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

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Translations of graphs Problem set B

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

Each question matches the question with the same number in set A. Do the ones you got wrong there.

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 \(y=f(x)+3\) and give the new coordinates of A, B, C and D.
    2. (b) Sketch \(y=f(x)-5\) 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 \(y=f(x+4)\) and give the new coordinates of A, B, C and D.
    2. (b) Sketch \(y=f(x-3)\) and give the new coordinates of A, B, C and D.

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

    1. (a) \(y=4x^2\) translated 3 units down
    2. (b) \(y=x^5\) translated 8 units up

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

    1. (a) \(y=x^6\) translated 5 units to the right
    2. (b) \(y=3x^3\) translated 4 units to the left

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

    1. (a) The graph of \(y=x^{2} - 8 x + 3\) is translated 3 units to the left.
    2. (b) \(f(x)=3 x^{2} - x + 2\). Find \(f(x-2)\).

In context

  1. A firework is launched upwards. Its height after t seconds is h(t) = −5t² + 30t metres. A second firework is launched 3 seconds later in the same way, so its height after t seconds from the start is h₂(t) = h(t − 3).

    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 temperature T(t), in °C, in a greenhouse, t hours after midnight (negative t is before midnight). A second greenhouse has temperature U(t) = T(t+2)−3.

    −5510−6−4−2246tTABCD
    1. (a) Give the new coordinates of the four marked points.
    2. (b) Describe the translation in words.

Stretch

  1. The graph of \(y=x^{2} - 6 x + 4\) is translated 2 units to the left and 5 units up. Find the equation of the new graph in the form \(y=ax^2+bx+c\).

  2. The graph of \(y=f(x)\), where \(f(x)=x^{2} - 10 x + 21\), is translated horizontally. The new graph crosses the x-axis at \(x=-2\) and \(x=2\).

    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