Translations of graphs Problem set B Worked solutions

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

1. (a) Sketch \(y=f(x)+3\) and give the new coordinates of A, B, C and D.

   • Every y-value goes up 3; the x-values do not change.
   • A'(−4, 3), B'(−1, 6), C'(2, 2), D'(5, 5)

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

   • Every y-value goes down 5; the x-values do not change.
   • A'(−4, −5), B'(−1, −2), C'(2, −6), D'(5, −3)

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

1. (a) Sketch \(y=f(x+4)\) and give the new coordinates of A, B, C and D.

   • Replace x by x + 4: the graph moves 4 to the left.
   • A'(−8, 0), B'(−5, 3), C'(−2, −1), D'(1, 2)

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

   • Replace x by x − 3: the graph moves 3 to the right.
   • A'(−1, 0), B'(2, 3), C'(5, −1), D'(8, 2)

Write down the equation of each graph after the translation.

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

   • Down 3 means subtract 3 from the output.
   • \(y=4x^2-3\)

2. (b) \(y=x^5\) translated 8 units up

   • Up 8 means add 8 to the output.
   • \(y=x^5+8\)

Write down the equation of each graph after the translation.

1. (a) \(y=x^6\) translated 5 units to the right

   • Right 5 means replace x by x − 5.
   • \(y=(x-5)^6\)

2. (b) \(y=3x^3\) translated 4 units to the left

   • Left 4 means replace x by x + 4.
   • \(y=3(x+4)^3\)

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.

   • \(f(x+3)=(x+3)^{2} - 8 (x+3) + 3\)
   • \(y=x^{2} - 2 x - 12\)

2. (b) \(f(x)=3 x^{2} - x + 2\). Find \(f(x-2)\).

   • \(f(x-2)=3 (x-2)^{2} - (x-2) + 2\)
   • \(y=3 x^{2} - 13 x + 16\)

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 \(at^2+bt+c\).

   • \(h_2(t)=h(t-3)=- 5 (t-3)^{2} + 30 (t-3)\)
   • \(h_2(t)=- 5 t^{2} + 60 t - 135\)

2. (b) State the time at which the new maximum height is reached.

   • The original maximum is at t = 3. The graph moves 3 to the right.
   • t = 3 + 3 = 6

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.

1. (a) Give the new coordinates of the four marked points.

   • Replace t by t + 2: each t-value moves 2 left.
   • Take 3 from each T-value.
   • A'(−5, 0), B'(−2, −4), C'(1, −1), D'(4, −5)

2. (b) Describe the translation in words.

   • 2 left and 3 down

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 \(y=ax^2+bx+c\).

• \(y=f(x+2)+5\)
• \(y=(x+2)^{2} - 6 (x+2) + 4+5\)
• \(y=x^{2} - 2 x + 1\)

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.

   • \(f(x)=0 \text{ at } x=3, 7\)
   • The roots move to −2 and 2.
   • Each root moves 5 units to the left.

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

   • \(y=f(x+5)\)
   • \(y=x^{2} - 4\)