This page presents clear answers and explanations for the questions in the sine function tutorial. The focus is on understanding how the parameters affect the graph of the sine function.
The sine function studied in this tutorial has the form:
\[ f(x) = a \sin(bx + c) + d \]Each parameter controls a specific transformation of the graph.
Set \( a = 1 \), \( b = 1 \), \( c = 0 \), and \( d = 0 \). Write down \( f(x) \) and determine the amplitude, period, and phase shift. Then change \( a \) and observe the effect on the graph.
Answer:
With these values,
\[ f(x) = \sin(x) \]The amplitude is:
\[ |a| = 1 \]The period is:
\[ 2\pi \]The phase shift is:
\[ 0 \]As \( |a| \) increases or decreases, the amplitude changes accordingly. The maximum value of \( f(x) \) is always equal to \( |a| \).
Set \( a = 1 \), \( c = 0 \), and \( d = 0 \), then change \( b \). Measure the period from the graph and compare it with \( \frac{2\pi}{|b|} \).
Answer:
The period of the sine function is given by:
\[ \text{Period} = \frac{2\pi}{|b|} \]As \( |b| \) increases, the graph is horizontally compressed. As \( |b| \) decreases, the graph is horizontally stretched.
Set \( a = 1 \), \( b = 1 \), and \( d = 0 \). Increase \( c \) gradually from zero to positive values. Observe the direction of the shift.
Answer:
The graph of \( f(x) \) shifts to the left.
Set \( a = 1 \), \( b = 1 \), and \( d = 0 \). Decrease \( c \) gradually from zero to negative values. Observe the direction of the shift.
Answer:
The graph of \( f(x) \) shifts to the right.
Repeat the previous steps for \( b = 2, 3, \) and \( 4 \). Measure the shift and compare it with the phase shift formula.
Answer:
The phase shift is given by:
\[ -\frac{c}{b} \]If \( -\frac{c}{b} > 0 \), the graph shifts to the left. If \( -\frac{c}{b} < 0 \), the graph shifts to the right.
Set \( a \), \( b \), and \( c \) to nonzero values and change \( d \). Observe how the graph moves.
Answer:
The parameter \( d \) controls the vertical shift: