Problem 25 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Graph the function. \(g(x)=\sin x+2\)

Step-by-Step Solution

Answer

Start by drawing a basic graph of the sine function, then shift the graph upwards by 2 units on the y-axis. The points on this shifted graph will be \((0,2)\), \((\frac{\pi}{2},3)\), \((\pi,2)\), \((\frac{3\pi}{2},1)\) and \((2\pi,2)\).

Step 1 - Recognize the function

The function \(g(x) = \sin x + 2\) is a sine function \(\sin x\) shifted upwards by 2 units on the y-axis.

Step 2 - Create basic sine graph points

To graph any sine function, we usually start by marking five points: \(x = 0\), \(x = \frac{\pi}{2}\), \(x = \pi\), \(x = \frac{3\pi}{2}\) and \(x = 2\pi\). For the function \(\sin x\), \(\sin 0 = 0\), \(\sin \frac{\pi}{2} = 1\), \(\sin \pi = 0\), \(\sin \frac{3\pi}{2} = -1\), and \(\sin 2\pi = 0\).

Step 3 - Apply vertical shift to sine graph points

Now we apply the +2 vertical shift to each of the y-values we calculated. This means adding 2 to each y-value: \(sin0 + 2 = 2\), \(sin\frac{\pi}{2} + 2 = 3\), \(sin\pi + 2 = 2\), \(sin\frac{3\pi}{2} + 2 = 1\), and \(sin2\pi + 2 = 2\).

Step 4 - Draw the function

Now we plot these points on the graph and draw the curve that goes through them.

Key Concepts

Sine Function

The sine function, represented as \( \sin x \), is a periodic function that describes a smooth wave-like pattern. This type of function is fundamental in trigonometry and models various natural phenomena such as sound waves and tides. Its graph is known as a sine wave.

The basic properties of the sine function include:

• Periodicity: The sine function has a period of \( 2\pi \). This means that it repeats its pattern every \( 2\pi \) units along the x-axis.
• Range: The range of \( \sin x \) is between -1 and 1.
• Symmetry: The sine function is an odd function, which means it’s symmetric with respect to the origin. This symmetry makes part of its graph mirrored along the x-axis.
• Key Points: At specific points like \( x = 0, \pi, 2\pi, \), the sine function value is 0. At \( x = \frac{\pi}{2} \), the sine value reaches its maximum, and at \( x = \frac{3\pi}{2} \), it reaches its minimum.

Understanding these properties helps simplify the graphing of the function by providing predictable points and shapes to work from.

Vertical Shift

The vertical shift is a transformation that moves the graph of a function up or down along the y-axis. This shift occurs by adding or subtracting a constant from the function's equation. In the function \( g(x) = \sin x + 2 \), the "+2" indicates a vertical shift.

Some important aspects of vertical shifts are:

• Positive Shift: Adding a positive number like in \( g(x) = \sin x + 2 \) moves the sine wave upward. Each point on the sine wave is elevated by 2 units from its original position.
• Negative Shift: Conversely, subtracting a number would move the graph downward.
• Effect on the Range: While the shape of the sine wave remains unchanged, the range changes. In this case, the range shifts from [-1, 1] to [1, 3].

This vertical shift doesn't affect the period or frequency of the function, only its vertical positioning. By understanding how the graph changes, you can accurately draw and interpret variations of the sine function.

Graph Transformations

Graph transformations allow us to alter the position and shape of a graph without changing its fundamental nature. These include translations, reflections, stretches, and compressions. For the sine function \( g(x) = \sin x + 2 \), the transformation is a vertical translation.

When graphing transformations, consider:

• Vertical Transformations: These include vertical shifts like our function \( g(x) = \sin x + 2 \), as well as vertical stretches and compressions which affect how 'tall' or 'flat' the graph appears.
• Horizontal Transformations: Affecting the period and phase of the wave, though not present in our current example.

Graph transformations provide flexibility in modeling real-world scenarios by changing a graph's size or position to fit data accurately. With the right transformations, the simple sine function can represent a wide range of oscillating phenomena.