Problem 47 - Precalculus with Calculus Previews

Question

Sketch the graph of the given function. $$ y=\arccos (x-1) $$

Step-by-Step Solution

Answer

The graph is a decreasing curve from \((0, \pi)\) to \((2, 0)\) in the domain \([0, 2]\).

Step 1: Understand the Domain of the Function

The function \( y = \arccos(x-1) \) is defined for those values of \( x \) where the expression inside the arc cosine function, \( x-1 \), is within the range \([-1, 1]\). So, solve the inequality \(-1 \leq x-1 \leq 1\). This leads to \(0 \leq x \leq 2\). Thus, the domain of the function is \([0, 2]\).

Step 2: Determine the Range of the Function

The range of the arccosine function, \( \arccos u \), is \([0, \pi]\) for \(-1 \leq u \leq 1\). Therefore, since \( x-1 \) ranges from \(-1\) to \(1\) in the given domain of \([0, 2]\), the range of \( y = \arccos(x-1) \) is \([0, \pi]\).

Step 3: Identify Key Points

It is useful to find some key points that lie on the graph. For \( x=0\), \( y = \arccos(0-1) = \arccos(-1) = \pi \). For \( x=1\), \( y = \arccos(1-1) = \arccos(0) = \frac{\pi}{2} \). For \( x=2\), \( y = \arccos(2-1) = \arccos(1) = 0 \). These points are \((0, \pi)\), \((1, \frac{\pi}{2})\), and \((2, 0)\).

Step 4: Sketch the Graph

Plot the points found in Step 3 on a coordinate plane. The graph will start at \((0, \pi)\), pass through \((1, \frac{\pi}{2})\), and end at \((2, 0)\). The graph is a decreasing function, adhering to the shape typical for an arccosine graph, forming a smooth curve downward from left to right within the indicated points and ranges.

Key Concepts

Arccosine Function

The arccosine function, often represented as \(\arccos(x)\), is the inverse of the cosine function on a restricted domain. Unlike the regular cosine function, which takes an angle (in radians) as input and outputs a value between

• -1 and 1,

the arccosine function takes a value within this range and outputs an angle between

• 0 and \(\pi\).

The arccosine results in the principal value of the angle whose cosine is the provided number.
This means if you know the cosine of an angle is a certain value, the arccosine helps you find that angle.
Understanding the behavior of the arccosine function is crucial when trying to solve problems involving trigonometric equations and is essential when graphing functions like \(y = \arccos(x-1)\), where transformations are involved.

Domain and Range

The domain and range are core concepts not only in trigonometry but throughout mathematics. For a function \(y = \arccos(x-1)\), the domain is determined by ensuring the expression \(x - 1\) falls within the interval

• \([-1, 1]\).

By solving the inequality \(-1 \leq x-1 \leq 1\), we find that the domain of \(x\) is

• \([0, 2]\).

This means \(x\) can take any value from 0 to 2, inclusive.

On the other hand, the range of the function \(y = \arccos(x-1)\) reflects the outputs or possible \(y\)-values of the function.
Given that "arccos" refers to angles between

• \(0\) and \(\pi\),

the range for this function turns out to be

• \([0, \pi]\).

In practical terms, this tells us the possible vertical span of the graphed line based on the selected domain.

Sketching Graphs

Sketching graphs involves plotting key points from a function to visualize its behavior across its domain. Let's look at the function \(y = \arccos(x-1)\).

First, select important values of \(x\) within the domain

• \([0, 2]\).

These points include:

• \((0, \pi)\),\(\arccos(-1)\)
• \((1, \frac{\pi}{2})\),\(\arccos(0)\)
• \((2, 0)\),\(\arccos(1)\)

Plot these points on a co-ordinate plane to start shaping the graph.

Next, connect these points smoothly. Since the arccosine function is

• naturally decreasing,

make sure to draw a smooth, downward-bending curve.

The graph should reflect that as \(x\) increases from 0 to 2, the corresponding \(y\)-value decreases from \(\pi\) down to 0—illustrating this function’s behavior accurately.