Problem 47 - College Algebra 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 function \(g(x) = x^3 + 2\) is the graph of \(f(x) = x^3\) shifted upwards by 2 units.

Step 1: Plot points for the function \(f(x) = x^3\)

Choose integer values for \(x\) from -2 to 2. Calculate \(f(x)\) for each chosen \(x\) value, resulting in pairs of \((x, y)\) that can be plotted. For instance, if \(x = -2\), \(f(x) = (-2)^3 = -8\). Repeat this process for all \(x\) values.

Step 2: Plot points for the function \(g(x) = x^3 + 2\)

Now use the same \(x\) values to determine corresponding \(y\) values for the function \(g(x)\), using the formula \(g(x) = x^3 + 2\). For example, if \(x = -2\), \(g(x) = (-2)^3 + 2 = -6\). Repeat this process for all \(x\) values.

Step 3: Draw Function Graphs

After calculating the corresponding \(y\) values for both \(f(x)\) and \(g(x)\), plot these as points on the same coordinate system and connect them to draw smooth curves for each function.

Step 4: Analyzing the Relationship

Examine the two function graphs and explain how they are related. In this case, it can be seen that the function \(g(x) = x^3 + 2\) is simply the graph of the function \(f(x) = x^3\), translated 2 units upwards.

Key Concepts

Coordinate System

When we talk about graphing functions like f(x) = x^3 and g(x) = x^3 + 2, the coordinate system—often referred to as the Cartesian coordinate system—is the stage on which all the action takes place. It consists of two number lines that intersect at a right angle. The horizontal line is called the x-axis, and the vertical line is the y-axis. Together, these axes divide the plane into four quadrants, each representing a unique combination of positive and negative values of x and y.

For a clear visualization, you should start by marking evenly spaced points along both axes, ensuring that the units are consistent. These marks serve as reference points to accurately plot the points that represent the solutions to the function for different values of x. Through plotting these points and connecting them, we transform numerical data into a visual graph that provides insights into the behavior of the function.

Function Translation

When observing changes between graphs of different functions, 'function translation' is a key concept to understand. In algebra, translation refers to shifting a graph horizontally or vertically without altering its shape. For the functions at hand, f(x) = x^3 forms a base graph, and g(x) = x^3 + 2 is its translated counterpart.

In this specific instance, since g(x) is f(x) plus 2, it results in a vertical translation. Every point on the graph of f(x) is moved up by two units to find its corresponding point on g(x). This means the shape and direction of the graph remain consistent, but its position on the coordinate plane changes, resulting in a visible and measurable shift of the entire graph.

Plotting Points in Algebra

Plotting points is a fundamental skill in algebra that allows us to visually represent functions. When we plot points, we translate an (x, y) pair from a numerical expression into a visual representation on the coordinate plane. Each point corresponds to a solution of the function where 'x' is the input and 'y' is the output.

To plot points, you start with an x value, then compute the corresponding y value using the function. For f(x) = x^3, picking integer values for x and cubing them gives us our y values. Consequently, the pair (x, y) can be plotted on the coordinate system. Remembering to mark each point clearly and then connect them with a smooth curve helps to visually convey the behavior of the function across different values of x.

Analyzing Relationships Between Functions

Analyzing the relationship between functions such as f(x) and g(x) is pivotal in understanding how algebraic manipulation can affect the graph of a function. Comparing these graphs aids in recognizing patterns and transformations, such as translations, reflections, and stretches.

For the functions given, we start by noting that both have similar terms except for the constant +2 in g(x). On a graph, this introduces a consistent vertical shift, but it does not affect the curvature or orientation of the original function's graph. Recognizing this type of relationship assists in predicting the graphs of more complex functions and understanding concepts such as function composition and inverse functions. It also forms a basis for solving real-world problems where changes in one variable can directly impact the value of another.