Problem 77 - A Graphical Approach to College Algebra

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

The regions are shaded between \(-1 < x < 0\) and \(0 < x < 1\) at points (-1,-1), (0,0), and (1,1).

Step 1: Find Points of Intersection

To find the points of intersection of the curves represented by the equations \(y = x^3\) and \(y = x\), we set them equal to each other: \(x^3 = x\). Rearranging gives \(x^3 - x = 0\). Factoring out \(x\) gives \(x(x^2 - 1) = 0\). This can be factored further as \(x(x-1)(x+1) = 0\). Therefore, the points of intersection are when \(x = 0,\ x = 1,\ x = -1\).

Step 2: Determine Corresponding y-values for each Intersection Point

With the solutions from Step 1, we calculate the \(y\)-values using \(y = x\). If \(x = 0\), then \(y = 0\); if \(x = 1\), then \(y = 1\); and if \(x = -1\), then \(y = -1\). Therefore, the full coordinates of the points of intersection are \((0, 0),\ (1, 1),\ (-1, -1)\).

Step 3: Analyze the Graphs to Identify Shaded Regions

Consider the intervals that the intersection points divide the x-axis into: 1. \(x < -1\): Here, \(x^3 < x\), so no shading.2. \(-1 < x < 0\): Here, \(x^3 > x\), so shading is above the line and below the cubic curve.3. \(0 < x < 1\): Here, \(x^3 < x\), so shading is below the line and above the cubic curve.4. \(x > 1\): Here, \(x^3 > x\), so no shading.Thus, the shaded regions are where \(-1 < x < 0\) and \(0 < x < 1\).

Step 4: Sketch the Graph and Shade the Regions

Plot the curves \(y = x^3\) and \(y = x\) on a coordinate plane. Mark the points of intersection \((0, 0),\ (1, 1),\ (-1, -1)\). For \(-1 < x < 0\), shade the region above the line \(y = x\) and below the curve \(y = x^3\). Similarly, for \(0 < x < 1\), shade the region below the line \(y = x\) and above the curve \(y = x^3\).

Key Concepts

Factoring Equations

Factoring is a key skill in algebra that lets us simplify and solve complex equations easily. In the problem, we're looking at the equation \(x^3 = x\). To find where this curve intersects with another curve, we need to solve this equation by setting it to zero: \(x^3 - x = 0\). This is known as putting the equation into standard form. Once in standard form, factoring becomes much easier.
In this example, factor out \(x\) from the left side of the equation to make it \(x(x^2 - 1) = 0\). Recognize the quadratic expression \(x^2 - 1\) as a difference of two squares, which can be further factored to \((x - 1)(x + 1)\).
Thus, the fully factored form becomes \(x(x - 1)(x + 1) = 0\). We find the solutions, or roots, by setting each factor equal to zero:\

• \(x = 0\)
• \(x - 1 = 0\,\text{gives}\, x = 1\)
• \(x + 1 = 0\,\text{gives}\, x = -1\)

Factoring helps us identify exactly where the intersection points occur on the x-axis.

Graph Analysis

Graph analysis involves studying the graph's shape and behavior to understand the relationship between equations. To do this, start by plotting the two equations, \(y = x^3\) and \(y = x\).
The graph of \(y = x^3\) is a cubic curve that passes through the origin and curves upwards for positive \(x\) while curving symmetrical downwards for negative \(x\). In contrast, \(y = x\) is a straight line that passes directly through the origin with a slope of 45 degrees relative to either axis. This line will continue diagonally across the graph.
When you examine these two graphs together, notice that they intersect at three points: \((-1, -1)\), \((0, 0)\), and \((1, 1)\).
The intervals between these intersection points can tell us where one curve lies above the other, helping us determine where any shading will occur.

• For \(-1 < x < 0\), the cubic curve \(y = x^3\) is above the line \(y = x\).
• For \(0 < x < 1\), the line \(y = x\) is above the cubic curve \(y = x^3\).

Graph analysis helps in visualizing where the equations intersect and overlap on a graph.

Curve Shading

Shading regions on a graph serves to visually represent solutions or areas of interest between curves. This often happens in calculus or algebra to show where one function dominates another.
To shade correctly, first understand which intervals to look at from your intersection points. In our case, they segment the x-axis into four parts. Focus on where the order or dominance of the functions changes.
Now, based on graph analysis, identify the regions of interest:

• Where \(-1 < x < 0\): Since \(x^3 > x\), shade above the line and below the curve.
• Where \(0 < x < 1\): Since \(x^3 < x\), shade below the line and above the curve.

Be sure to capture these regions visually on your plot; marking these distinct areas adds clarity to the solution by showing the intervals of overlap between the two graphs. These regions illustrate the scenarios or solutions where one equation's output is larger than the other's within given ranges on the real number line.