91Ó°ÊÓ

The constant of proportionality plays a crucial role in understanding how graphs transform in relation to each other. In the context of the given functions, the constant of proportionality, often denoted as `k`, determines the steepness and orientation of the graph.

Consider the equation for each function in the form of \( y = k \times x^3 \). Here are the respective values of `k` for each function in the exercise:

• \( f(x) = x^3 \) has \( k = 1 \)
• \( g(x) = -x^3 \) has \( k = -1 \)
• \( h(x) = \frac{1}{2} x^3 \) has \( k = \frac{1}{2} \)
• \( j(x) = -2x^3 \) has \( k = -2 \)

Understanding `k` helps us determine how the graph will look. For instance, a `k` value greater than 1 indicates a steeper graph, while a `k` value between 0 and 1 results in a flatter graph. Moreover, the sign of `k` indicates whether the graph will be flipped across the x-axis.

Reflection across the x-axis occurs when the graph is flipped upside down. This happens when the constant of proportionality, `k`, is negative.

In the given exercise, the function \( g(x) = -x^3 \) serves as the reflection of \( f(x) = x^3 \) across the x-axis because its `k` value is -1.

Mathematically, the reflection is shown when \( y = -f(x) \). This means every positive `y` value from \( f(x) = x^3 \) becomes negative in \( g(x) = -x^3 \). Similarly, negative `y` values become positive.

A function stretch occurs when the graph of the function is elongated vertically. This happens when the absolute value of `k` is greater than 1.

For the given functions, \( j(x) = -2x^3 \) experiences both stretch and reflection. In this case, \( k = -2 \), which is greater than 1 in magnitude, indicates a vertical stretch by a factor of 2. The negative sign also indicates a reflection.

When stretching a graph, any value of \( y \) at a given \( x \) point in the function \( f(x) = x^3 \) becomes `2y` in the function \( j(x) = -2x^3 \). Thus, the graph appears taller and reflected across the x-axis.

A function compression makes the graph flatter or squished down vertically. This happens when the constant of proportionality, `k`, lies between 0 and 1 (exclusive).

In the exercise, \( h(x) = \frac{1}{2}x^3 \) represents a compressed version of \( f(x) = x^3 \) because `k` is \( \frac{1}{2} \), lying within this range.

To visualize compression, consider that for each point \( (x, y) \) on the graph of \( f(x) = x^3 \), the corresponding point on \( h(x) = \frac{1}{2}x^3 \) will be \( (x, \frac{y}{2}) \). This makes the graph appear flatter, as the values of `y` are halved.