© Daniel Holloway

Transforming graphs is not too dissimilar from transforming shapes. Whereas you can translate, rotate, reflect and enlarge shapes; you can translate, stretch and reflect graphs. We use the notation f(x) to denote a function of x. A function of x is any algebraic expression where x is the only variable.

There are six rules you need to learn about transforming graphs. To show these rules, we will use the following graph. This is the graph y = f(x)

Rule 1: This is a translation of the graph in the vector ( ) in the y-direction a 0 y = f(x) + a

Rule 2: This is a translation of the graph in the vector ( ) in the x-direction y = f(x – a) a 0

Rule 3: This is a stretch of the graph by a scale factor of k in the y-direction Note that they cross at the x axis y = kf(x)

Rule 4: This is a stretch of the graph by a scale factor of 1 / t in the x-direction Note that they cross at the y axis y = f(tx)

Rule 5: This is a reflection of the graph in the x-axis y = -f(x)

Rule 6: This is a reflection of the graph in the y-axis y = f(-x)

y x y x The grid shows the graph of y=x 2 for -2 ≤ x ≤ 2 The dotted line shows the same graph. Describe the transformation taking it to the new line on the grid. y = (x + 3) 2

y x y x The grid shows the graph of y=x 2 for -2 ≤ x ≤ 2 The dotted line shows the same graph. Describe the transformation taking it to the new line on the grid. y = x 2 - 2

y x y x The grid shows the graph of y=x 2 for -2 ≤ x ≤ 2 The dotted line shows the same graph. Describe the transformation taking it to the new line on the grid. y = 2x 2