The graph of y = f(x) – c is obtained by shifting the graph of y = f(x) downward a distance of c units.

If c > 0, then the graph of y = f(x) + c is obtained by shifting the graph of y = f(x) upward a distance of c units.

Vertical Shifting of the Graph of a Function

2.6 – Transformations of Graphs

If c > 0, the graph of y = f(x + c) is obtained by shifting the graph ofy = f(x) to the left a distance of c units.

If c > 0, the graph of y = f(x – c) is obtained by shifting the graph of y = f(x) to the right a distance of c units.

Horizontal Shifting of the Graph of a Function

2.6 – Transformations of Graphs

Describe how the graph of y = |x + 2| − 6 would be obtained by translating the graph of y = |x|.

Horizontal shift: 2 units left

Vertical shift: 6 units down

𝑦=|𝑥| 𝑦=|𝑥+2|− 6

2.6 – Transformations of GraphsVertical and Horizontal Shifts

2.6 – Transformations of Graphs

𝑦=√𝑥 𝑦=√𝑥+4

𝑦=√𝑥− 3 𝑦=√𝑥+7 −5

Write the equation of each graph using the appropriate transformations.

2.6 – Transformations of Graphs

If a point (x, y) lies on the graph of y = f(x) then the point (x, cy) lies on the graph of y = cf(x).

If c > 1, then the graph of y = cf(x) is a vertical stretching of the graph of y = f(x) by applying a factor of c.

Vertical Stretching and Shrinking of the Graph of a Function

If 0 < c < 1, then the graph of y = cf(x) is a vertical shrinking of the graph of y = f(x) by applying a factor of c.

𝑦=𝑥2− 3 𝑦=2 𝑥2−3 𝑦=0.2 𝑥2−3

2.6 – Transformations of GraphsHorizontal Stretching and Shrinking of the Graph of a Function

If a point (x, y) lies on the graph of y = ƒ(x), then the point (x/c, y) lies on the graph of y = ƒ(cx).

(a) If c > 1, then the graph of y = ƒ(cx) is a horizontal shrinking of the graph of y = ƒ(x).

(b) If 0 < c < 1, then the graph of y = ƒ(cx) is a horizontal stretching of the graph of y = ƒ(x).

𝑦=𝑥2− 3 𝑦=(3 𝑥 )2− 3 𝑦=(0.3 𝑥 )2− 3

2.6 – Transformations of GraphsReflections Across the x and y Axes

For a function, y = f(x), the following are true.

(a) the graph of y = –f(x) is a reflection of the graph of f across the x-axis.

(b) the graph of y = f(– x) is a reflection of the graph of f across the y-axis.

Given the graph of a function y = f(x) sketch the graph of:

y = –f(x) f(– x)

f(x) f(x)

2.6 – Transformations of GraphsReflections Across the x and y Axes

𝑦=− 3 (𝑥− 4 )2+5

5)4(3 2 xy

2( 4)y x

23( 4)y x

23( 4)y x horizontal shift 4 units right

vertical stretch by a factor of 3

reflect across the x-axis

vertical shift 5 units up

2.6 – Transformations of GraphsTransformations

y = f(x) + C C > 0 moves it upC < 0 moves it down

y = f(x + C) C > 0 moves it leftC < 0 moves it right

y = C·f(x) C > 1 stretches it in the y-direction0 < C < 1 compresses it

y = f(Cx)C > 1 compresses it in the x-

direction0 < C < 1 stretches it

y = -f(x) Reflects it about x-axisy = f(-x) Reflects it about y-axis

2.6 – Transformations of GraphsSummary of Transformations