Problem 99 - College Algebra

Question

Begin by graphing the standard cubic function, \(f(x)-x^{3} .\) Then use transformations of this graph to graph the given function. $$ h(x)=-x^{3} $$

Step-by-Step Solution

Answer

The graph of the function \(h(x) = -x^3\) is a reflection of the original cubic function \(f(x) = x^3\) over the x-axis. So, it's a curve that passes through the points (-2, 8), (-1, 1), (0, 0), (1, -1), (2, -8).

Step 1: Graph the Original Function

Plot the graph of the original cubic function. Let the x-values range from -2 to 2. Find the corresponding y-values by evaluating \(f(x) = x^3\) for each x-value. The graph of \(f(x) = x^3\) will be a curve that passes through the points (-2, -8), (-1, -1), (0, 0), (1, 1), (2, 8). Plot these points and sketch the curve.

Step 2: Identify the Transformation

From \(f(x) = x^3\) to \(h(x) = -x^3\), the transformation is a reflection over the x-axis. This changes the sign of each y-coordinate of the points in the original graph.

Step 3: Graph the Transformed Function

Now, graph the transformed function \(h(x) = -x^3\). Reflect all the points of the original graph over the x-axis. This means the points (-2, -8), (-1, -1), (0, 0), (1, 1), (2, 8) now become (-2, 8), (-1, 1), (0, 0), (1, -1), (2, -8). Plot these points and sketch the curve of \(h(x) = -x^3\).

Key Concepts

Standard Cubic Function

Understanding the standard cubic function is foundational for graphing more complex cubic equations. A standard cubic function has the form of f(x) = x^3. This equation represents a curve that's known for its distinctive 'S' shape, which stretches infinitely in both the positive and negative directions on the y-axis as the x-values move away from zero.

When graphed, the function passes through the origin (0,0) and is symmetric with respect to the origin due to its odd-power nature. The points where the function moves from increasing to decreasing or decreasing to increasing are known as inflection points, which for the standard cubic function occurs at the origin.

Any cubic function can be considered a transformation of this basic graph. Clearly understanding this fundamental shape aids in visualizing how transformations like stretching, compressing, or reflecting will affect the graph.

Transformations of Graphs

Transformations of graphs are modifications that alter the appearance of the original graph without changing its nature or function type. There are several kinds of transformations, including translations (shifting the graph up, down, left, or right), reflections (flipping the graph over an axis), and scaling (stretching or compressing the graph in the x or y directions).

For cubic functions, common transformations include vertical and horizontal shifts, vertical and horizontal stretches and compressions, and reflections. To apply these transformations to the standard cubic function, you'll often be working with an equation that adds, subtracts, or multiplies the x or f(x) (y-values) by constants.

By understanding how each type of transformation affects the graph, students can easily predict and construct the new shape of a transformed function, which is a key skill for graphing more advanced functions.

Reflection over the X-axis

Reflection over the x-axis is one of the transformations that can have a dramatic effect on the graph of a function. When you reflect a graph over the x-axis, each point on the graph is flipped to the opposite side of the x-axis, while its x-coordinate remains the same. As a result, the sign of the y-coordinate of each point is changed.

For the cubic function, a reflection over the x-axis requires us to multiply the entire function by -1, resulting in h(x) = -x^3. This means that if you have a point (a, b) on the original graph, the corresponding point on the reflected graph will be (a, -b). So, if the original graph passes through (2, 8), after reflection, it will pass through (2, -8).

Being able to visualize this reflection helps in graphing functions that include negative coefficients before the variable term and is a core skill for understanding symmetry and transformations in algebraic graphing.