Problem 47 - 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 \(f .\) $$f(x)=x^{3}, g(x)=x^{3}+2$$

Step-by-Step Solution

Answer

The graph of function \(g(x)\) is a vertical shift upward by 2 units from the graph of function \(f(x)\).

Step 1: Define the Functions and Range

First of all, recognize the two functions are \(f(x) = x^3\) and \(g(x) = x^3 + 2\). The range of values for \(x\) that need to be considered are from -2 to 2.

Step 2: Calculate the Function Values

To plot the graph for each function, calculate the corresponding y-values by substituting the x-values (-2, -1, 0, 1, 2) into each function respectively.

Step 3: Plot the Points and Draw the Graph

Now, plot the pairs of x and y values in the Cartesian plane for each function respectively, interlink the plotted points with smooth lines to draw the curve.

Step 4: Compare and Describe the Relations

Compare the two graphs. In particular, focus on how the function \(g(x)\) is a vertical shift of the function \(f(x)\). Describe the relation as part of the solution.

Key Concepts

Rectangular Coordinate System

Imagine facing a map where two lines cross each other: one running from left to right and another from top to bottom. In mathematics, this map is known as the rectangular coordinate system or Cartesian plane, named after René Descartes. It is a two-dimensional plane divided by two perpendicular number lines: the horizontal axis (x-axis) and the vertical axis (y-axis).

The intersection of these axes is called the origin, where both x and y have a value of zero. Every point on this plane can be described by an ordered pair of numbers, \( (x, y) \), known as coordinates. The first number, x, corresponds to a position along the horizontal axis, and the second number, y, corresponds to a position along the vertical axis.

For graphing polynomial functions like \( f(x) = x^3 \), we use this system to plot the values of x and their corresponding y-values, which are the function's outputs. As a vital tool, understanding this coordinate system allows for visualizing how functions behave.

Cubic Functions

Polynomial functions include a variety of functions characterized by terms which are powers of x. A cubic function is a specific type of polynomial function where the highest degree of the variable x is three. The general form can be written as \( f(x) = ax^3 + bx^2 + cx + d \), where a, b, c, and d are constants with \( a \eq 0 \).

The graph of a cubic function is known for its distinctive S-shape, which can have one or two turns depending on the values of its coefficients. For the basic function \( f(x) = x^3 \), the graph will pass through the origin and exhibit symmetry around the origin, indicating that it's an odd function.

Understanding the shape and properties of cubic functions is important for graphing and analyzing their behavior on a rectangular coordinate system. They are used to model real-world situations with three variables and can show complex relationships between variables.

Function Transformation

When a function undergoes a transformation, its graph is altered in specific ways. Transformations include shifts, stretches, reflections, and compressions. In the context of graphing polynomial functions, grasping the concept of transformations is key to understanding how changes in a function's equation impact its graph.

There are a few basic types of transformations:

• Horizontal shifts occur when we add or subtract a constant from the x-variable, moving the graph left or right.
• Vertical shifts happen when we add or subtract a constant from the entire function, moving the graph up or down.
• By multiplying the x-variable by a constant, we induce a horizontal stretch or compression, widening or narrowing the graph horizontally.
• When the entire function is multiplied by a constant, a vertical stretch or compression occurs, affecting the graph's height.

Each transformation brings a new layer to the function's graph, revealing the intricate ways in which the function's formula shapes its graphical representation.

Vertical Shift

A vertical shift is a specific type of function transformation where every point on the graph of a function slides the same distance in the same direction along the y-axis. In other words, for the function \( f(x) \), if we add or subtract a constant value, \( k \), to form a new function \( g(x) = f(x) + k \), the graph of \( g(x) \) will simply be the graph of \( f(x) \) moved up or down.

In the initial exercise, the function \( g(x) = x^3 + 2 \) represents a vertical shift of the function \( f(x) = x^3 \) by two units upwards. This is because each output value \( y \) from \( f(x) \) has been increased by two to get the corresponding output value for \( g(x) \).

If you can master identifying such shifts in functions, you will be better prepared for both analyzing real-world data and grasifying more complex mathematical concepts. A vertical shift doesn't change the shape or orientation of a graph; it simply translates it, preserving its original features and making it a handy tool to compare related functions.