Transformations of Functions and Graphs

The video tutorial explores the effects of composing functions with linear functions, focusing on translation and scaling. It demonstrates how these transformations affect the graphs of functions, using examples like quadratic and sine functions. The tutorial concludes with applying these concepts to sine waves, explaining terms like amplitude, period, phase, and equilibrium position.

What is the main challenge in describing the graphs of composed functions?

• They are always linear.

• They are too complex to describe easily.

• They are identical to their component functions.

• They do not exist.

What are the two main characteristics of linear functions?

• Translation and reflection

• Scaling and rotation

• Reflection and rotation

• Scaling and translation

What happens to the graph of g(x) = x^2 when composed with f(x) = x + 5 as f(g(x))?

• It shifts 5 units downwards.

• It shifts 5 units to the right.

• It remains unchanged.

• It shifts 5 units upwards.

When g(f(x)) = (x+5)^2, how is the graph transformed?

• Shifted 5 units downwards

• Shifted 5 units upwards

• Shifted 5 units to the left

• Shifted 5 units to the right

What is the effect of composing f(x) = 2x with g(x) = sin(x) as f(g(x))?

• The graph is reflected.

• The graph is shifted horizontally.

• The graph is compressed vertically.

• The graph is stretched vertically.

How does the graph of g(f(x)) = sin(2x) differ from sin(x)?

• It is shrunk horizontally.

• It is shifted vertically.

• It is reflected.

• It is stretched horizontally.

What does g(x) + a do to the graph of g(x)?

• Shifts it a units to the right

• Shifts it a units upwards

• Stretches it vertically by a factor of a

• Shrinks it horizontally by a factor of a

What transformation does g(x + a) apply to the graph of g(x)?

• Shifts it a units upwards

• Shrinks it horizontally by a factor of a

• Stretches it vertically by a factor of a

• Shifts it a units to the left

What is the effect of ag(x) on the graph of g(x)?

• Vertical shift by a

• Horizontal shift by a

• Horizontal scaling by a factor of a

• Vertical scaling by a factor of a

What does g(ax) do to the graph of g(x)?

• Shifts it a units to the right

• Shifts it a units upwards

• Stretches it vertically by a factor of a

• Shrinks it horizontally by a factor of 1/a