Problem 83 - 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. \([0,100]\)

Step-by-Step Solution

Answer

Range is [0, 10]; graph starts at (0, 0) and ends at (100, 10).

Step 1: Identify the function and the viewing window

We are given the function \(y = \sqrt{x}\) and the viewing window \([0, 100]\). This means we need to graph the function for \(x\) values ranging from 0 to 100.

Step 2: Understand the behavior of the square root function

The square root function, \(y = \sqrt{x}\), is defined for \(x \geq 0\) and always produces non-negative outputs. As \(x\) increases, \(y\) increases but at a decreasing rate.

Step 3: Calculate the range of the function over the given window

Since \(x\) ranges from 0 to 100, calculate \(y_{\text{min}} = \sqrt{0} = 0\) and \(y_{\text{max}} = \sqrt{100} = 10\). Hence, the range of \(y\) over this window is \([0, 10]\).

Step 4: Sketch the graph of y = sqrt(x)

Plot the function \(y = \sqrt{x}\) on a graph with \(x\) on the horizontal axis ranging from 0 to 100. The corresponding \(y\) values should cover the range from 0 to 10. The graph will start at the origin (0, 0) and curve upwards to the right, flattening out as it approaches (100, 10).

Key Concepts

Graphing Functions

Graphing functions is a fundamental concept in mathematics. It involves visually representing a mathematical equation on a coordinate system. In our case, we need to graph the function \(y = \sqrt{x}\) within a specific range. The horizontal axis (x-axis) will display values from 0 to 100, while the vertical axis (y-axis) will represent the results of the square root function for these x-values.

• Start by creating an axis system with appropriately labeled scales.
• Identify key points on the graph: for instance, \(x = 0\) gives \(y = 0\), and \(x = 100\) results in \(y = 10\).
• Plot these points and draw a smooth curve to connect them, reflecting the function's gradual increase.
• The graph of \(y = \sqrt{x}\) steadily rises but becomes less steep as x increases, creating a characteristic curve.

This process provides insights into how the function behaves, allowing for a better understanding of its properties.

Range of a Function

The range of a function represents all possible output values (y-values) that the function can produce. For the function \(y = \sqrt{x}\), we determine the range by examining the minimum and maximum outputs within the provided x-interval, which in this case is from 0 to 100.

• Calculate the square root of the smallest x-value (\(0\)), which is \(0\) for y.
• Find the square root of the largest x-value (\(100\)), resulting in \(10\) for y.
• Thus, the range of the function in the given window is \([0, 10]\).

This means the function produces y-values from 0 to 10 for x-values between 0 and 100. Understanding the range is crucial for setting up appropriate graphing windows and analyzing function behavior.

Square Root

The square root function, symbolized as \(\sqrt{x}\), is a special type of mathematical operation. It answers the question: "What number squared (multiplied by itself) results in x?" Here are some important points about this function:

• Defined only for non-negative x-values (\(x \geq 0\)) since square roots of negative numbers are not real.
• The output, y, is also non-negative because of the nature of the function.
• Produces a curve that begins at the origin (0, 0) and smoothly increases, but at a decreasing rate.
• The bigger the x-value, the slower the rate of increase for y.

These characteristics shape how the square root function graph appears and determine its outputs.

Function Behavior

Understanding the behavior of a function is crucial for interpreting its graph. The square root function \(y = \sqrt{x}\) has particular traits:

• Begins at the origin (0, 0), ensuring both x and y are non-negative.
• As x increases, y increases as well, but at a decreasing rate (slope gets flatter).
• Known as a non-linear function, because the graph is not a straight line.

This behavior indicates that initial changes in x produce more significant changes in y compared to larger values of x. Recognizing this variable rate of increase helps in anticipating and graphing the overall shape of the function on the coordinate plane.