Problem 48 - Precalculus Essentials

Question

Graph the given functions, \(f\) and \(g,\) in the same rectangular coordinate system. Select integers for \(x,\) starting with -2 and ending with \(2 .\) Once you have obtained your graphs, describe how the graph of \(g\) is related to the graph of \(f .\) $$f(x)=x^{3}, g(x)=x^{3}-1$$

Step-by-Step Solution

Answer

The graph of the function \(g(x) = x^{3} - 1\) is identical in shape to the graph of the function \(f(x) = x^{3}\), but it is shifted down by 1 unit.

Step 1: Calculate the values

To start, calculate the values of each function for the given range of \(x\) values, i.e., from -2 to 2. To do so, substitute each of these values into both equations to get the corresponding \(y\) values. For \(f(x)=x^{3}\), the \(y\) values will be for \(x = -2, -1, 0, 1, 2\) are -8, -1, 0, 1, 8 respectively. For \(g(x)=x^{3}-1\), the \(y\) values when \(x = -2, -1, 0, 1, 2\) are -9, -2, -1, 0, 7 respectively.

Step 2: Plot the points

The second step is to plot these \(x, y\) coordinates on the graph. For \(f(x)=x^{3}\), plot the points (-2, -8), (-1, -1), (0,0), (1,1), and (2,8). For \(g(x)=x^{3}-1\), plot the points (-2, -9), (-1, -2), (0,-1), (1,0), and (2,7).

Step 3: Draw the graphs

The next step is to draw a curve through the plotted points for each function.

Step 4: Compare the graphs

Based on the observed graphs for \(f(x)\) and \(g(x)\), we can note that the graph of \(g(x)=x^{3}-1\) is identical in shape to the graph of \(f(x)=x^{3}\). It has just been translated downward by 1 unit. This is because the effect of subtracting a constant from a function is to shift the graph of the function vertically.

Key Concepts

Rectangular Coordinate System

To properly graph a function, you need to understand the rectangular coordinate system, also known as the Cartesian plane. This system is a two-dimensional plane consisting of two perpendicular lines called axes. The horizontal axis is known as the x-axis, and the vertical axis is known as the y-axis. Together, they divide the plane into four quadrants. Each point on the plane is represented by a pair of numbers either (x, y). The first number, x, indicates the position to the left or right of the vertical axis, and the second number, y, represents the position above or below the horizontal axis. When graphing functions, like the cubic functions in our example, we calculate the y-values for each chosen x-value, and then plot these points on the Cartesian plane. Once plotted, we can connect the dots to form the graph of the function.

Function Transformation

Function transformation involves changing a graph's position or shape through various operations, such as shifting, stretching, or reflecting. In the given exercise, we see one of the simplest forms of transformation: a vertical shift. The function g(x) = x^{3} - 1 can be thought of as the parent function f(x) = x^{3} that has been shifted downward by one unit. This shift does not alter the shape or orientation of the graph; it simply moves it up or down. Such transformations are essential in understanding how different parts of a function's equation affect its graphical representation.

Polynomial Graph

A polynomial graph represents functions that include terms like x^{3}, x^{2}, x, and constants. The graph of a cubic function, which is a type of polynomial function with a degree of three, typically has an 'S' shaped curve. In the provided exercise, we are asked to graph the functions f(x) = x^{3} and g(x) = x^{3} - 1. The resulting graphs will resemble each other in shape with one key difference: the g(x) function is a translated version of f(x), shifted vertically. This is evidenced by their equations – g(x) is simply f(x) subtracted by 1. By mastering the graphing of polynomial functions, students can visualize the effects that different polynomial terms have on the graph's overall shape and position on the rectangular coordinate system.