Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-30, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\)-intercepts. $$ y=x^{3}-x $$

Step-by-Step Solution

Answer

The graph is symmetrical about the origin, with intercepts at (-1,0), (0,0), and (1,0).

Step 1: Check for Symmetries

To check for symmetry, consider the original equation \(y = x^3 - x\) and replace \(x\) with \(-x\) to test for symmetry with respect to the y-axis: \(y = (-x)^3 - (-x) = -x^3 + x\). Since this is not equal to the original equation, there is no symmetry with respect to the y-axis.For symmetry with respect to the origin, test if \((-x,-y)\) satisfies the equation. Replace \(y\) with \(-y\) in the equation, giving \(-y = x^3 - x\), or \(y = -x^3 + x\). This is the same as changing both \(x\) and \(y\) signs in the original equation, indicating symmetry with respect to the origin.

Step 2: Find the x-Intercepts

Find the x-intercepts by setting \(y=0\) and solving \(x^3 - x = 0\). Factor the expression: \(x(x^2 - 1) = 0\). Further factoring gives \(x(x-1)(x+1) = 0\). This means the x-intercepts are at \(x = 0\), \(x = 1\), and \(x = -1\).

Step 3: Find the y-Intercept

Find the y-intercept by setting \(x=0\) in the equation: \(y = 0^3 - 0 = 0\). Thus, the y-intercept is at \(y = 0\).

Step 4: Plot Key Points and Sketch the Graph

Identify key points from previous steps: symmetry at the origin, x-intercepts at \((-1,0)\), \((0,0)\), \((1,0)\), and the y-intercept at \((0,0)\). Plot these points on a graph.Since the function is asymmetrical to the y-axis and symmetrical to the origin, the graph should show a cubic curve that passes through these intercepts. The curve should increase as \(x\) becomes more positive and decrease as \(x\) becomes more negative, resembling an 'S' shape due to the cubic term.

Key Concepts

Symmetry in Graphs

When analyzing the symmetry of a graph, you're essentially checking how the graph reflects or rotates with respect to an axis or the origin. For functions such as \(y = x^3 - x\), evaluating symmetry is crucial. Here, we tested for a couple of types of symmetries:

• Y-Axis Symmetry: Replacing \(x\) with \(-x\) in the function gives \(-x^3 + x\). Since this is not the same as our original equation, there is no symmetry with respect to the y-axis. This means the graph doesn’t mirror around the vertical y-axis.
• Origin Symmetry: To test this, replace \(x\) with \(-x\) and \(y\) with \(-y\). If the equation holds, the graph is symmetric around the origin. Here, this condition is met, indicating that rotating the graph 180 degrees around the origin yields the same graph. This kind of symmetry suggests that the function is an odd function.

Understanding symmetry aids in predicting the general shape of the graph. It simplifies the process of sketching complex functions, particularly cubic ones.

Finding Intercepts

Intercepts are points where the graph crosses the axes. They provide valuable information about the behavior of the function. Let's break down the process:

X-Intercepts

To find where the graph intersects the x-axis, set the equation equal to zero and solve for \(x\). For \(y = x^3 - x\), you set \(y = 0\), leading to solving the equation \(x^3 - x = 0\). Factoring it as \(x(x^2 - 1) = 0\) further simplifies to \(x(x - 1)(x + 1) = 0\). Therefore, the x-intercepts are at points \(x = 0\), \(x = 1\), and \(x = -1\). These are the points where the graph touches or crosses the x-axis.

Y-Intercept

The y-intercept is found by setting \(x = 0\) and solving for \(y\). In this case, \(y = 0^3 - 0\), or simply \(y = 0\). Therefore, the y-intercept is the point \((0, 0)\), which is also an x-intercept. Graphing these intercepts assists in forming the foundational structure of the graph. With this information, you can accurately depict the function's behavior across its domain.

Understanding Cubic Functions

Cubic functions frequently appear in mathematics, characterized by the degree of 3 in their polynomial expression. For the function \(y = x^3 - x\), the highest power of \(x\) is 3, making it a cubic function.

General Shape and Behavior

Cubic functions can represent a wide variety of shapes, but they typically resemble an 'S' curve. Here’s what you can generally expect:

• One peculiarity is that they can have up to three real roots or solutions, which correspond to the x-intercepts found during the analysis.
• The graph has both increasing and decreasing intervals, giving it that distinct wave-like shape.
• As \(x\) approaches positive infinity, the function tends to positive infinity, and as \(x\) moves towards negative infinity, the function heads towards negative infinity. This dual behavior is the result of the cubed term \(x^3\), dominating the graph’s end behavior.

Points of Inflection

In a cubic function, the point of inflection is where the graph changes from being concave up to concave down or vice versa. For \(y = x^3 - x\), the point of inflection occurs at the origin, \((0, 0)\). Understanding these attributes ensures you comprehend the graph’s entire demeanor from simple plots to intricate movements. This knowledge forms a foundation for tackling other complex polynomial functions.