Translations of graphs Problem set A One question at a time

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

1. (a) Sketch \(y=f(x)+2\) and give the new coordinates of A, B, C and D.

   • Every y-value goes up 2; the x-values do not change.
   • A'(−5, 1), B'(−2, 4), C'(1, 0), D'(4, 3)

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

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

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

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

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

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

   • Replace x by x − 2: the graph moves 2 to the right.
   • A'(−3, −1), B'(0, 2), C'(3, −2), D'(6, 1)

Write down the equation of each graph after the translation.

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

   • Up 4 means add 4 to the output.
   • \(y=3x^2+4\)

2. (b) \(y=x^4\) translated 6 units down

   • Down 6 means subtract 6 from the output.
   • \(y=x^4-6\)

Write down the equation of each graph after the translation.

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

   • Left 2 means replace x by x + 2.
   • \(y=(x+2)^4\)

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

   • Right 1 means replace x by x − 1.
   • \(y=5(x-1)^2\)

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.

   • \(f(x+2)=(x+2)^{2} - 6 (x+2) + 11\)
   • \(y=x^{2} - 2 x + 3\)

2. (b) \(f(x)=2 x^{2} + x - 3\). Find \(f(x-1)\).

   • \(f(x-1)=2 (x-1)^{2} + (x-1) - 3\)
   • \(y=2 x^{2} - 3 x - 2\)

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

   • \(h_2(t)=h(t-2)=- 5 (t-2)^{2} + 20 (t-2)\)
   • \(h_2(t)=- 5 t^{2} + 40 t - 60\)

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

   • The original maximum is at t = 2. The graph moves 2 to the right.
   • t = 2 + 2 = 4

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.

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

   • Replace t by t − 2: each t-value moves 2 right.
   • Add 3 to each d-value.
   • A'(−2, 5), B'(1, 1), C'(4, 4), D'(7, 2)

2. (b) Describe the translation in words.

   • 2 right and 3 up

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

• \(y=f(x-3)-4\)
• \(y=(x-3)^{2} - 4 (x-3) + 7-4\)
• \(y=x^{2} - 10 x + 24\)

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.

   • \(f(x)=0 \text{ at } x=2, 4\)
   • The roots move to −1 and 1.
   • Each root moves 3 units to the left.

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

   • \(y=f(x+3)\)
   • \(y=x^{2} - 1\)