 21\), where x and y are in metres.

    1. (a) Write \(\frac{x^{2}}{2} - 6 x + 21\) in the form \(a(x-h)^2+k\).

      • \(\frac{x^{2}}{2} - 6 x + 21=\frac{1}{2}(x^2-12x)+21\)
      • \(=\frac{1}{2}\big[(x-6)^2-36\big]+21\)
      • \(=\frac{1}{2}(x-6)^2+3\)

    2. (b) Describe two transformations that map the graph of \(y=x^2\) to this curve.

      • With \(f(x)=x^2\): \(y=\frac{1}{2}f(x-6)+3\)
      • Vertical stretch with scale factor 0.5, then translate 6 in the x direction and 3 in the y direction

  2. A graph shows the temperature C in °C of a room against time. The temperature in °F is \(F=1.8C+32\).

    1. (a) Describe the transformations that map the graph of C to the graph of F.

      • F\(=1.8\times\)(old value)\(+32\)
      • A multiplication comes first, then the addition: stretch, then translate

    2. (b) The same graph can be made by translating first and stretching second. Describe that translation and that stretch.

      • \(1.8y+32=1.8\big(y+160/9\big)\)
      • So translate up 160/9 first, then stretch with scale factor 1.8

Stretch

  1. Describe the transformations that map \(y=\sqrt{x}\) to each graph, in order.

    1. (a) \(y=\sqrt{5x-3}\)

      • \(\sqrt{5x-3}=g(5x)\) where \(g(x)=\sqrt{x-3}\)
      • First \(g\): translate by 3. Then replace x by 5x: stretch with scale factor \(1/5\)

    2. (b) \(y=\sqrt{5(x-3)}\)

      • \(\sqrt{5(x-3)}=h(x-3)\) where \(h(x)=\sqrt{5x}\)
      • First \(h\): stretch with scale factor \(1/5\). Then translate by 3

    3. (c) Where does the point \((0,0)\) of \(y=\sqrt{x}\) end up in each graph?

      • First graph: x = 3 before the stretch, then 3 ÷ 5 = 0.6
      • Second graph: the stretch leaves 0 where it is, then the translation gives 3
      • The two graphs are different: the order matters

  2. Find two different sequences of two transformations that map \(y=f(x)\) to \(y=f(6x+3)\).

    • As written: \(f(x+3)\) first, then replace x by 6x
    • Sequence 1: translate −3, then stretch with scale factor \(1/6\)
    • Factor: \(6x+3=6\big(x+0.5\big)\)
    • Sequence 2: stretch with scale factor \(1/6\), then translate −0.5

  3. The graph of \(y=ax^2+bx+c\) is translated 1 unit in the negative x direction and then stretched horizontally with scale factor \(\tfrac12\). The resulting graph is \(y=4x^2+cx+17\). Find a, b and c.

    • The new graph is \(y=f(2x+1)\) with \(f(x)=ax^2+bx+c\)
    • \(a(2x+1)^2+b(2x+1)+c=4 a x^{2} + a + b + c + x \left(4 a + 2 b\right)\)
    • Compare: \(4 a=4\) gives a = 1; the x term gives \(4 a + 2 b=c\); the constant gives \(a + b + c=17\)
    • Solving these three equations: a = 1, b = 4, c = 12

  4. The graph of \(y=f(x)\) is stretched vertically with scale factor 5 away from the line \(y=-1\) (the line \(y=-1\) does not move).

    1. (a) Write down the equation of the new graph in terms of f.

      • Translate so the line is the x-axis: \(y=f(x)+1\)
      • Stretch: \(y=5\big(f(x)+1\big)\)
      • Translate back: \(y=5\big(f(x)+1\big)-1=5f(x)+4\)

    2. (b) The point (3, 4) is on the graph of \(y=f(x)\). Find its image.

      • The x-coordinate stays 3
      • y: \(5\times4+4=24\)

jmaths.xyz · AI HL · 12.5.7 Composite transformations