Question

A family of sinusoidal graphs with equations of the form \(y=a \sin b(x-c)+d\) is created by changing only the vertical displacement of the function. If the range of the original function is \(\\{y |-3 \leq y \leq 3, y \in \mathrm{R}\\}\) determine the range of the function with each given value of \(d .\) a) \(d=2\) b) \(d=-3\) c) \(d=-10\) d) \(d=8\)

Short answer

For \(d=2\), range is \{-1 \leq y \leq 5\}; for \(d=-3\), range is \{-6 \leq y \leq 0\}; for \(d=-10\), range is \{-13 \leq y \leq -7\}; for \(d=8\), range is \{5 \leq y \leq 11\}.

Step-by-step solution

- Understand the effect of vertical displacement

The given equation is of the form \(y=a \, \sin \, b(x-c)+d\). The vertical displacement \(d\) shifts the entire graph up or down by \(d\) units. This means the range of the function will be shifted vertically by \(d\).

- Identify the range of the original function

The range of the original function \(y=a \, \sin \, b(x-c)\) without vertical displacement is \(\{-3 \, \leq \, y \, \leq \, 3, \, y \, \in \, \mathbb{R}\}\).

- Apply the vertical displacement for each given \(d\) value

Now, we'll apply the vertical displacement \(d\) to the range of the original function:

Step 4a - Calculate new range when \(d=2\)

Add \(2\) to both the lower and upper bounds of the original range: \[-3 + 2 \leq y \leq 3 + 2\]Thus, the new range is: \{-1 \leq y \leq 5\}.

Step 4b - Calculate new range when \(d=-3\)

Subtract \(3\) from both the lower and upper bounds of the original range: \[-3 - 3 \leq y \leq 3 - 3\]Thus, the new range is: \{-6 \leq y \leq 0\}.

Step 4c - Calculate new range when \(d=-10\)

Subtract \(10\) from both the lower and upper bounds of the original range: \[-3 - 10 \leq y \leq 3 - 10\]Thus, the new range is: \{-13 \leq y \leq -7\}.

Step 4d - Calculate new range when \(d=8\)

Add \(8\) to both the lower and upper bounds of the original range: \[-3 + 8 \leq y \leq 3 + 8\]Thus, the new range is: \{5 \leq y \leq 11\}.

Key concepts

vertical displacement

In trigonometric functions, vertical displacement refers to the movement of the entire graph up or down the y-axis. For sinusoidal functions such as the sine function, the vertical displacement is expressed with the parameter \(d\) in the equation \(y = a \, \sin \, b(x-c) + d\). This can be thought of as adding (or subtracting) a constant value to every point on the graph. Imagine the sine wave as a wave oscillating around the horizontal axis (y = 0). When you introduce a vertical displacement, the whole wave moves up or down. For example:

• If \(d\) is positive, the wave shifts upward.
• If \(d\) is negative, the wave shifts downward.

This affects the range of the function, which we'll discuss next.

range of a function

The range of a function is the set of all possible output values (y-values) that the function can produce. For the sine function, \(y = a \, \sin \, b(x-c)\), without any vertical displacement (\(d = 0\)), the range depends on the amplitude and is bound between the negative and positive values of the amplitude. For example:

• If the amplitude \(a\) is 3, then the range is \{-3 \, \leq \, y \, \leq \, 3, \, y \, \in \, \mathbb{R}\}\>.

When we introduce a vertical displacement \(d\), the entire range is shifted up or down by \(d\) units. For example:

• If \(d = 2\), the range will shift up by 2: \{-3 + 2 \leq y \leq 3 + 2\} or \{-1 \, \leq \, y \, \leq \, 5\}\.
• If \(d = -3\), the range will shift down by 3: \{-3 - 3 \leq y \leq 3 - 3\} or \{-6 \, \leq \, y \, \leq 0\}\.
• If \(d = -10\), the range will shift down by 10: \{-3 - 10 \leq y \leq 3 - 10\} or \{-13 \, \leq \, y \, \leq -7\}\.
• If \(d = 8\), the range will shift up by 8: \{-3 + 8 \leq y \leq 3 + 8\} or \{5 \, \leq \, y \, \leq 11\}\.

Therefore, understanding vertical displacement is crucial for determining the new range of a function.

sine function

The sine function is one of the fundamental trigonometric functions and is commonly written as \(y = \, \sin(x)\). It features a periodic wave that oscillates between its maximum and minimum values repetitively.The general form of the sine function is \(y = a \, \sin \, b(x-c) + d\), which includes several parameters that transform the graph:

• \(a\): Amplitude. It affects the height of the peaks and depth of the troughs.
• \(b\): Frequency. It determines how many cycles occur within a given interval.
• \(c\): Horizontal shift. It moves the graph left or right.
• \(d\): Vertical displacement. It shifts the entire graph up or down.

In this exercise, we focused on vertical displacement, \(d\). When \(d\) is applied to the sine function, it results in a vertical shift of the entire graph, impacting the function's range but not its shape or periodicity. Understanding the sine function's properties and how each parameter transforms its graph is fundamental to working with trigonometric equations.