In Exercises 95-106, begin by graphing the standard cubic function, f(x) = x³. Then use transformations of this graph to graph the given function. h(x) = x³/2

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1. Graph the standard cubic function \( f(x) = x^3 \). This graph is a curve that passes through the origin (0,0) and has points such as (1,1), (-1,-1), (2,8), and (-2,-8). It is symmetric about the origin.
2. Identify the transformation needed to graph \( h(x) = \frac{x^3}{2} \). This function is a vertical compression of the standard cubic function by a factor of \( \frac{1}{2} \).
3. To apply the vertical compression, take each y-coordinate of the points on the graph of \( f(x) = x^3 \) and multiply it by \( \frac{1}{2} \).
4. Plot the transformed points. For example, the point (1,1) on \( f(x) \) becomes (1,0.5) on \( h(x) \), and the point (2,8) becomes (2,4).
5. Draw the new graph of \( h(x) = \frac{x^3}{2} \) using the transformed points. The shape of the graph remains the same, but it is vertically compressed.

Key Concepts

Cubic Functions

A cubic function is a polynomial function of degree three, typically expressed in the form f(x) = ax³ + bx² + cx + d. The standard cubic function, f(x) = x³, has a characteristic S-shaped curve that passes through the origin and extends to infinity in both the positive and negative directions. Understanding the basic shape and properties of cubic functions is essential for analyzing transformations.

Graph Transformations

Graph transformations involve altering the position or shape of a function's graph through operations such as translations, reflections, stretches, and compressions. For example, the function h(x) = x³/2 represents a vertical compression of the standard cubic function f(x) = x³ by a factor of 2. Recognizing how these transformations affect the graph is crucial for accurately sketching the new function.

Vertical Compression

Vertical compression occurs when the output values of a function are multiplied by a factor less than one, resulting in a 'squished' appearance along the y-axis. In the case of h(x) = x³/2, each y-value of the cubic function f(x) = x³ is halved, leading to a graph that is closer to the x-axis. This concept is vital for understanding how the shape of the graph changes in response to the transformation applied.