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Graph Transformations0%

Functions & Graphs · Topic 5 of 9

Graph Transformations

Video lesson · from 1:20:132 worked examples

One lesson video covers all of Functions & Graphs, so it opens at 1:20:13 for this topic — not from the beginning.

Theory

Given any graph f(x)f(x), we can reflect, move, or scale it. Each transformation influences the points on the graph.

NotationDescriptionChange to Points (x,y)(x,y)
f(x)+af(x) + aGraph moves up by aa units→(x,y+a)\to (x, y + a)
f(x)−af(x) - aGraph moves down by aa units→(x,y−a)\to (x, y - a)
af(x)af(x)Vertical stretch by aa→(x,ay)\to (x, ay)
−f(x)-f(x)Reflect over x-axis→(x,−y)\to (x, -y)
f(x+a)f(x + a)Graph moves left by aa units→(x−a,y)\to (x - a, y)
f(x−a)f(x - a)Graph moves right by aa units→(x+a,y)\to (x + a, y)
f(−x)f(-x)Reflect over y-axis→(−x,y)\to (-x, y)

Remember:

Changes inside the bracket affect the x-coordinates (horizontally) and do the opposite of what you might expect.

Changes outside the bracket affect the y-coordinates (vertically) and do exactly what they say.

⚠️ Common Examiner Traps

  • Label the point you were asked for: a very common loss is failing to identify clearly which coordinates are the maximum of the transformed graph, or naming the wrong ones. Mark and name it explicitly.
  • Inside the bracket does the opposite: f(x+3)f(x+3) moves the graph left 3, not right. This is the most common transformation error there is.
  • Apply transformations in order: for something like 3f(2x)+13f(2x)+1, deal with each change separately and track what happens to a known point at each stage.
  • Transform the key points, not the whole curve: take the turning points and intercepts, move each one, then draw a curve of the same shape through them.
  • Practise on the axes given: both reports ask for practice at sketching on provided axes — use the scale printed on them.

Worked examples

Example 1

Example 1

Sketch the graph of y=−f(x−2)y = -f(x - 2).

(-3, 0)(-2, 10)(1, -17)y = f(x)

The transformation involves two steps:

  1. −2-2 inside the bracket: Move right 2 units. (x→x+2x \to x + 2).
  2. Negative sign outside: Reflect over the x-axis. (y→−yy \to -y).
Original Point (x,y)(x,y)Move right 2 (x+2,y)(x+2,y)Reflect x-axis (x,−y)(x,-y)
(-3, 0)(-1, 0)(-1, 0)
(-2, 10)(0, 10)(0, -10)
(1, -17)(3, -17)(3, 17)

Plot the final points and connect them with a curve similar to the original shape.

Example 2

Example 2

Sketch the graph of y=3f(2x)+1y = 3f(2x) + 1.

(-1, 0)(0, 2)(1, 0)(3, 0)y = f(x)

The transformation involves three operations:

  1. 2x2x inside: Compress horizontally by a factor of 2. (x→x2x \to \frac{x}{2}).
  2. 33 outside: Vertical stretch by 3. (y→3yy \to 3y).
  3. +1+ 1 outside: Move up 1 unit. (y→y+1y \to y + 1).
Original (x,y)(x,y)Horiz compress (x2,y)\left(\frac{x}{2},y\right)Vertical stretch (x,3y)(x, 3y)Move up 1 (x,y+1)(x, y+1)
(-1, 0)(-0.5, 0)(-0.5, 0)(-0.5, 1)
(0, 2)(0, 2)(0, 6)(0, 7)
(1, 0)(0.5, 0)(0.5, 0)(0.5, 1)
(3, 0)(1.5, 0)(1.5, 0)(1.5, 1)