Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

, 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 function has origin symmetry, intercepts at (-1,0), (0,0), and (1,0), and is a cubic graph.

Step 1: Determine Symmetry

To check for symmetry, we evaluate the equation for symmetry about the y-axis, x-axis, and origin. First, for y-axis symmetry, replace \(x\) with \(-x\): \(y = (-x)^{3} - (-x) = -x^{3} + x\), which is not equal to the original equation, therefore, there is no symmetry about the y-axis. Second, for x-axis symmetry, replace \(y\) with \(-y\): \(-y = x^3 - x\), which is not equal to the original equation, hence no symmetry about the x-axis. Third, for origin symmetry, replace \(y\) with \(-y\) and \(x\) with \(-x\): \(-y = (-x)^3 - (-x) = -x^3 + x\). Distributing the negative to both sides, we get \(y = x^3 - x\), indicating symmetry about the origin.

Step 2: Find x-intercepts

To find the x-intercepts, set \(y = 0\) in the equation: \(0 = x^3 - x\). Factor the equation: \(x(x^2 - 1) = 0\). This gives \(x = 0\), \(x^2 = 1\), thus \(x = 1\) and \(x = -1\). The x-intercepts are \((-1, 0)\), \((0, 0)\), and \((1, 0)\).

Step 3: Find y-intercept

To find the y-intercept, set \(x = 0\) in the equation: \(y = 0^3 - 0 = 0\). Thus, the y-intercept is \((0, 0)\).

Step 4: Plot the Points and Graph

With the points found: \((-1, 0)\), \((0, 0)\), and \((1, 0)\), and symmetry about the origin, plot these intercepts on a coordinate plane. Because the function \(y = x^3 - x\) is a cubic equation, it will have a cubic curve shape. It passes through these points, and the behavior of the curve is decreasing from the left, passing through the origin, and increasing to the right.

Key Concepts

Symmetry of Functions

Understanding the symmetry of a function can simplify the process of graphing it and provide insights into its behavior. There are three main types of symmetry to consider: y-axis symmetry, x-axis symmetry, and origin symmetry.

Y-Axis Symmetry: A function is symmetric about the y-axis if replacing \(x\) with \(-x\) yields the same equation. However, in our function \y = x^3 - x\, replacing \(x\) with \(-x\) results in \(y = -x^3 + x\), which is not the same as the original equation. Thus, it is not y-axis symmetric.

X-Axis Symmetry: A function is symmetric about the x-axis if replacing \(y\) with \(-y\) results in the original equation. Doing this for our function gives \(-y = x^3 - x\), which again, does not match the original equation, ruling out x-axis symmetry.

Origin Symmetry: If replacing \(x\) with \(-x\) and \(y\) with \(-y\) yields the original equation, the function is symmetric about the origin. Replacing gives \(-y = -x^3 + x\), which simplifies back to the original \(y = x^3 - x\). Thus, our function has origin symmetry.

X-Intercepts

The x-intercepts of a function occur where the graph crosses the x-axis. At these points, the y-value is zero. To determine the x-intercepts of the function \(y = x^3 - x\), we set \(y = 0\) and solve for \(x\). This gives us the equation: \[0 = x^3 - x\].

To solve, factor out \(x\) to get: \[x(x^2 - 1) = 0\]. This tells us that either \(x = 0\) or \(x^2 - 1 = 0\). Solving \(x^2 - 1 = 0\) results in \(x^2 = 1\), which means \(x = 1\) or \(x = -1\).

Thus, the x-intercepts of the function are located at \((-1, 0), (0, 0),\) and \( (1, 0)\). These points help us understand the points at which the graph crosses the x-axis.

Y-Intercepts

Finding the y-intercept of a function involves identifying where the graph crosses the y-axis. At the y-intercept, the x-value is always zero. To find the y-intercept of \(y = x^3 - x\), substitute \(x = 0\) into the equation. This calculation results in: \[y = 0^3 - 0 = 0\].

Thus, the y-intercept is simply at the origin point \(0, 0\). Since it is also one of the x-intercepts, this point holds particular significance as a point where both intercepts intersect.

Composition of Functions

The composition of functions involves combining two or more functions into a single function. Although not directly part of graphing a cubic function, understanding function composition is essential in broader mathematical contexts.

Let's say we have two functions, \(f(x)\) and \(g(x)\). The composition of these functions is denoted as \(f(g(x))\) and involves substituting \(g(x)\) into \(f(x)\).
Example: If \(f(x) = x^2\) and \(g(x) = x-2\), then the composition \(f(g(x)) = (x-2)^2\).

This process can reveal relationships between functions and helps in transforming and analyzing complex functions by understanding their simpler components.