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Magazine Chapter 5: Problem 8
Graph the function. $$f(x)=2-\cos x$$
Short Answer
Expert verified
The function \( f(x) = 2 - \cos x \) is a vertically shifted and reflected cosine wave oscillating between y=1 and y=3.
Step by step solution
01
Identify the Parent Function
The given function is \( f(x) = 2 - \cos x \). The parent function here is \( \cos x \). We will analyze how the graph of \( \cos x \) changes due to transformations.
02
Determine Transformation - Vertical Shift
The function \( f(x) = 2 - \cos x \) shows a vertical transformation: it shifts the graph of \( \cos x \) up by 2 units. This is because of the +2 added to the function after the \( \cos x \) part.
03
Determine Transformation - Vertical Reflection
The function \( f(x) = 2 - \cos x \) also involves a reflection. \( \cos x \) is reflected vertically over the x-axis because of the negative sign before the \( \cos x \). So \( \cos x \) becomes \( -\cos x \).
04
Sketch the Parent Function
Start by sketching the graph of the parent function \( \cos x \). It has a period of \( 2\pi \), maximum at 1, and minimum at -1, oscillating between these values symmetrically about the x-axis.
05
Reflect the Graph
Reflect the graph of \( \cos x \) vertically, switching the maximum and minimum points. Thus, the peaks are at -1 and troughs at 1, for the graph \( -\cos x \).
06
Apply Vertical Shift to the Transformed Graph
Take the reflected graph from the previous step and shift it up by 2 units. This results in the maximum point being at 1 (i.e., \(-1+2\)) and the minimum point being at 3 (i.e., \(1+2\)). The function completes a cycle every \( 2\pi \) as usual.
07
Finalize the Graph
Plot the final graph, correctly marking the new amplitude, midline, and period. The midline is y=2, and the function oscillates between y=1 and y=3, completing one cycle every \( 2\pi \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Trigonometric Functions
When graphing trigonometric functions like \( f(x) = 2 - \cos x \), understanding the parent function can help us. The parent function in this case is \( \cos x \). The graph of \( \cos x \) is a wave that repeats every \( 2\pi \) units along the x-axis. As it moves, it touches a maximum of 1 and a minimum of -1.To graph a specific trigonometric function, start by noting these traits:
- The period of the cosine function, which is \( 2\pi \).
- The maximum and minimum values of 1 and -1.
- How it oscillates symmetrically about the x-axis.
Vertical Shift
A vertical shift in a function moves it up or down on the graph. In the function \( f(x) = 2 - \cos x \), the "2" indicates a vertical shift. Specifically, the entire graph of \( \cos x \) is shifted up 2 units.Here鈥檚 how to identify and implement a vertical shift:
- Look for a constant added or subtracted outside of the trigonometric function, such as "+2".
- This constant shifts all points on the graph vertically.
Vertical Reflection
A vertical reflection flips parts of a graph over the x-axis, creating a mirror effect. In \( f(x) = 2 - \cos x \), the factor "-" before \( \cos x \) causes this reflection. The graph of \( \cos x \) flips, making what used to be peaks into troughs, and vice versa.To see the vertical reflection:
- Identify negative signs outside the trigonometric function, just like the "-" before \( \cos x \).
- This flips all points over the x-axis.
