Problem 82 - College Algebra with Corequisite Support

Question

For the following exercises, graph \(y=\sqrt{x}\) on the given viewing window. Determine the corresponding range for each viewing window. Show each graph. \([0,0.01]\)

Step-by-Step Solution

Answer

The range for the viewing window [0,0.01] is [0,0.1].

Step 1: Define the Function

The function we need to graph is given by \( y = \sqrt{x} \). This function represents the square root of \( x \).

Step 2: Identify the Domain

The domain of the function \( y = \sqrt{x} \) is all non-negative numbers because square roots are only defined for non-negative values. For this exercise, the domain is restricted to the interval \([0, 0.01]\).

Step 3: Calculate the Range

To find the range, we need to consider the outputs of \( y \) when \( x \) varies from 0 to 0.01. - At \( x=0 \), \( y = \sqrt{0} = 0 \).- At \( x=0.01 \), \( y = \sqrt{0.01} = 0.1 \).Therefore, the range is \([0, 0.1]\).

Step 4: Plot the Graph

Create a graph with \( x \)-axis ranging from 0 to 0.01 and \( y \)-axis ranging from 0 to 0.1. Plot the curve \( y = \sqrt{x} \). - At \( x=0 \), the point is \( (0,0) \). - At \( x=0.01 \), the point is \( (0.01, 0.1) \). The graph will be a curve starting at \( (0,0) \) and growing upward to \( (0.01, 0.1) \). This part of the curve indicates that as \( x \) increases, \( y \) also increases gently.

Key Concepts

Understanding the Domain and Range

The domain and range are vital concepts when working with functions, such as square root functions like \( y = \sqrt{x} \). Understanding the domain helps us know the set of inputs the function can accept.
In the case of our function \( y = \sqrt{x} \), the domain includes all non-negative numbers: \( [0, \infty) \). This is because the square root of a negative number isn't defined within the real numbers.
For the specific viewing window \([0, 0.01]\), our domain is limited to just this interval. This guides us on where to focus when graphing.
The range, on the other hand, tells us the set of possible outputs or \( y \)-values. Given our restricted domain of \([0, 0.01]\), the corresponding range of our function \( y = \sqrt{x} \) is \([0, 0.1]\).

• When \( x = 0 \), \( y = \sqrt{0} = 0 \).
• When \( x = 0.01 \), \( y = \sqrt{0.01} = 0.1 \).

This means as \( x \) changes between 0 and 0.01, the \( y \)-values move from 0 to 0.1. Remember, the range can be affected by how we choose to limit the domain.

Exploring Graphing Techniques

Graphing square root functions can seem tricky at first, but using systematic graphing techniques can make this easier.
For \( y = \sqrt{x} \), our primary goal is to visually represent how \( y \) changes as \( x \) increases. Start by setting up your axes:

• The \( x \)-axis will run from 0 to 0.01.
• The \( y \)-axis will cover the range from 0 to 0.1.

Plot key points:

• At \( x = 0 \), point \( (0, 0) \).
• At \( x = 0.01 \), point \( (0.01, 0.1) \).

Draw a smooth curve between these points. The plot starts at the origin and gently curves upward towards \( (0.01, 0.1) \). As you graph, you see \( y \) increase more slowly for small increases in \( x \). This gentle slope characterizes the square root function.
Taking note of the curve's shape helps in understanding the function's behavior.

Delving into Function Analysis

Function analysis goes beyond just plotting points; it involves understanding the behavior and properties of the function. For \( y = \sqrt{x} \), a few aspects are noteworthy.
First, consider the **increasing nature** of the function. As \( x \) increases from 0 to 0.01, \( y \) consistently goes up from 0 to 0.1. This is a typical feature of square root functions; they are always increasing for positive \( x \). Next, focus on identifying the **curve's gradient**. In our limited domain, the graph of \( y = \sqrt{x} \) is a small part of what would be a larger curve.
As \( x \) very slowly increases, \( y \) displays a slower rate of increase. This slower angle tells us how quickly \( y \) catches up compared to other parts of the function where \( x \) is larger. Additionally, this knowledge aids in predicting the function's real-world behavior and applications.
By fully understanding the nature of \( y = \sqrt{x} \), we grasp how it behaves in both mathematical and practical contexts. Through this function analysis, students can better relate these mathematical concepts to empirical tasks.