Question

From the curve y = |x|, draw y = |x − 1| + 1

Answer 1

y = |x|

y = `{{:(x,  "if"  x ≥ 0),(- x,  "if"  x < 0):}`

x	0	1	2	3	− 1	− 2	− 3
y	0	1	2	3	1	2	3

(a) Consider y = |x – 1|

y = `{{:((x - 1),  "if"  x - 1 ≥ 0),(-(x - 1),  "if"  x - 1 < 0):}`

y = `{{:(x - 1,  "if"  x  ≥ 1),(-x + 1,  "if"  x < 1):}`

x = 0 ⇒ y = – x + 1 ⇒ y = 1

x = 1 ⇒ y = x – 1 ⇒ y = 0

x = 2 ⇒ y = x – 1 ⇒ y = 1

x = 3 ⇒ y = x – 1 ⇒ y = 2

x = – 1 ⇒ y = – x + 1 ⇒ y = 2

x = – 2 ⇒ y = – x + 1 ⇒ y = 3

x	0	1	2	3	− 1	− 2
y	1	0	1	2	2	3

The graph of y = |x – 1| causes the graph y = |x| a shift to the right by 1 unit.

The graph of y = f(x – c), c > 0 causes the graph y = f(x) a shift to the right by c units.

(b) Consider y = |x – 1| + 1

y = `{{:((x - 1) + 1,  "if"  x - 1 ≥ 0),(-(x - 1) + 1,  "if"  x - 1 < 0):}`

y = `{{:(x,  "if"  x ≥ 1),(- x + 2,  "if"  x < 1):}`

x = 0 ⇒ y = – x + 2 ⇒ y = 2

x = 1 ⇒ y = x ⇒ y = 1

x = 2 ⇒ y = x ⇒ y = 2

x = 3 ⇒ y = x ⇒ y = 3

x = – 1 ⇒ y = – x + 2 ⇒ y = 3

x = – 2 ⇒ y = – x + 2 ⇒ y = 4

x	0	1	2	3	− 1	− 2
y	2	1	2	3	3	4

The graph of y = |x – 1| causes the graph y = |x| a shift to the right by 1 unit.

The graph of y = f(x – c), c > 0 causes the graph y = f(x) a shift to the right by c units.

The graph of y = f(x) + d, d > 0 causes the graph y = f(x) a shift to the upward by d units.