Problem 49 - 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 \(f .\) $$f(x)=3, g(x)=5$$

Step-by-Step Solution

Answer

The functions \(f(x) = 3\) and \(g(x) = 5\) are both straight horizontal lines. The graph of \(g(x)\) lies 2 units above the graph of \(f(x)\), hence a vertical shift upward by 2 units.

Step 1: Determine function values

Function \(f(x) = 3\) which implies that, regardless of the value for \(x\), the result will always be 3. Similarly, for function \(g(x) = 5\), it will always be 5 regardless of the value of \(x\). Now let's substitute \(x\) with the values from -2 to 2 into both functions for plotting purposes.

Step 2: Plot the functions

Create a graph with \(x\) ranging from -2 to 2. Plot the function \(f(x) = 3\) as a straight line parallel to the x-axis at point y=3. Similarly, plot \(g(x) = 5\) as another straight line parallel to the x-axis at y=5.

Step 3: Describe the relationship between \(f\) and \(g\)

Having obtained the graphs, observe and describe their relationship. From the plots, we can see that both \(f\) and \(g\) are horizontal straight lines, however, \(g\) spreads above \(f\). This is because the constant value '5' of \(g(x)\) is greater than the constant value '3' of \(f(x)\). Therefore, \(g(x)\) is a vertical shift of \(f(x)\) up by 2 units.

Key Concepts

The Rectangular Coordinate System

Understanding the basics of the rectangular coordinate system, often referred to as the Cartesian coordinate system, is essential in graphing functions. It is made up of two perpendicular lines, called axes, which intersect at a point known as the origin. The horizontal axis is labeled the x-axis, and the vertical axis is labeled the y-axis.

Each point in this system is defined by an ordered pair of numbers, \( (x, y) \), representing its horizontal and vertical positions. For example, the point \( (2, -3) \) indicates that we move 2 units to the right of the origin along the x-axis and 3 units down along the y-axis. This system allows us to plot mathematical functions like lines, parabolas, and others with precision and clarity.

Function Plotting

When plotting functions, each type of equation will correspond to a particular shape or pattern on the graph. To plot a function, you start by identifying key points called 'coordinates' that satisfy the function's equation, and then you mark these points on your coordinate system.

To plot a horizontal line, like \( f(x) = 3 \) or \( g(x) = 5 \) from the given exercise, you simply find all the points where the y-value is equal to the function's constant value. Since these lines do not change elevation as we move along the x-axis, every point on the line \( f(x) \) will have a y-value of 3, and for \( g(x) \) it will be 5, regardless of the x-value. This results in a straight, horizontal line across the graph.

Vertical Shift

A vertical shift in graphing is when a function moves up or down on a coordinate system, without changing its shape. Vertical shifts are typically achieved by adding or subtracting a constant to the function's equation. A positive constant will move the graph upward, while a negative constant will shift it downward.

In the step-by-step solution provided, \( g(x) = 5 \) is a vertical shift of \( f(x) = 3 \) because it takes the horizontal line for \( f(x) \) and shifts it up 2 units. As such, these two lines remain parallel to each other and to the x-axis. Plotting \( f(x) \) and \( g(x) \) helps visualize that while both functions are horizontal lines, \( g(x) \) is always situated above \( f(x) \) due to this vertical shift.