91Ó°ÊÓ

Understanding the domain and range of inverse trigonometric functions is crucial for sketching their graphs. For the function \(y = 2\arccos x\), identifying these elements is your first step. Since you're dealing with an inverse cosine function, the domain is restricted between \(-1\) and \(1\). This is because the cosine function outputs values in that interval.

The range of \(\arccos x\) is between \(0\) and \(\pi\). However, in this case, the function is multiplied by \(2\), doubling the range. Thus, the range of \(y = 2\arccos x\) extends from \(0\) to \(2\pi\).

• Domain: \(-1 \leq x \leq 1\)
• Range: \(0 \leq y \leq 2\pi\)

Understanding these boundaries helps you know where the function is defined and where it reaches its minimum and maximum values.

Sketching the graph of \(y = 2\arccos x\) involves plotting the behavior of the function over its defined domain. The graph of the \(\arccos x\) function starts at \((1, 0)\) and decreases to \((-1, \pi)\). When multiplied by \(2\), the behavior stretches vertically without altering the domain.

To begin sketching:

• Start at the point \((1, 0)\), where the function value begins.
• Draw the graph gradually rising as you move leftwards from \(1\) to \(-1\).
• Ensure the graph smoothly reaches \((-1, 2\pi)\).

The key is to show a decreasing curve moving downward from left to right, reflecting the properties of the inverse cosine function while accounting for the multiplication factor.

A vertical stretch impacts how the graph of a function appears without changing its domain. For \(y = 2\arccos x\), the presence of the coefficient \(2\) affects the graph by stretching it vertically.

To understand this, consider the original function, \(y = \arccos x\), which outputs values from \(0\) to \(\pi\). The multiplication by \(2\) transforms each point on the graph to double its distance from the x-axis.

• If \((x, y)\) is a point on \(y = \arccos x\), then the corresponding point on \(y = 2\arccos x\) is \((x, 2y)\).
• This makes the curve appear taller and more stretched out vertically.

This vertical stretch alters the range of the function, doubling it from \(0\) to \(2\pi\), allowing the graph to fully reflect its new amplitude.