Calculus, 8th Edition,stewart; page 42 problem 2

Question

Explain how each graph is obtained from the graph of y = f(x).

(a) y = f(x) + 8     (b) y = f(x + 8)     (c) y = 8 f(x)      (d) y = f (8x)     (e) y = - f(x) - 1      (f) y = 8f (1/8x)

Answer 1

Step:1

(a)

The function is .

Vertical shift : , shift the graph of  a distance  units upward.

Here .

 is the graph of  shift to a distance of  units upward.

Step:2

(b)

The function is .

Horizontal shift : , shift the graph of  a distance  units to the left.

Here .

 is the graph of  shift to a distance of  units to the left.

Step:3

(c)

The function is .

Vertical stretch : , stretch the graph of  vertically by a factor of .

Here .

 is the graph of  vertically stretched by a factor of .

Step:4

(d)

The function is .

Horizontal shrink: , shrink the graph of  horizontally by a factor of .

Here .

 is the graph of  horizontally shrink by a factor of .

Step:5

(e)

The function is .

Vertical shift : , shift the graph of  a distance  units downward.

Reflection : , reflect the graph of  about the axis.

 is shifted  unit down and reflect about the axis.

Step:6

(f)

The function is .

Vertical stretch: , stretch the graph of  vertically by a factor of .

Horizontal stretch: , stretch the graph of  horizontally by a factor of .

 is stretched vertically and horizontally by a factor of .

Solution:

(a) The graph of  shifted to a distance  units upward.

(b) The graph of  shifted to a distance  units to the left.

(c) The graph of  stretched vertically by a factor of .

(d) The graph of  shrinks horizontally by a factor of .

(e) The graph of  shifted  unit down and reflect about the axis.

(f) The graph of  is stretched vertically and horizontally by a factor of .