Problem 133 - Precalculus

Question

Use a graphing calculator to graph \(Y_{1}=\sin x\) and \(Y_{2}=\sin x+c,\) where a. \(c=1,\) and explain the relationship between \(Y_{2}\) and \(Y_{1}\) b. \(c=-1,\) and explain the relationship between \(Y_{2}\) and \(Y_{1}\)

Step-by-Step Solution

Answer

For \(c=1\), \(Y_2\) is \(Y_1\) shifted up 1 unit; for \(c=-1\), \(Y_2\) is \(Y_1\) shifted down 1 unit.

Step 1: Graph the basic function

Enter the function \(Y_1 = \sin x\) into the graphing calculator. This is the standard sine wave, which oscillates between -1 and 1 with a period of \(2\pi\). Record its appearance for reference.

Step 2: Graph with a positive vertical shift

Input the function \(Y_2 = \sin x + 1\) into the graphing calculator. Compare this graph with that of \(Y_1 = \sin x\). Notice that \(Y_2\) shifts the entire graph of \(Y_1\) upward by 1 unit. This means that all y-values of \(Y_1\) are increased by 1 in the \(Y_2\) graph, so it still oscillates but between 0 and 2.

Step 3: Analyze the positive shift relationship

The function \(Y_2 = \sin x + 1\) is a vertical shift of \(Y_1 = \sin x\) by 1 unit upwards. This does not affect the period or frequency of the oscillation, only the vertical position.

Step 4: Graph with a negative vertical shift

Clear the previous function and input \(Y_2 = \sin x - 1\). Compare this with the original \(Y_1 = \sin x\). You should see that \(Y_2\) shifts the graph downward by 1 unit. This results in the wave oscillating between -2 and 0.

Step 5: Analyze the negative shift relationship

For \(Y_2 = \sin x - 1\), the graph is a vertical shift of \(Y_1 = \sin x\) by 1 unit downwards. Like the positive shift, this affects the vertical position but not the wave's period or frequency.

Key Concepts

Vertical Shift

A vertical shift occurs when a constant is added to or subtracted from a function, affecting its vertical position on the graph without altering the shape of the graph. When we talk about vertical shifts in the context of sine functions, this usually involves a change in the baseline of the wave.
For example, in the function \(Y_2 = \sin x + c\):

• If \(c > 0\), the entire sine wave shifts upwards by c units. This means its peaks and troughs shift higher on the y-axis.
• If \(c < 0\), the sine wave shifts downwards. So, each point on the wave is moved c units lower.

The key aspect of vertical shifts is that while they change the wave's starting and ending vertical positions, the period and frequency remain unchanged. This makes the concept of vertical shifts quite useful in transforming sine waves without affecting their timing.

Sine Wave

A sine wave is a smooth and periodic oscillation that is fundamental in mathematics and many applications, such as physics and engineering. It's described mathematically by the function \(Y = \sin x\), which repeats its patterns every \(2\pi\) units along the horizontal axis.
Sine waves have several essential characteristics:

• Amplitude: This is the "height" of the wave, from the centerline to the peak. For the basic sine wave, this is 1.
• Period: The distance over which the wave's shape repeats. For sine, the period is \(2\pi\).
• Frequency: How often the wave repeats over a set interval. Inverse of the period.

A sine wave is perfectly symmetrical, meaning its pattern remains consistent, providing predictable results. By understanding the sine wave, you can predict oscillating behaviors in natural systems and engineered designs.

Graphing Calculator

A graphing calculator is a powerful tool for visualizing mathematical functions, such as sine waves, and their transformations, like vertical shifts. These devices allow students to input equations and see their graphical representations quickly and with precision.
Here's how to effectively use a graphing calculator for graphing sine functions:

• Enter the base function \(Y_1 = \sin x\) to see the standard sine wave on your screen, providing a reference for comparison.
• Use the calculator to input transformed functions, such as \(Y_2 = \sin x + 1\) or \(Y_2 = \sin x - 1\), and observe the vertical shifts instantly.
• Be sure to adjust the viewing window to accommodate changes in the wave's range due to vertical shifts, ensuring you see the full extent of these transformations.

With a graphing calculator, you gain immediate feedback and deeper understanding of mathematical concepts, helping you grasp transformations like vertical shifts more intuitively and effectively.