Problem 93 - A Graphical Approach to Precalculus with Limits

Question

Shade the region(s) contained inside the graphs and give any points of intersection of the equations. $$\begin{aligned}&y=x^{3}\\\&y=x\end{aligned}$$

Step-by-Step Solution

Answer

Shade the region between \( y = x^3 \) and \( y = x \) from \( x = -1 \) to \( x = 1 \); intersection points: \((0, 0), (1, 1), (-1, -1)\).

Step 1: Graph the Equations

First, graph the given equations. The equation \( y = x^3 \) is a cubic function, which is a curve that starts low, increases, and then drops again as \( x \) goes to negative infinity. The equation \( y = x \) is a straight line passing through the origin with slope 1.

Step 2: Find Points of Intersection

Set the two equations equal to each other to find their points of intersection. This gives the equation \( x^3 = x \). Solving for \( x \), we factor it as \( x(x^2 - 1) = 0 \). Hence, the solutions are \( x = 0 \), \( x = 1 \), and \( x = -1 \).

Step 3: Determine Corresponding y-Values

For the points found, substitute each into one of the original equations. For \( x = 0 \), \( y = 0 \). For \( x = 1 \), \( y = 1 \). For \( x = -1 \), \( y = -1 \). Thus, intersection points are \((0, 0)\), \((1, 1)\), and \((-1, -1)\).

Step 4: Analyze the Regions

Determine which regions are fully or partially encompassed by each graph. The line \( y = x \) is above the curve \( y = x^3 \) in the interval \((-1, 0)\) and \((0, 1)\), and below the curve outside these intervals.

Step 5: Shade the Region

Shade the region contained between \( y = x^3 \) and \( y = x \) from \( x = -1 \) to \( x = 1 \) since \( y = x \) is above \( y = x^3 \) within this interval.

Step 6: Verify and Conclude

Check the solution by plotting both functions and ensuring that the shaded region between the curves is consistent. The shading is correct between the roots at \( x = -1, 0, \) and \( 1 \). Therefore, the region of interest is between these intersections.

Key Concepts

Understanding Cubic Functions

Cubic functions are mathematical expressions that include a variable raised to the third power, written in the form \( y = ax^3 + bx^2 + cx + d \). In the simplest form, like \( y = x^3 \), they are characterized by their noticeable 'S' shaped curve known as an inflection point. This point is where the graph changes its curvature direction. The graph of a cubic function can have many features based on its coefficients:

• One real root or solution.
• Can cross the x-axis up to three times.
• Curves can appear in different positions or orientations on the graph.

In our exercise, the cubic function \( y = x^3 \) has roots at \( x = 0 \), strikingly making it pass through the origin. It starts from negative infinity, gracefully rises as it crosses point \( (0,0) \), and continues positively skyward. Understanding how these attributes work can help you better interpret graphs of cubic functions in various contexts.

Finding Points of Intersection

Points of intersection are the coordinates where two graphs meet or cross. Identifying these points is crucial as they often represent solutions to a system of equations. To find them for two functions, we set their equations equal to each other and solve for \( x \).
For instance, with our functions \( y = x^3 \) and \( y = x \), setting \( x^3 = x \) gives us a straightforward polynomial to solve. We can simplify it by factoring:

• First, we factor it as \( x(x^2 - 1) = 0 \).
• This reveals three solutions: \( x = 0 \), \( x = 1 \), and \( x = -1 \).

To find the corresponding \( y \) values, we substitute back into one of the original equations, preferably the simpler one, \( y = x \), giving points \((0,0)\), \((1,1)\), and \((-1,-1)\).
Understanding where these points are located helps determine how the graphs interact, which is essential when analyzing intersections and overlaps.

Practical Graph Shading

Graph shading involves coloring the area between two curves or lines to illustrate the region of interest. This process is visual and helps pinpoint solutions on the graph. In the exercise, shading highlights where one graph is above or below the other.
For the given equations \( y = x^3 \) and \( y = x \), we find:

• Between the points of intersection \( x = -1 \), \( x = 0 \), and \( x = 1 \), the line \( y = x \) is above the curve \( y = x^3 \).
• This segment is where we shade since it contains all values fulfilling the conditions of being between these intersecting graphs.

To successfully shade, plot both equations on the same graph. Identify their intersection points and observe which graph lies above the other in the relevant intervals. Shading these sections correctly ensures understanding of solutions related to inequalities or bounded regions between curves.
It's a powerful tool for visual learners and facilitates a deeper comprehension of function relationships.