Problem 130 - 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 obtained by reflecting \(f(x)=x^{3}\) around the x-axis and then shifting 3 units to the left.

Step 1: Sketch the graph of \(f(x)=x^{3}\)

The function \(f(x)=x^{3}\) is a basic cubic function. It passes through the origin (0,0), curves upwards to the right and downwards to the left, with a slope that increases as you move to the right and decreases as you move to the left.

Step 2: Understanding the transformation

The function \(g(x)=-(x+3)^{3}\) is a transformed version of \(f(x)=x^{3}\). The negative sign in front of the function reflects the graph about the x-axis. Meanwhile, the addition of 3 inside the function (i.e. replacing \(x\) with \(x+3\) ) causes a horizontal shift of 3 units to the left.

Step 3: Sketch the graph of \(g(x)=-(x+3)^{3}\)

Having noted the transformations, we mirror the graph of \(f(x)=x^{3}\) about the x-axis and then shift the mirrored graph 3 units to the left to get the graph of \(g(x)=-(x+3)^{3}\). This results in a graph that passes through point (-3,0), curves downwards to the right and upwards to the left.

Key Concepts

Graph Transformations

When we talk about graph transformations, we mean altering the appearance or position of a graph in a systematic way. In the case of polynomial functions such as cubics, these transformations can be quite visually striking. When you take a basic cubic function like \(f(x) = x^3\), it presents a signature shape known for its snazzy curve that passes through the origin, rises upward to the right, and dips downward to the left.

Now, applying transformations changes this default appearance. There are several types of transformations, including:

• Reflection: Flipping the graph over a particular axis.
• Translation: Shifting the graph left, right, up, or down.
• Stretching or Compressing: Altering the steepness or breadth of the graph.

For instance, a reflection across the x-axis takes every point on the graph and flips it to the opposite side of the x-axis. Translation, on the other hand, would take the entire graph and slide it horizontally or vertically without altering its shape.

Polynomial Functions

Polynomial functions are expressions that involve sums of powers of variables. They can be of different degrees, and here we focus on degree three, making it a cubic function. Consider \(f(x) = x^3\), which is foundational in analyzing the behavior of higher-degree polynomials too.

What makes cubic functions special is their unique curve that can pass through the origin — displaying one local maximum and one local minimum or simply continuously increasing or decreasing. The function \(f(x) = x^3\) is known for its smoothness and continuous curve which doesn't sharply turn but instead smoothly transitions in its ascent and descent to infinity.

Some key properties of polynomials include:

• The degree indicated by the highest exponent, dictating the number of turning points.
• They are continuous, smooth without breaks or gaps.
• Exhibit symmetrical behavior in certain forms, particularly cubes with \(x^3\).

These characteristics help delineate why transformations result in predictable shifts and reflections rather than chaotic alterations.

Reflection and Translation

Two primary transformations involved in our exercise are reflection and translation. Let's start with reflection, which gives our graph a mirror image. This is achieved by multiplying the function by -1. In our example of \(g(x) = -(x+3)^3\), this results in the graph flipping over the x-axis.

Translation involves moving the graph horizontally or vertically. For \(g(x) = -(x+3)^3\), the addition inside the cubed term \((x+3)\) signals a horizontal movement. Because it's addition inside the function, it actually shifts the curve horizontally to the left, not the direction you might expect! Specifically, moving the entire curve 3 units left.

With reflection and translation combined, the graph that looked up and to the right (for \(f(x)=x^3\)) now looks down and to the right (for \(g(x)=-(x+3)^3\)), beginning at point (-3,0). Each transformation is important as it repositions or reshapes the graph while maintaining the integrity of the original curve's properties.