Problem 101 - College Algebra Essentials

Question

Begin by graphing the standard cubic function, \(f(x)=x^{3} .\) Then use transformations of this graph to graph the given function. $$h(x)=\frac{1}{2} x^{3}$$

Step-by-Step Solution

Answer

The graph of \( h(x) = 0.5x^3 \) is a vertically shrunk version of the standard cubic function \( f(x) = x^3 \) by a factor of 0.5. So, for every corresponding point (x, y) on \( f(x) \), there exists a point (x, 0.5y) on \( h(x) \). The graph passes through the origin and extends towards both positive and negative infinity - the same as \( f(x) = x^3 \), but is more flattened due to the shrinking.

Step 1: Graph the standard cubic function

Draw the graph of the standard cubic function \( f(x) = x^3 \). It will be a curve that passes through the origin (0, 0) and extends to both positive and negative infinity as \( x \) increases and decreases.

Step 2: Apply the transformation

Apply the transformation to graph the function \( h(x) = 0.5x^3 \) that is described in the exercise. Since it is a vertical shrink by a factor of 0.5, the y-coordinates of each corresponding point on \( f(x) \) are halved. For instance, if a point on \( f(x) \) is (2, 8), the corresponding point on \( h(x) \) will be (2, 4).

Step 3: Sketch the final graph

Once all transformations of the original function have been applied, sketch the graph of the new function. It should still pass through the origin as well as points formed from the transformed coordinates of \( f(x) \). The graph of \( h(x) = 0.5x^3 \) should be the same shape as \( f(x) \) but will appear more flattened because of the vertical shrink.

Key Concepts

Standard Cubic Function

The standard cubic function, represented by the equation \(f(x) = x^3\), is one of the fundamental shapes in graph theory. Understanding this basic form is essential because it establishes the foundation for more complex graph transformations.

The graph of a standard cubic function exhibits a distinctive 'S' shape, with the curve passing through the origin point (0,0) and demonstrating symmetry with respect to the origin. As the value of \(x\) increases or decreases, the \(y\) value of the function increases or decreases at a cubic rate, making the slope of the graph steepen away from the origin. This function is odd, meaning that \(f(-x) = -f(x)\), which is a key property reflected in its symmetry about the origin.

The cubic function can model various real-world scenarios such as volume calculations and the behavior of certain physical systems. It's also the simplest example of a function with a single inflection point, where the concavity changes from upwards to downwards or vice versa as \(x\) crosses zero.

Transformations of Graphs

Transformations of graphs involve altering the basic shape of a function in order to accommodate changes in an equation. These alterations are achieved through a series of operations such as translations, reflections, stretches, and shrinks.

There are specific mathematical methods to apply these transformations. For instance, adding or subtracting a constant to the \(x\) or \(y\) variable translates the graph vertically or horizontally, respectively. Multiplying \(x\) or \(y\) by a constant greater than 1 stretches the graph, while a constant between 0 and 1 shrinks it.

A vertical shrink, specifically, impacts the \(y\)-values directly, reducing the height of the curve without altering the \(x\)-values, resulting in a flatter appearance to the graph. Transformations allow us to graph complex equations by adjusting a standard function, such as \(x^3\), step by step to achieve the desired shape, like going from \(f(x) = x^3\) to \(h(x) = 0.5x^3\).

Vertical Shrink

A vertical shrink in the context of graph transformations is when each point on the graph of a function is moved closer to the \(x\)-axis. This effect is carried out by multiplying the \(y\)-component of the function by a constant that is between 0 and 1.

For example, consider the function \(h(x) = \frac{1}{2}x^3\) as described in the original problem. The \(\frac{1}{2}\) factor before \(x^3\) means that every \(y\)-value of the standard cubic function, \(f(x) = x^3\), is halved. If \(f(x)\) approached 8 when \(x\) was 2, \(h(x)\) will only reach 4 at the same \(x\) value—demonstrating the vertical shrink. The net effect of this transformation is a compressed graph along the \(y\)-axis without changing the \(x\)-values.

This type of transformation is useful for modeling scenarios where a reduction in magnitude is observed without a shift in the horizontal component, such as in damping effects in physics or adjustments in economic projections.