Question

Determine the period, amplitude, and shifts (both horizontal and vertical) and draw a graph over the interval \(-5 \leq x \leq 5\) for the functions listed. $$ y=21+7 \sin (2 x+3) $$

Short answer

Amplitude: 7, Period: \( \pi \), Horizontal Shift: \( -\frac{3}{2} \), Vertical Shift: 21.

Step-by-step solution

Identify the Amplitude

In the function \( y = 21 + 7 \sin(2x + 3) \), the amplitude is the coefficient of the sine function. Here, the amplitude is \( 7 \).

Calculate the Period

The period of a sine function \( y = a \sin(bx + c) \) is given by \( \frac{2\pi}{b} \). In this function, \( b = 2 \), so the period is \( \frac{2\pi}{2} = \pi \).

Determine the Horizontal Shift

The horizontal shift, also called phase shift, is calculated by solving \( bx + c = 0 \). Here, \( 2x + 3 = 0 \) gives the horizontal shift \( x = -\frac{3}{2} \). The shift is to the left by \( \frac{3}{2} \).

Identify the Vertical Shift

The vertical shift is the constant added to the sine function. In this equation, \( 21 \) is the vertical shift, moving the graph upwards by \( 21 \) units.

Graph the Function

Draw the sine function starting from the horizontal shift moved left by \( \frac{3}{2} \). Consider the vertical shift up by 21 units to adjust the midline of the graph. Plot key points for one period, noting the amplitude of 7 from this new midline. Repeat the pattern until covering \( -5 \leq x \leq 5 \).

Key concepts

Amplitude

In trigonometric functions, the amplitude measures how far the wave peaks and troughs vary from its midline. It's essentially the height of the wave crest above or below this central line.
For a function of the form \( y = a \sin(bx + c) + d \), the amplitude is given by the absolute value of \(a\). For the function in our example, \( y = 21 + 7 \sin(2x + 3) \), the amplitude is \( |7| = 7 \).
The amplitude does not affect the position of the wave along the x-axis but instead affects how tall or short the graph is relative to the midline. Think of it as a scaling factor that stretches or compresses the graph vertically. This means with an amplitude of 7, the sine wave will reach 7 units above and 7 units below its central position.

Period

The period of a trigonometric function indicates how long it takes for the function to complete one full cycle. In simpler terms, it tells you the length needed on the x-axis before the wave pattern repeats itself.
For sine functions, such as \( y = a \sin(bx + c) + d \), the period can be calculated using the formula \( \frac{2\pi}{b} \). Here, \( b = 2 \), so the period is \( \frac{2\pi}{2} = \pi \).
This means every \( \pi \) units along the x-axis, the sine wave begins a new cycle. Understanding the period helps in plotting the function, as you can expect it to repeat itself at regular intervals. It's akin to knowing the rhythm of a beat that repeats over specified counts.

Phase Shift

Phase shift, also known as horizontal shift, describes how the graph of a function moves left or right from its usual start point. This shift occurs due to the \( c \) term in the sine function \( y = a \sin(bx + c) + d \).
To calculate the phase shift, solve the equation \( bx + c = 0 \). In our example, solving \( 2x + 3 = 0 \) results in \( x = -\frac{3}{2} \).
This means the entire graph is shifted \( \frac{3}{2} \) units to the left. It’s important to adjust for this when sketching the graph, as the standard cycle will not start at the origin. It’s as if you've moved where the wave begins its rise and fall on the x-axis.

Vertical Shift

The vertical shift occurs when a constant is added or subtracted from a function, raising or lowering the entire graph. In the equation \( y = a \sin(bx + c) + d \), the \( d \) is responsible for this shift.
In our given function, \( y = 21 + 7 \sin(2x + 3) \), the vertical shift is \( 21 \). This means the sine wave is moved upward 21 units from the standard central axis (usually the x-axis).
Visualize this as lifting the base level of the wave, which affects how it looks but not its shape. So the middle of the wave, rather than being around 0, is at 21. This shift allows for the wave to fluctuate around a new central line at \( y = 21 \), making all the peaks and troughs adjust accordingly.