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Chapter 5: Problem 10

Graph the function. $$f(x)=-1+\cos x$$

Short Answer

Expert verified
The graph of \( f(x) = -1 + \cos x \) is a vertically shifted cosine wave oscillating between -2 and 0.

Step by step solution

01

Identify the Function Components

The function given is \( f(x) = -1 + \cos x \). This is a transformation of the basic cosine function \( \cos x \). The transformation involves shifting the graph vertically by -1 unit.
02

Determine Key Points of Cosine Function

The cosine function \( \cos x \) has a period of \( 2\pi \) and typically starts at \( (0, 1) \). Other key points include \( (\frac{\pi}{2}, 0) \), \( (\pi, -1) \), \( (\frac{3\pi}{2}, 0) \), and \( (2\pi, 1) \).
03

Apply Vertical Shift to Key Points

Adjust each of the key points of the \( \cos x \) function by shifting them 1 unit downward. This gives the transformed points: \( (0, 0) \), \( (\frac{\pi}{2}, -1) \), \( (\pi, -2) \), \( (\frac{3\pi}{2}, -1) \), and \( (2\pi, 0) \).
04

Sketch the Transformed Graph

Plot the points identified in the previous step on a coordinate plane. Draw a smooth curve passing through these points to complete one cycle of the transformed cosine function. The graph will oscillate between -2 and 0, peaking at 0 and reaching a trough at -2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Function
The cosine function, denoted as \( \cos x \), is one of the fundamental trigonometric functions. It describes the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Mathematically, it is periodic and oscillates between the values of 1 and -1 over its domain of all real numbers.

Here are some essential characteristics of the basic cosine function:
  • Amplitude: The maximum height (from the midline) of the wave is 1.
  • Periodicity: The function repeats its pattern every \(2\pi\) units; hence, the period is \(2\pi\).
  • Key Points: The typical points in one period are \((0, 1)\), \((\frac{\pi}{2}, 0)\), \((\pi, -1)\), \((\frac{3\pi}{2}, 0)\), and \((2\pi, 1)\).
The smooth wave of \(\cos x\) is symmetric around the y-axis, making it an even function.
Vertical Shift
A vertical shift in trigonometric functions involves moving the entire graph up or down on the coordinate plane. This is done by adding or subtracting a constant from the function.

Let's see how this applies to our function \( f(x) = -1 + \cos x \):
  • Effect of Vertical Shift: The graph of \( \cos x \) is moved downward by 1 unit.
  • Impact on Key Points: Each point on the original cosine graph is lowered by 1 unit.
    - The point \((0, 1)\) becomes \((0, 0)\).
  • New Range: The range of the transformed function is now from -2 to 0.
The overall shape and pattern of the cosine wave remain unchanged; only its position along the y-axis is adjusted.
Periodicity of Trigonometric Functions
Trigonometric functions like cosine exhibit periodicity, which means they repeat their values in a predictable pattern.

For the cosine function:
  • Period: The time it takes for the cosine function to complete one full cycle is \(2\pi\). This is the length along the x-axis after which the function starts to repeat.
  • Understanding Periodicity: Knowing the period of a function helps in plotting the graph efficiently, as you can expect the function to reproduce its shape over each interval of length \(2\pi\).
  • Application to Transformed Functions: Even when transformations like vertical shifts are applied, the period of the function remains unaffected.
Recognizing the periodic nature of the cosine function is crucial for accurately graphing it over any range and understanding how transformations affect the graph's appearance without changing its periodicity.

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Most popular questions from this chapter

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