Graph Transformations and Period Changes

The video tutorial explains the concept of horizontal stretch in graph transformations. It covers two cases: when the scaling factor 'a' is greater than 1, resulting in a horizontal shrink, and when 'a' is a fraction between 0 and 1, leading to a horizontal stretch. The tutorial uses examples with sine functions to illustrate these transformations, showing how the number of cycles changes with different values of 'a'. The video concludes with a generalization of these transformations for any function f(x).

What is the primary focus of the horizontal stretch transformation?

• Changing the amplitude of the graph

• Shifting the graph vertically

• Reflecting the graph over the x-axis

• Altering the period of the graph

In the transformation of sin(x) to sin(2x), what happens to the period of the graph?

• It becomes zero

• It halves

• It remains the same

• It doubles

When a > 1, how does the graph of f(ax) compare to f(x)?

• It is compressed horizontally

• It is stretched vertically

• It is reflected over the y-axis

• It is stretched horizontally

What is observed when the value of 'a' is increased in the graph of sin(ax)?

• The graph shifts to the right

• The graph completes fewer cycles

• The graph completes more cycles

• The graph's amplitude increases

In the transformation of sin(x) to sin(x/2), what happens to the period of the graph?

• It doubles

• It becomes zero

• It halves

• It remains the same

When 0 < a < 1, how does the graph of f(ax) compare to f(x)?

• It is stretched vertically

• It is compressed horizontally

• It is stretched horizontally

• It is reflected over the y-axis

What is the effect of a horizontal stretch on the graph of a function when a is a fraction?

• The graph is compressed

• The graph is stretched

• The graph is reflected

• The graph is shifted

How many cycles does the graph of sin(x/2) complete in the interval 0 to 4π?

• One cycle

• Four cycles

• Three cycles

• Two cycles

What general rule can be applied to the graph of f(ax) when a > 1?

• Reflect it over the x-axis

• Stretch it horizontally by a times

• Compress it horizontally by a times

• Stretch it vertically by a times

What general rule can be applied to the graph of f(ax) when 0 < a < 1?

• Stretch it vertically by a times

• Reflect it over the x-axis

• Compress it horizontally by a times

• Stretch it horizontally by a times