Question

A certain function is vertically stretched by a factor of 2, horizontally compressed by 1/4, and translated down 6. The new function is represented by f(x) = 6 |8x| -2. Write the original function.

Answer 1

Answer:

The original function would be $g(x)=3|32x|+2$

Step-by-step explanation:

Recall the following transformation rules:

(i) vertically stretched by a factor of c ( c must be greater than 1):

f(x) is transformed to cf(x).

(ii) horizontally compressed by a factor of a ( a must be greater than 0):

f(x) is transformed to f(ax).

(iii) translated down a unit:

f(x) is transformed to f(x)-a

Consider the original function is g(x).

After vertically stretched by a factor of 2,

Transformed function is 2g(x).

After horizontally compressed by a factor of 1/4,

Transformed function is 2g(x/4).

Finally, translate down 6 units,

Transformed function is 2g(x/4) - 6.

According to the question,

$2g(\frac{x}{4}) - 6 = f(x) = 6 |8x| -2$

$2g(\frac{x}{4})-6=6 |8x| -2$

$2g(\frac{x}{4})=6 |8x| -2+6$

$2g(\frac{x}{4})=6 |8x| +4$

$g(\frac{x}{4})=3 |8x| +2$

$\implies g(y)=3 |32y| +2$    ( Say x/4 = y )

Substitute x for y,

$g(x) = 3 |32x| +2$

Hence, the original function would be $g(x) = 3 |32x| +2$ .