Problem 66 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(y=x^{3}-1\)

Step-by-Step Solution

Answer

The x-intercept of the function \(y=x^{3}-1\) is at (1, 0) and the y-intercept is at (0, -1). The graph is not symmetric with respect to the y-axis. After plotting the intercepts and connecting more points on the graph, the sketch shows a positive cubic function starting from the negative direction, going through the intercepts, and then rising towards the positive direction.

Step 1: Identify X-Intercept

Set \(y=0\) and solve the equation for \(x\). For \(y=x^{3}-1\), we get:\n\(0=x^{3}-1\)\nAdd 1 to both sides:\n\(x^{3}=1\)\nTaking cube root on both sides, we get \(x=1\) as the x-intercept. Thus, the x-intercept is at (1, 0).

Step 2: Identify Y-Intercept

Set \(x=0\) and solve the equation for \(y\). For \(y=x^{3}-1\), we get:\n\(y=0^{3}-1=-1\)\nSo the y-intercept is (0, -1).

Step 3: Test for Symmetry

Symmetry is tested by replacing \(x\) with \(-x\). If the equation remains the same, it is symmetric with respect to the y-axis. If the sign of the equation changes, the graph is symmetric with respect to the origin. For the equation \(y=x^{3}-1\), replacing \(x\) with \(-x\) gives \(y=(-x)^{3}-1 = -x^{3}-1\), which is not the same as the original equation, hence it's not symmetric to y-axis.

Step 4: Sketch the Graph

Now, with the intercepts and properties of the function (as it's not symmetric with respect to y-axis), the graph of the equation can be sketched which shows the behavior of the cubic function. Choose a variety of points around the x-intercept and the y-intercept, plot them, and connect the points to get the rough sketch. The graph rises from the negative direction, passes through the x-intercept at (1, 0) and the y-intercept at (0, -1), and continues to rise due to being a cubic function.

Key Concepts

Intercepts

Finding the intercepts of a cubic function such as the equation \( y = x^3 - 1 \) is an important step in graphing. There are two types of intercepts to consider: the x-intercept and the y-intercept.

• X-Intercept: This occurs where the graph crosses the x-axis. To find this, set \( y = 0 \) in the equation and solve for \( x \). For our equation, setting \( y = 0 \) gives \( 0 = x^3 - 1 \), leading to \( x^3 = 1 \). By taking the cube root of both sides, we find \( x = 1 \), so the x-intercept is at the point (1, 0).


• Y-Intercept: This happens where the graph intersects the y-axis. Set \( x = 0 \) and solve for \( y \). Substituting \( x = 0 \) into the equation \( y = x^3 - 1 \) results in \( y = 0^3 - 1 = -1 \). Therefore, the y-intercept is located at (0, -1).

Identifying these intercepts is crucial, as they provide key points that help shape the sketch of the cubic function's graph.

Symmetry Testing

Symmetry testing of a cubic function involves examining if the graph remains unchanged when transformed in certain ways. Testing symmetry can help identify patterns and reduce the effort needed to graph functions.

• Symmetry with Respect to the Y-Axis: A graph is symmetric with respect to the y-axis if replacing \( x \) with \( -x \) yields an identical equation. For our function \( y = x^3 - 1 \), substituting \( -x \) gives \( y = (-x)^3 - 1 = -x^3 - 1 \), which is not identical to the original function. Hence, the graph is not symmetric with respect to the y-axis.


• Symmetry with Respect to the Origin: Another type of symmetry we might consider is with respect to the origin. This occurs if substituting \( x \) with \( -x \) results in the exact negative of the original equation. In this case, since changing \( x \) to \( -x \) results in \( y = -x^3 - 1 \), which doesn't completely negate the original equation \( y = x^3 - 1 \), the graph is also not symmetric with respect to the origin.

While testing indicates this particular cubic function lacks symmetry, understanding symmetry is still useful, especially when graphing other types of functions.

Cubic Equations

Cubic equations are polynomial equations of degree three and have the general form \( ax^3 + bx^2 + cx + d = 0 \). In our example, the cubic equation is \( y = x^3 - 1 \).

• Behavior of Cubic Functions: Typical features of cubic functions include one to three real roots (x-intercepts) and the ability to change direction of curvature at least once. These functions can turn once or twice, making them more dynamic and interesting to graph compared to linear or quadratic functions.


• End Behavior: The leading coefficient (the coefficient of the term with the highest exponent, here it is 1) determines if the graph will rise or fall as \( x \) moves towards positive or negative infinity. For positive leading coefficients, as in our equation, the graph rises in both directions. This explains why our function appears to rise steeply, continuing upwards as it moves away from the center of the graph.


• Distinctiveness: The absence of symmetry in this particular cubic function highlights its uniqueness, as cubic graphs don't necessarily possess symmetry as a standard trait, unlike parabolas.

By understanding these properties, graphing cubic equations becomes less daunting and more of a straightforward process, allowing us to predict and sketch their curves accurately.