Transformation Notes

Summary S/n Type of Transformation General Equation Description Results 1 Translation (along y-axis) y = f(x) + a Translate up along y-axis by a units. y co-ordinates change. y = f(x) - a Translate down along y-axis by a units. y co-ordinates change. 2 Translation (along x-axis) y = f(x + b) Translate left along x-axis by b units. x co-ordinates change. y = f(x ̶ b) Translate right along x-axis by b units. x co-ordinates change. 3 Stretching (along y-axis) y = pf(x), p > 1 Stretch y = f(x), parallel to the y-axis. Factor p. y co-ordinates change. Multiply the y co-ordinates by p. y = pf(x), p < 1 y = ̶ pf(x), |p|> 1 Stretch y = f(x) parallel to the y-axis to obtain y = pf(x). Factor p. Reflect y = pf(x) on the x-axis. y co-ordinates change. Multiply the y co-ordinates by -p. y = ̶ pf(x), |p|<1 4 Stretching (along x-axis) y = f(kx), k > 1 Stretch y = f(x), parallel to the x-axis. Factor k 1 . x co-ordinates change. Multiply the x co-ordinates by k 1 . y = f(kx), k < 1 y = f(-kx), |k| > 1 Stretch y = f(x), parallel to the x-axis to obtain y = f(kx). Factor k 1 . Reflect y = f(kx) on the y-axis. x co-ordinates change. Multiply the x co-ordinates by k− 1 . y = f(-kx), |k| < 1 1. Translations change the location of a graph. 2. Stretchings change the appearance of a graph. S/n Graph Transformation 1 y = f(x ̶ 2) Translate y = f(x) right along x-axis by 2 units to obtain y = f(x ̶ 2). 2 y = f(x) – 2 Translate y = f(x) down along y-axis by 2 units to obtain y = f(x) ̶ 2. 3 y = 3f(x) Stretch y = f(x) parallel to the y-axis by a factor of 3 to obtain y = 3f(x). 4 y = f(3x) Stretch y = f(x) parallel to the x-axis by a factor of 1 3 to obtain y = f(3x). 5 y = 2f(x + 3) – 5 First, translate y = f(x) left along x-axis by 3 units to obtain y = f(x + 3). Next, stretch y = f(x + 3) parallel to the y-axis by a factor of 2 to obtain y = 2f(x + 3). Lastly, translate y = f(x) down along y-axis by 5 units to obtain y = 2f(x + 3) – 5. 6 y = ̶ f(x) Reflect y = f(x) on the x-axis. 7 y = f( ̶ x) Reflect y = f(x) on the y-axis.