Problem 64 - Precalculus: A Concise Course

Question

\(g\) is related to a parent function \(f(x)=\sin (x)\) or \(f(x)=\cos (x)\) (a) Describe the sequence of transformations from \(f\) to \(g\). (b) Sketch the graph of \(g\). (c) Use function notation to write \(g\) in terms of \(f\). $$ g(x)=1+\cos (x+\pi) $$

Step-by-Step Solution

Answer

The parent function for \(g(x)\) is \(f(x) = \cos (x)\). The function \(g(x) = 1+ \cos (x+ \pi)\) can be obtained by shifting \(f(x)\) up by one unit and to the left by \(\pi\) units. The function \(g(x)\) in terms of \(f(x)\) is given by \(g(x) = 1 + f(x + \pi)\). The graph of \(g(x)\) is a cosine wave shifted as described.

Step 1: Identify the Parent Function

The parent function in this case is \(f(x) = \cos (x)\) since cosine function is mentioned in the exercise.

Step 2: Describe the Transformations

Here two transformations can be observed: (a) Vertical shift: the '+' sign in '1+ \cos' translates the function \(f (x)\) upward by 1 unit. (b) Horizontal shift: \(x + \pi\) inside the cosine function shifts the graph of the function \(f (x)\) to the left by \(\pi\) units.

Step 3: Sketch the Graph of \(g(x)\)

Begin with the original graph of \(f(x)\) which is a cosine wave, starting at the maximum value of 1 at x = 0, falling to a minimum of -1 at \(x = \pi\), and returning to the maximum of 1 at \(x = 2\pi\). For the graph of \(g(x)\), vertically shift this original graph upward by one unit and then shift it left by \(\pi\) units.

Step 4: Write \(g\) in Terms of \(f\)

In terms of the parent function, the function \(g\) can be written as \(g(x) = 1 + f(x + \pi)\)

Key Concepts

Parent Trigonometric Functions

Understanding the basic parent trigonometric functions is essential for mastering transformations. Trigonometric functions such as sine, cosine, and tangent exhibit specific shapes and properties that form the foundation of their respective family of functions.
For example, the parent cosine function, denoted as \( f(x) = \text{cos}(x) \), is a wave that starts at a maximum value, decreases to a minimum value, and then returns to the maximum, completing a full cycle over an interval of \( 2\pi \) radians. These functions are periodic, meaning they repeat their values at regular intervals, and are used to model a variety of real-world phenomena.

Cosine Function

The cosine function is one of the primary trigonometric functions and is related to the sine function through the phase shift of \( \pi/2 \) radians. The standard cosine function, \( f(x) = \cos(x) \), has a peak at 1 when \( x = 0 \), crosses the origin at \( \pi/2 \), reaches a minimum of -1 at \( x = \pi \), and completes the cycle at \( x = 2\pi \).
Its graph is a smooth, continuous wave, and it's even, meaning it's symmetrical about the Y-axis. This symmetry is an important concept, particularly when the function undergoes transformations.

Graphing Cosine Functions

Graphing cosine functions involves plotting the wave-like pattern of cosines across a coordinate plane. The key properties to consider are the amplitude, which is the height of the wave, the period, which is the distance over which the wave repeats, the phase shift, which is the horizontal displacement, and the vertical shift.

Example of Graphing \( \cos(x) \):

• Identify the maximum and minimum values (1 and -1 for the parent function).
• Determine the period, which for the parent cosine function is \( 2\pi \).
• Mark critical points that represent the beginning, peak, trough, and end of a cycle.
• Plot the points and draw the smooth, periodic wave.

When graphing transformations, you apply shifts and stretches to these key points.

Horizontal Shift

Horizontal shifts occur when the trigonometric function's input is adjusted by adding or subtracting a constant value. This mainly affects the phase of the function.
For instance, the transformation \( \cos(x + \pi) \) shifts the graph of the parent cosine function to the left by \( \pi \) units. This kind of shift is also known as a phase shift. In practical scenarios, horizontal shifts do not change the shape of the graph: they merely translate it left or right on the X-axis. Recognizing this shift is essential, as it can fundamentally change the starting point and repetition interval of the wave.

Vertical Shift

A vertical shift involves adding or subtracting a constant to the entire function, effectively moving the graph up or down on the coordinate plane.
In the function \( 1 + \cos(x) \), for example, a vertical shift of 1 unit up is applied. This raises the maximum and minimum values of the cosine wave by 1, transforming the standard range of \( [ -1, 1 ] \) to \( [ 0, 2 ] \). Unlike horizontal shifts, vertical shifts change the vertical position of the entire graph without affecting the period or horizontal placement. It's critical to account for vertical shifts when analyzing or graphing a transformed trigonometric function.