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IB Mathematics: Applications and Interpretation HL · 12.5: Functions · 12.5.5

Name Date

Translations of graphs Problem set A

Check each answer: outside the bracket changes y, inside the bracket changes x (and works backwards). Sketch graphs by moving the marked points.

Difficulty: first skills applying them stretch

Skills

  1. The graph of y = f(x) is made of straight lines through the points A, B, C and D.

    −10−5510−6−4−2246xyABCD
    1. (a) Sketch \(y=f(x)+2\) and give the new coordinates of A, B, C and D.
    2. (b) Sketch \(y=f(x)-4\) and give the new coordinates of A, B, C and D.

  2. The graph of y = f(x) is made of straight lines through the points A, B, C and D.

    −10−5510−6−4−2246xyABCD
    1. (a) Sketch \(y=f(x+3)\) and give the new coordinates of A, B, C and D.
    2. (b) Sketch \(y=f(x-2)\) and give the new coordinates of A, B, C and D.

  3. Write down the equation of each graph after the translation.

    1. (a) \(y=3x^2\) translated 4 units up
    2. (b) \(y=x^4\) translated 6 units down

  4. Write down the equation of each graph after the translation.

    1. (a) \(y=x^4\) translated 2 units to the left
    2. (b) \(y=5x^2\) translated 1 unit to the right

  5. Find each equation, in the form y = ax² + bx + c.

    1. (a) The graph of \(y=x^{2} - 6 x + 11\) is translated 2 units to the left.
    2. (b) \(f(x)=2 x^{2} + x - 3\). Find \(f(x-1)\).

In context

  1. A ball is thrown upwards. Its height after t seconds is h(t) = −5t² + 20t metres. A second ball is thrown 2 seconds later with the same speed, so its height after t seconds from the start is h₂(t) = h(t − 2).

    1. (a) Write the new height function \(h_2(t)\) in the form \(at^2+bt+c\).
    2. (b) State the time at which the new maximum height is reached.

  2. The graph shows the water level d(t), in centimetres above or below a mark on a post beside a rock pool, t hours after noon (negative t is before noon). A second pool has water level D(t) = d(t−2)+3.

    −6−4−22468−4−2246tdABCD
    1. (a) Give the new coordinates of the four marked points.
    2. (b) Describe the translation in words.

Stretch

  1. The graph of \(y=x^{2} - 4 x + 7\) is translated 3 units to the right and 4 units down. Find the equation of the new graph in the form \(y=ax^2+bx+c\).

  2. The graph of \(y=f(x)\), where \(f(x)=x^{2} - 6 x + 8\), is translated horizontally. The new graph crosses the x-axis at \(x=-1\) and \(x=1\).

    1. (a) Find the translation.
    2. (b) Write the equation of the new graph in the form \(y=ax^2+bx+c\).

jmaths.xyz · AI HL · 12.5.5 Translations of graphs