Problem 16 - Thomas Calculus

Question

In Exercises 5–30, determine an appropriate viewing window for the given function and use it to display its graph. $$ y=\left|x^{2}-x\right| $$

Step-by-Step Solution

Answer

Use viewing window: \(-2 \le x \le 3\); \(-1 \le y \le 3\).

Step 1: Understanding the Function

The function is defined as \( y = |x^2 - x| \). This is an absolute value function that may change its form based on the condition where the expression inside the absolute value changes sign.

Step 2: Identify Key Points

Find the points where the expression inside the absolute value, \( x^2 - x \), equals zero. Solve \( x^2 - x = 0 \) to find \( x(x - 1) = 0 \), which gives roots at \( x = 0 \) and \( x = 1 \). These points are important as they might represent turning points in the graph.

Step 3: Analyze the Expression

Consider the behavior of \( y = x^2 - x \) without the absolute value. For \( x < 0 \) or \( x > 1 \), \( x^2 - x > 0 \), making \( y = x^2 - x \). For \( 0 < x < 1 \), \( x^2 - x < 0 \), making \( y = -(x^2 - x) = x - x^2 \). This affects how the function looks within these ranges.

Step 4: Determine the Range for the Graph

Since the equation \( y = |x^2 - x| \) consists of quadratic forms, a viewing window that captures interesting points, intersections, and behavior changes around \( x = 0 \) and \( x = 1 \) is needed. Consider \( x \) values ranging from \(-2\) to \(3\) and \( y \) values from \(-1\) to \(3\). This range captures oscillations, turning points, and the intersection at the roots.

Step 5: Describe the Graph

The graph should reflect the quadratic nature where the function appears as a parabola opening upwards for \( x < 0 \) and \( x > 1 \) and opens downwards between \( 0 \) and \( 1 \). Make sure the graph shows a clear vertex at \( x=0.5 \).

Step 6: Verify the Viewing Window

Plot the graph using the identified viewing window to ensure it captures all the behavior points described. Adjust the window if any important feature of the graph is not visible or is clipped in the plot.

Key Concepts

Absolute Value Function

The absolute value function is a fundamental concept in calculus. It is denoted by two vertical bars surrounding an expression, like this:

• If the inside expression is positive or zero, the absolute value function does not change it.
• If the inside expression is negative, the absolute value function turns it into its positive counterpart.

This means the function effectively creates a 'reflection' of negative values across the x-axis, making them positive.
In our original problem, we have the function:

• \(y = |x^2 - x|\)

The expression inside the bars, \(x^2 - x\), can both be positive or negative, depending on the value of \(x\). Evaluating an absolute value function requires checking when the inside changes sign.
For better understanding, simplify by solving \(x^2 - x = 0\), where it crosses the x-axis, which has solutions at \(x = 0\) and \(x = 1\).
These are your critical points where the behavior described above changes.

Quadratic Functions

Quadratic functions are polynomial functions of degree 2 and have the general form:

• \(f(x) = ax^2 + bx + c\)

The feature of quadratics is their "U" shape, called a parabola. It can open upwards if the leading coefficient \(a > 0\) or downwards if \(a < 0\).
In our exercise, the expression \(x^2 - x\) is a quadratic function. Without the absolute value, its parabola opens upwards as the coefficient of \(x^2\) is 1.A key property of quadratic functions is their vertex, which is the highest or lowest point of the function. It can be calculated using the formula:

• \(x = \frac{-b}{2a}\)

For \(x^2 - x\), this occurs when \(b = -1\) and \(a = 1\), giving the vertex:

• \(x = \frac{1}{2}\)

However, the absolute value changes this as it flips the part of the curve below the x-axis above it, creating a different visual path between the critical points we analyzed.

Graphing Functions

Graphing functions helps us visualize the behavior of mathematical expressions across different intervals and conditions.
Understanding graphing is crucial for functions like absolute value and quadratics, as these can exhibit complex changes quickly. Here’s a basic guide:

• Identify key points, such as where the function crosses the axes and any turning points.
• Consider the intervals where different rule sets apply, especially for piecewise functions like our absolute value example.
• Determine the range and domain to be represented and choose an appropriate viewing window.
• Plot these points and sketch the curve that naturally connects them.
• For absolute values, ensure to reflect portions of the graph as described above.

In our scenario, plotting \(y = |x^2 - x|\) reveals a curve that changes direction at critical points and is symmetric about the y-axis due to the nature of the absolute value.

Function Intervals

Analyzing function intervals involves partitioning the domain based on characteristics of the function, enabling a deeper understanding of its behavior.
In our exercise, function intervals are significant because they show the regions over which the expression inside the absolute value sign varies:

• For \(x < 0\) and \(x > 1\), \(x^2 - x > 0\), so the function behaves as \(y = x^2 - x\), with normal quadratic properties.
• For \(0 < x < 1\), \(x^2 - x < 0\), modifying the behavior to \(y = x - x^2\) due to the absolute value.

These intervals explain why the function appears to flip at \(x = 0\) and \(x = 1\), showing different graph sections.
Effectively graphing and solving these requires recognizing these switching points and ensuring the viewing window encompasses these behaviors.