Problem 18 - 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 graph of the function \(g(x) = x^{3} - 3\) is the graph of the function \(f(x) = x^{3}\) shifted down by 3 units on the y-axis.

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

Sketch the graph of the basic cubic function \(f(x) = x^{3}\). This function crosses the origin (0,0), and increases in value as x moves away from zero both in the positive and negative directions. This will give a curve that looks like an S on its side.

Step 2: Understand the Transformation

Examine the function \(g(x) = x^{3} - 3\). The '-3' in this function tells us that the graph of \(f(x)\) will be shifted vertically downwards by 3 units. In terms of transformations, this is known as a vertical translation.

Step 3: Sketch \(g(x) = x^{3} - 3\)

Sketch the graph of function \(g(x) = x^{3} - 3\). Start with the graph of \(f(x)\), then shift every point on that graph downwards by 3 units. This moves the curve of the graph down, so while it still has the general shape of a sideways S, it now crosses the y-axis at (0,-3) instead of at the origin.

Key Concepts

Cubic Function Transformations

When it comes to understanding cubic functions, one of the fundamental concepts is transformations. A transformation alters the appearance of a graph without changing its fundamental shape. In the case of cubic functions, such as the basic function f(x) = x^3, transformations can stretch or compress the graph, reflect it across an axis, or translate it up, down, left, or right.

For instance, if we begin with f(x) = x^3 and add a constant after the cubic term, like in g(x) = x^3 - 3, we are applying a vertical translation. The '-3' translates the graph of f(x) downward by 3 units. It's crucial to note that the curve's general 'sideways S' shape remains intact; the graph simply moves up or down in its entirety based on the value and sign of the constant.

Sketching Graphs

Sketching the graph of a cubic function can be a visual journey of understanding its behavior. The graph of a basic cubic function, f(x) = x^3, has distinct characteristics: it passes through the origin and has a point of inflection at the origin, where the curve changes from concave down to concave up.

In order to sketch the graph accurately, start by plotting key points such as the origin and then draw a smooth curve that extends infinitely in both directions, making sure to reflect the proper end behavior of the function. The curve starts in the bottom left quadrant, increasing at a decreasing rate until the origin, then it increases at an increasing rate into the top right quadrant. This visual representation can greatly aid in understanding further transformations applied to the function.

Horizontal and Vertical Translations

Horizontal and vertical translations are like sliding a graph across the coordinate plane. A horizontal translation shifts the graph left or right, while a vertical translation moves it up or down. To perform a vertical translation, you add or subtract a constant to the entire function as seen in g(x) = x^3 - 3. The '-3' indicates a downward shift.

Every point on the original graph moves in the same direction by the same amount. No stretching or compressing occurs, and the general shape and orientation of the graph do not change. It's as if the entire graph is picked up and repositioned along the vertical or horizontal axis. When sketching, especially in homework exercises, pay close attention to these translations to ensure the resulting graph reflects the correct movement from the original function.