Problem 79 - College Algebra

Question

For the following exercises, graph \(y=x^{3}\) on the given viewing window. Determine the corresponding range for each viewing window. Show each graph. $$[-0.1,0.1]$$

Step-by-Step Solution

Answer

The range for the window \([-0.1, 0.1]\) is \([-0.001, 0.001]\).

Step 1: Understand the Function

The function we are given is a cubic function, \( y = x^3 \). This function has a unique characteristic: it is symmetric about the origin and increases/decreases rapidly as \( x \) moves away from zero.

Step 2: Determine the Viewing Window

We're working with a viewing window of \([-0.1, 0.1]\), which means we will graph the function for values of \( x \) from \(-0.1\) to \(0.1\).

Step 3: Calculate Values in the Window Range

Evaluate \( y = x^3 \) at key points within the given range. For example, at \( x = -0.1 \), \( y = (-0.1)^3 = -0.001 \). At \( x = 0 \), \( y = 0^3 = 0 \). And at \( x = 0.1 \), \( y = (0.1)^3 = 0.001 \).

Step 4: Determine the Range

Based on the values calculated, the range of \( y \) for \( x \) in the viewing window \([-0.1, 0.1]\) is \(-0.001 \leq y \leq 0.001\).

Step 5: Graph the Function

Plot the calculated points to sketch the graph of \( y = x^3 \) within the window \([-0.1, 0.1]\). The curve will be very shallow, showing a slight increase from left to right, passing through the origin.

Key Concepts

Graphing Functions

Graphing functions involves visually representing mathematical relationships on a coordinate plane. When we graph a function like \( y = x^3 \), we're interested in how the variable \( y \) changes as \( x \) moves across a specified interval. Here, we focus on the interval \([-0.1, 0.1]\), an incredibly small range. This choice helps us observe that cubic functions, although typically known for steep, S-shaped curves, can appear flat when zoomed in tightly.

• Graphing begins with selecting key points; in this example, \( x = -0.1, 0, \text{ and } 0.1 \).
• We calculate the corresponding \( y \) values using the equation \( y = x^3 \).
• With these points, the graph is plotted, sketching a gentle curve that passes right through the origin due to the symmetry of cubic functions.

Understanding how functions transform on graphs, even under magnified views, is essential for analyzing their behavior closely.

Range of a Function

The range of a function is the set of possible output values \( y \) it can produce. For the cubic function \( y = x^3 \), when viewed across the interval \([-0.1, 0.1]\), its range is determined by the smallest and largest \( y \) values it achieves.This involves calculating the function at critical points:

• When \( x = -0.1 \), \( y = (-0.1)^3 = -0.001 \).
• As \( x = 0 \), \( y = 0^3 = 0 \).
• For \( x = 0.1 \), \( y = (0.1)^3 = 0.001 \).

Thus, within our specific window, the range is \([-0.001, 0.001]\). This small range shows us just a fragment of how the cubic function behaves. The narrow span of \( y \) values reflects the function's behavior near the origin, emphasizing the vertical flattening at this interval.

Symmetric About the Origin

A function symmetric about the origin can be rotated 180 degrees around the origin point and remain unchanged. This property is termed "odd symmetry." For cubic functions like \( y = x^3 \), this inherent symmetry is evident because for every point \((x, y)\), there is a corresponding point \((-x, -y)\).

• Odd functions satisfy the condition \( f(-x) = -f(x) \).
• For \( y = x^3 \), substituting \(-x\) gives \((-x)^3 = -x^3\), demonstrating its symmetry.
• Graphically, this reflects as a balance between the function's rising segment in the positive \( x \) direction and its falling segment in the negative \( x \) direction.

Understanding symmetry helps in predicting the function's behavior without calculating every detail, offering a form of pattern recognition for efficient sketching and analysis.