Problem 9 - Algebra 2

Question

State the vertical shift, amplitude, period, and phase shift for each function. Then graph the function. $$ y=3 \sin \left[2\left(\theta-30^{\circ}\right)\right]+10 $$

Step-by-Step Solution

Answer

Vertical shift: 10; Amplitude: 3; Period: 180°; Phase shift: 30° right.

Step 1: Identify the vertical shift

The vertical shift of the function can be obtained by looking at the constant term outside the sine function. Here, the constant is 10, which indicates that the graph is shifted vertically upwards by 10 units.

Step 2: Determine the amplitude

The amplitude of a sine function is given by the coefficient of the sine term. In the function \( y = 3 \sin\left[2(\theta - 30^\circ)\right] + 10 \), the coefficient of \( \sin \) is 3, so the amplitude is 3.

Step 3: Calculate the period

The period of a sine function \( y = a \sin(bx + c) + d \) is found by the formula \( \frac{360^\circ}{b} \). For this function, \( b = 2 \), so the period is \( \frac{360^\circ}{2} = 180^\circ \).

Step 4: Find the phase shift

The phase shift of the function is determined by the horizontal translation \( \frac{c}{b} \) where the function is \( y = a \sin(bx + c) + d \). Here, \( c = -2 \times 30^\circ = -60^\circ \) and \( b = 2 \), so the phase shift is \( \frac{-60^\circ}{2} = -30^\circ \). This indicates the graph shifts right by \( 30^\circ \) since \( \theta - 30^\circ \) is equivalent to a right-shift of \( 30^\circ \).

Step 5: Draw the graph

To graph the function, start by plotting the basic sine curve with amplitude 3 for one full period of \( 180^\circ \), shifted vertically up by 10 units and horizontally to the right by \( 30^\circ \). Each period of the sine wave will fit within \( 180^\circ \) on the x-axis (\( \theta \)), repeating after \( 180^\circ \). Apply the vertical shift and phase shift to see the final transformation.

Key Concepts

Vertical Shift

When discussing vertical shifts in trigonometric functions, imagine sliding the entire graph up or down the y-axis. In the given function, the vertical shift can be identified by looking at the constant term added outside of the trigonometric expression. Here, the term is "+10." This means that every point on the basic sine curve is moved 10 units upwards.

• This vertical shift does not affect the wave's shape; it purely slides it vertically.
• In practical terms, if the centerline of the original sine curve was at 0 on the y-axis, it is now at 10 due to this upward shift.

Remember, the constant directly outside of the sine function indicates how the graph moves vertically without changing the fundamental behavior of the sine wave.

Amplitude

Amplitude refers to how "tall" or "short" the waves of the graph are, determining their peaks and valleys' reach from the centerline. In our function, which is of the form \( y = a \sin(bx + c) + d \), the amplitude is given by the absolute value of "a." Here, "a" is 3, so the amplitude is 3.

• The sine waves will have a maximum height of 3 units above and below the centerline.
• Amplitude impacts the "intensity" of the wave—higher values mean taller waves.

Amplitude does not change with the vertical shift; it always measures from the centerline to the peak of the wave.

Period

In trigonometric functions, the period describes how long it takes for the wave to complete one full cycle. For sine functions of the form \( y = a \sin(bx + c) + d \), the period is calculated using \( \frac{360^\circ}{b} \). For our particular function, "b" is 2, making the period \( \frac{360^\circ}{2} = 180^\circ \).

• This means the graph repeats its full cycle every \( 180^\circ \).
• The period affects the horizontal stretching or compressing of the graph.

The period helps determine how "tight" or "stretched" the sine waves appear along the horizontal axis. Smaller period values indicate more waves appearing in the same horizontal space.

Phase Shift

Phase shift tells us how much the graph is moved horizontally along the x-axis. It's like shifting the starting point of the wave either to the left or to the right. The phase shift is determined by the expression inside the sine function. For a function \( y = a \sin(bx + c) + d \), the shift is calculated by \( \frac{-c}{b} \).

• In our equation, with \( c = -60^\circ \) (from \(-2 \times 30^\circ\)) and \( b = 2 \), the phase shift is \( \frac{-60^\circ}{2} = -30^\circ \).
• A negative phase shift here means the graph moves to the right.

A positive result means a left shift, whereas a negative result, like in our case, results in a right shift. With this function, the phase indicates a start \( 30^\circ \) further than usual, aligning the sine waves differently along the x-axis. This is crucial when overlaying or comparing multiple sine graphs.