Question

Graph each function using transformations or the method of key points. Be sure to label key points and show at least two cycles. Use the graph to determine the domain and the range of each function. $$ y=5-3 \sin (2 x) $$

Short answer

The domain is \( (-\infty, \infty) \) and the range is \( [2, 8] \). The graph reflects a sine wave scaled, shifted, and reflected as described.

Step-by-step solution

- Identify the base function

The base function for this transformation is the sine function, \( y = \sin(x) \). We will be modifying this function with transformations.

- Determine the amplitude and period

The coefficient 3 in \( y = 5 - 3\sin(2x) \) affects the amplitude, and the coefficient 2 affects the period. Amplitude is \( |A| = 3 \). The period is calculated using the formula \( \frac{2\pi}{B} \), so \( B = 2 \) hence the period is \( \frac{2\pi}{2} = \pi \).

- Vertical shift

The vertical shift is given by the constant added or subtracted outside the trigonometric function. Here, the graph is shifted up by 5 units because of the +5.

- Reflection

The coefficient -3 indicates a reflection across the horizontal axis. This means that the sine curve will be flipped upside down.

- Identify key points

Starting from the sine function key points (0,1,0,-1,0), apply the transformations. Shift these points up by 5, and multiply the sine value by -3: \( (0,5),(\frac{\pi}{2},5+3),(\pi,5),(\frac{3\pi}{2},5-3),(2\pi,5) \). This should be calculated for at least two cycles.

- Sketch the graph

Plot the transformed points on the graph. To show two cycles, repeat the above steps for \( 2\pi \leq x \leq 4\pi \). Connect the points smoothly to reflect the sine wave.

- Determine the domain and range

The domain of a sine function and its transformations is all real numbers \( (-\infty,\infty) \). The range can be found from the minimum and maximum values after transformation: Minimum value = 2 (5 - Abs(-3)) and Maximum value = 8 (5 + Abs(-3)), thus the range is \( [2,8] \).

Key concepts

sine function transformations

Transformations make sine functions versatile because we can shape and move them easily. Here, the function is transformed from its base, which is the familiar sine function, \(y = \sin(x)\).
Transformations include changes like vertical shifts, reflections, adjusting amplitudes, and tweaking periods. It's important to identify the transformations to understand how the graph will look.

To tackle transformations, recognize the modifications in the function given, which are visible in the coefficients and constants around the sine function.
These coefficients will affect the graph's amplitude, period, vertical shift, and reflection, giving a unique look compared to the original sine curve.

amplitude and period

Amplitude and period are crucial to understanding the shape of the sine graph.
Amplitude refers to how high and low the graph goes, measured from the midline. In our function \(y = 5 - 3\sin(2x)\), the amplitude is calculated from the coefficient in front of the sine function.
Here, it is \(3\) (amplitude is always positive). This tells us the graph will stretch vertically 3 units from its midline.

The period of the function tells us how long it takes for the graph to complete one cycle. It's calculated using the formula \( \frac{2\pi}{B}\), where \(B\) is the coefficient of \(x\) inside the sine function. Here, \(B = 2\), so the period is \( \frac{2\pi}{2} = \pi\).
This means one complete wave (or cycle) of the sine function will occur in \( \pi\) units along the x-axis.

vertical shift

Vertical shifts move the sine graph up or down. This is controlled by the constant added or subtracted outside the sine function. In our case, \(y = 5 - 3\sin(2x)\), the sine curve shifts up by 5 units. This is because of the \(+5\) at the beginning of the equation.

The vertical shift repositions the midline of the wave from the x-axis up to \( y = 5\). All points on the sine function are moved up equally by this amount.
Hence, the new midline, or the center of the sine wave's oscillation, will be at \( y = 5\) instead of on the x-axis.

reflection across axis

Reflection across the axis occurs when the sine function is multiplied by a negative number. Here, the equation \(y = 5 - 3\sin(2x)\) has a \(-3\) coefficient. This negative sign causes the sine curve to reflect or flip upside down across the x-axis.

Normally, \( \sin(x)\) starts at 0 and goes up to 1 and down to -1. However, with the reflection, the curve goes directly down from 0 to its minimum value and then back up to its maximum value.
In our example, because of the reflection and amplitude, the height of the wave will change as well and have an amplitude of 3, flipped across the axis.

determining domain and range

The domain and range are essential to know where the graph lives on the coordinate plane. The domain of the sine function and its transformations always remains \(( -\infty , \infty)\) because the sine function is defined for all real numbers.

The range is more specific as it depends on the amplitude, vertical shift, and reflection.
In \(y = 5 - 3\sin(2x)\), look at the new minimum and maximum values of the sine function to determine the range:

• The minimum value is \(5 - 3 = 2\)
• The maximum value is \(5 + 3 = 8\)

Hence, the range of this transformed function is \([2, 8]\) since the sine function spans from its lowest to its highest value within these limits.