Transformation of Functions

Question 1

Explain translations.

Answer 1

If c is a positive number, then the graph of y = f(x) + c is the graph of y = f(x) shifted upward a distance of c units.
The graph of y = f(x - c) is the graph of y = f(x) shifted c units to the right.

Question 2

Explain stretching and reflecting.

Answer 2

If c is positive, the graph of y = cf(x) is the graph of y = f(x) stretched by a factor of c in the vertical direction.

The graph of y = -f(x) is the graph of y = f(x) reflected about the x-axis.

The graph of y = f(cx) is the graph of y = f(x) compressed horizontally y a factor of c.

The graph of y = f(-x) is the graph of y = f(x) reflected about the y-axis.

Question 3

Explain the sum and difference functions and state the domain.

Answer 3

(f + g)(x) = f(x) + g(x)
(f - g)(x) = f(x) - g(x)

If the domain of f is A and the domain of g is B, then the domain of f + g is A n B.

Question 4

Explain the product and quotient functions and state the domains.

Answer 4

(fg)(x) = f(x)g(x)
(f/g)(x) = f(x)/g(x)

If the domain of f is A and the domain of g is B, then the domain of fg is A n B.
The domain of f/g is {x E A n B | g(x) /= 0}

Question 5

Explain the composition of functions and state the domain.

Answer 5

(f o g)(x) = f(g(x))

(f o g)(x) is defined whenever both g(x) and f(g(x)) are defined.