Problem 47 - Calculus for Biology and Medicine

Question

Use a graphing calculator to sketch the graphs of the functions. $$ y=x^{3 / 2}, x \geq 0 $$

Step-by-Step Solution

Answer

Sketch the curve for \( y = x^{3/2} \) from the origin, increasing smoothly for \( x \geq 0 \).

Step 1: Understand the Function

We are given the function \( y = x^{3/2} \) with the condition \( x \geq 0 \). This means the function is only defined for non-negative values of \( x \). The exponent \( 3/2 \) implies the cube of the square root of \( x \).

Step 2: Identify Key Features of the Graph

Identify key features such as the domain, which here is \( x \geq 0 \), and the behavior at key points. At \( x = 0 \), \( y = 0^{3/2} = 0 \). As \( x \) increases, the value of \( y \) will also increase.

Step 3: Plot and Analyze the Graph

Using a graphing calculator, plot the function starting from \( x = 0 \). Notice that at \( x = 0 \), the graph starts at the origin (0, 0) because \( y = 0^{3/2} = 0 \), and then the graph rises smoothly to the right as \( x \) increases.

Step 4: Interpret the Graph's Shape and Behavior

The graph of \( y = x^{3/2} \) for \( x \geq 0 \) is a curve that starts at the origin and increases as \( x \) does, but not as steeply as linear or quadratic curves. This reflects the increasing rate of growth due to the cubic and square root components.

Key Concepts

Understanding Domain and Range

When exploring functions, it's essential to first understand the concepts of domain and range. The domain of a function is the set of all possible input values (usually represented by \(x\)) that the function can accept. For the function \(y = x^{3/2}\) with the condition \(x \geq 0\), the domain consists of non-negative values of \(x\), meaning any value of \(x\) that is zero or greater can be used. This is important because it determines where the function exists on the coordinate plane.

On the other hand, the range of a function is the set of all possible output values (represented by \(y\)). For our function, as \(x\) starts at zero and increases, the value of \(y\) starts at zero and also increases. Thus, the range of this function is \(y \geq 0\). Understanding both domain and range is crucial when graphing functions as it outlines the boundaries within which the function operates.

Utilizing a Graphing Calculator

Graphing calculators are powerful tools that help visualize functions quickly and accurately. To graph the function \( y = x^{3/2} \) using a graphing calculator, you'll need to ensure that you input the function correctly and set the viewing window to display the relevant portion of the graph.

Here's how to use it effectively:

• Enter the function in the form \(y = x^{3/2}\) into your graphing calculator.
• Adjust the window settings so that the \(x\)-axis starts from zero, since \(x \geq 0\).
• Set an appropriate range for \(y\) to encompass the values you expect, starting from zero.
• Once these settings are in place, plot the graph to see the smooth curve rising from the origin.

Using a graphing calculator not only aids in visualizing the function but can also help in understanding the function’s behavior and verifying your manually calculated points.

Exploring Function Behavior

Understanding how a function behaves helps predict how it will act at different points on the graph. For the function \(y = x^{3/2}\), the behavior describes how \(y\) changes as \(x\) changes.

Let's explore some aspects of function behavior:

• **Initial Point**: The function starts at the origin (0,0). This is because at \(x = 0\), \(y = 0^{3/2} = 0\).
• **Growth Pattern**: As \(x\) increases, \(y\) increases as well, but the rate of increase is influenced by the power \(3/2\). The combination of a square root and a cube means the graph grows, but less steeply than a linear \(y = x\) or quadratic \(y = x^2\) function.
• **Shape of the Curve**: The curve is smooth and continues to rise as \(x\) increases, which shows a steady, albeit non-linear, growth.

By analyzing the behavior, you can anticipate the function's direction and pattern without needing to calculate every single point, making it easier to understand and graph functions accurately.