Problem 45 - 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 \(f .\) $$f(x)=|x|, g(x)=|x|-2$$

Step-by-Step Solution

Answer

The graph of \(g(x)\) is the graph of \(f(x)\) translated downwards by 2 units across the y-axis.

Step 1: Plot function \(f(x) = |x|\)

Start by plotting some \(x\) and \(f(x)\) values in a table. Considering -2, -1, 0, 1 and 2 for \(x\) would result in 2, 1, 0, 1 and 2 for \(f(x)\) respectively, forming a 'V’ shape if plotted on the coordinate system.

Step 2: Plot function \(g(x) = |x| - 2\)

Function \(g(x)\) adds an extra transformation to \(f(x)\), which is a vertical shift downwards by 2 units. So, with the same \(x\) values, \(g(x)\) results in 0, -1, -2, -1 and 0 respectively. If plotted on the same coordinate system, it would again form a 'V’ shape but instead centered at the point (0, -2).

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

From the plotted functions, it can be seen that the graph of \(g(x)\) is a translation of the graph of \(f(x)\) downwards by 2 units across the y-axis. Every point on \(f(x)\) is therefore moved down by 2 units to find the corresponding point on \(g(x)\).

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a two-dimensional plane consisting of two perpendicular axes—the horizontal x-axis and the vertical y-axis. The point where these axes intersect is called the origin, denoted as (0,0). The system is divided into four quadrants, each with a unique combination of sign (+/-) for the coordinates. When graphing functions like absolute value functions, we use this coordinate system to plot points that represent solutions to the function for different values of the 'x' variable. This visual representation helps students understand the behavior of the function across its domain.

Vertical Shifts

Vertical shifts are a type of function transformation that occurs when a constant is added or subtracted from the function's output. Specifically, adding a positive constant shifts the graph upwards, while subtracting a constant shifts it downwards. In the context of the exercise, the function transformation from f(x) = |x| to g(x) = |x| - 2 involves a vertical downward shift of 2 units. Every point on f(x) is simply moved down two units to find the corresponding point on g(x), which alters the position of the graph on the coordinate system but not its shape.

Function Transformation

Function transformation involves changing the appearance of the original graph of a function. This can be done through various means, including shifting, stretching, compressing, or reflecting. In our exercise, we perform a vertical translation. This concept is crucial for students to understand as it shows how a simple alteration to the function's equation can significantly affect its graph. Transformations follow specific rules and knowing them allows students to predict and plot the resulting graphs accurately without having to compute every single value.

Absolute Value Graph

The graph of an absolute value function typically takes the shape of a 'V' or an inverted 'V', depending on the coefficients and constants involved. The absolute value, denoted as |x|, defines the distance of a number from zero on the number line, and it is always non-negative. This results in the graph having a sharp corner at its vertex, which for the basic f(x) = |x| function is at the origin. Understanding the graph's characteristic shape is essential, as it provides a reference point for predicting the transformations that occur when additional terms are added to the function, such as in the g(x) function in the exercise.

Plotting Functions

Plotting functions involves calculating the outputs for a range of inputs and marking the resulting points on the coordinate plane. Starting with simple, crucial values makes it easier to see the general shape of the function's graph. In our exercise, selecting integers from -2 to 2 provides clear insight into the symmetry and shape of the absolute value graph. To facilitate a better understanding, plotting key points such as the vertex and intercepts helps construct the graph more accurately. This methodical approach simplifies the process for students, making the function's behavior more comprehensible.