Problem 17 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of \(f(x)=x^{3}\) and the graph of the function \(g .\) Describe the transformation from \(f\) to \(g .\) \(g(x)=(x-3)^{3}\)

Step-by-Step Solution

Answer

The function \(g(x)=(x-3)^{3}\) is a rightward shift of \(f(x)=x^{3}\) by 3 units.

Step 1: Plot the function \(f(x)=x^{3}\)

Begin by sketching a graph of the parent cubic function \(f(x)=x^{3}\). This graph passes through the origin (0,0) and increases to the right and decreases to the left.

Step 2: Understand the transformation

Take note that shifting \(f(x)\) to the right by 'c' units results in the new function \(f(x-c)\). Therefore, \(g(x)=(x-3)^{3}\) is a horizontal shift of function \(f\) 3 units to the right.

Step 3: Plot the function \(g(x)=(x-3)^{3}\)

Sketch the function \(g(x)\) using the transformation rule identified in Step 2. Every point on the graph of \(f(x)\) will shift 3 units to the right to form the graph of \(g(x)\). For example, the point at the origin on \(f(x)\) will now be at (3,0).

Key Concepts

Parent Cubic Function

The term 'parent cubic function' refers to the simplest form of cubic functions, which is given by the equation \( f(x) = x^3 \). This is the foundational graph from which other cubic function graphs are derived through various transformations.

The key characteristics of the parent cubic function graph include its shape, which resembles that of an elongated 'S' on its side, and the fact that it passes through the origin (0,0). This function exhibits symmetry with respect to the origin, meaning that it behaves the same in all quadrants. When graphing \( f(x) = x^3 \), you'll notice that as \( x \) takes on positive values, the \( f(x) \) values increase rapidly; similarly, as \( x \) becomes negative, \( f(x) \) decreases sharply. Recognizing this parent function is essential for understanding the effects of transformations applied to it.

Horizontal Shift

In the context of graph transformations, a 'horizontal shift' occurs when every point of a graph moves to the left or to the right by a certain number of units. This shift does not change the shape of the graph; it simply translates the entire graph horizontally.

In the equation \( g(x)=(x-h)^3 \), the value of 'h' determines the magnitude and direction of the horizontal shift from the parent function \( f(x)=x^3 \). If 'h' is positive, the graph shifts 'h' units to the right, and if 'h' is negative, the graph shifts 'h' units to the left. For the specific exercise given, the graph of \( g(x) \) is \( f(x) \) shifted 3 units to the right, as represented by the equation \( g(x)=(x-3)^3 \).

It's important for students to grasp that horizontal shifts do not affect the vertical position or the 'stretched' or 'shrunk' appearance of the graph - they solely influence its horizontal position.

Cubic Function Graphing

Graphing a cubic function involves understanding the effects of different transformations applied to the parent function. In our exercise, to graph \( g(x)=(x-3)^3 \) from \( f(x)=x^3 \), we apply a horizontal shift to the right by 3 units. Here's how we approach it:

- Begin with the parent function, recognizing its curve through the origin and its increase and decrease in the respective quadrants.
- Apply the transformation: Shift each point of the parent graph 3 units to the right. For example, the origin (0,0) on \( f(x) \) is moved to (3,0) on \( g(x) \).
- Retain the original shape of the curve while shifting, ensuring the graph of \( g(x) \) accurately reflects the transformation without distortion.

Graphing cubic functions in this manner, with a focus on systematic transformations, facilitates a better understanding of the effects that algebraic changes within the function’s equation have on the graph itself. This approach is key for students to visualize and predict the graph's behavior without plotting numerous points.