Question

a) Given the function \(y=x^{3},\) list the parameters of the transformed polynomial function \(y=\left(\frac{1}{2}(x-2)\right)^{3}-3\) b) Describe how each parameter in part a) transforms the graph of the function \(y=x^{3}\) c) Determine the domain and range for the transformed function.

Short answer

Parameters: \(a=\frac{1}{2}\), \(h=2\), \(k=-3\). Domain: \(( -\infty, +\infty )\). Range: \(( -\infty, +\infty )\).

Step-by-step solution

- Identify parameters in the transformed function

The given transformed polynomial function is \(y = \left( \frac{1}{2}(x-2) \right)^{3} - 3\). Here are the parameters: \(a=\frac{1}{2}\) (horizontal stretch/compression), \(h=2\) (horizontal shift), and \(k=-3\) (vertical shift).

- Describe the horizontal stretch/compression

The parameter \(a=\frac{1}{2}\) indicates a horizontal stretch by a factor of 2. This means that the graph of the function will be stretched away from the y-axis by a factor of 2.

- Describe the horizontal shift

The parameter \(h=2\) indicates a horizontal shift to the right by 2 units. This means that every point on the graph of the original function \(y = x^3\) will be moved 2 units to the right.

- Describe the vertical shift

The parameter \(k=-3\) indicates a vertical shift downward by 3 units. This means that every point on the graph of the function, after the horizontal stretch and shift, will be moved 3 units down.

- Determine the domain of the transformed function

The domain of the original function \(y = x^3\) is all real numbers, \( ( -\infty, +\infty ) \). Since neither horizontal stretches/compressions nor shifts affect the domain, the domain of the transformed function remains all real numbers: \(( -\infty, +\infty )\).

- Determine the range of the transformed function

The range of the original function \(y = x^3\) is also all real numbers, \( ( -\infty, +\infty ) \). Since vertical shifts by \(k\) units down still cover all real numbers, the range of the transformed function is also \( ( -\infty, +\infty ) \).

Key concepts

horizontal stretch

Understanding a horizontal stretch is key when working with polynomial transformations. A horizontal stretch occurs when you change the distance between the points on a graph and the y-axis. For example, in the function
\[ y = \left( \frac{1}{2}(x-2) \right)^3 - 3 \] the parameter \( a = \frac{1}{2} \) causes a horizontal stretch. Here, every point on the graph moves away from the y-axis by a factor of 2. Imagine pushing the graph horizontally, so each point is now twice as far from the y-axis as it was before.

horizontal shift

A horizontal shift moves the graph left or right along the x-axis. This transformation is controlled by the parameter \( h \). For the function
\[ y = \left( \frac{1}{2}(x-2) \right)^3 - 3 \] the parameter \( h = 2 \) shifts the graph 2 units to the right. This means every point on the original graph of \( y = x^3 \) is moved 2 units to the right. Think of sliding the entire graph along the x-axis without altering its shape.

vertical shift

Vertical shifts move the graph up or down along the y-axis. The parameter \( k \) determines the direction and magnitude of this shift. In our example
\[ y = \left( \frac{1}{2}(x-2) \right)^3 - 3 \] the parameter \( k = -3 \) shifts the graph 3 units downward. Every point on the function graph moves down by 3 units. Imagine dropping the whole graph down vertically by 3 spaces while maintaining its shape and horizontal position.

domain and range

The domain and range of transformed functions can sometimes be tricky to determine. The domain represents all possible x-values a function can take, and for the cubic function \( y = x^3 \), this is all real numbers, from \( -\infty \) to \( +\infty \). Transformations like horizontal stretches/compressions and shifts do not affect the domain. Therefore, the domain for
\[ y = \left( \frac{1}{2}(x-2) \right)^3 - 3 \]
remains \( (-\infty, +\infty) \). The range represents all possible y-values. Similar to the domain, the range for \( y = x^3 \) is all real numbers. Vertical shifts move the graph up or down but do not limit the y-values it can reach. Therefore, the range for our transformed function is also \( (-\infty, +\infty) \).