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

Step-by-Step Solution

Answer

The graph of \(g(x) = |x| - 2\) is the graph of \(f(x) = |x|\), shifted down by 2 units.

Step 1: Understanding the Functions

The first function \(f(x) = |x|\), is the absolute value function, which results in a 'V' shaped graph. The second function \(g(x) = |x| - 2\), is a transformation of the absolute value function, where the 'V' shape is vertically shifted down by 2 units.

Step 2: Selecting Values for x

Get the x values from -2 to 2, which are -2, -1, 0, 1, 2. Substitute these x-values into the functions.

Step 3: Substituting x into the function \(f(x)\)

Substitute x values into function \(f(x) = |x|\). The resulting outputs are \(f(-2) = 2\), \(f(-1) = 1\), \(f(0) = 0\), \(f(1) = 1\), \(f(2) = 2\). These give you the points (-2,2), (-1,1), (0,0), (1,1), (2,2) to plot on the graph.

Step 4: Substituting x into the function \(g(x)\)

Substitute x values into function \(g(x) = |x| - 2\). The resulting outputs are \(g(-2) = 0\), \(g(-1) = -1\), \(g(0) = -2\), \(g(1) = -1\), \(g(2) = 0\). These give you the points (-2,0), (-1,-1), (0,-2), (1,-1), (2,0) to plot on the graph.

Step 5: Plotting the Functions

On a graph, plot the points for \(f(x)\) and \(g(x)\). For \(f(x)\), the points should create the 'V' shape centered at origin, while for \(g(x)\), the 'V' shape should be centered 2 units below the origin.

Step 6: Describing the Relationship

By inspecting the graphs, it's evident that \(g(x) = |x| - 2\) is a vertical shift down by 2 units of the graph of \(f(x) = |x|\). Both graphs remain symmetrical to the y-axis.

Step 7: Writing the Final Answer

Conclude that the graph of \(g(x) = |x| - 2\) is the graph of \(f(x) = |x|\), shifted downward by 2 units.

Key Concepts

Absolute Value Function

The absolute value function is a fundamental concept in graphing functions. Represented by the notation \( f(x) = |x| \), this function involves taking the absolute value of any given input \( x \). The absolute value itself is defined as the non-negative value of \( x \). It turns the value positive if it is negative and leaves it unchanged if it is already positive.

In terms of graphing, the absolute value function creates a 'V' shaped plot on the rectangular coordinate system. This occurs because the function mirrors the negative side of the x-axis onto the positive side. The vertex of this 'V' is found at the origin point \((0, 0)\), where the function changes from decreasing to increasing.

• For \(x \geq 0\), the output is simply the input: \(f(x) = x\).
• For \(x < 0\), the output is the negation of the input: \(f(x) = -x\).

This basic 'V' shape and its symmetry are central characteristics of the absolute value function.

Vertical Shift

A vertical shift is a transformation that results in the movement of a graph up or down along the y-axis. This transformation does not affect the shape or horizontal symmetry of the graph, only its vertical position.

Let's consider the function \( g(x) = |x| - 2 \). Compared to the parent function \( f(x) = |x| \), the graph of \( g(x) \) is shifted downward by 2 units. This translation is understood by the \(-2\) in the function, which reduces every y-value of \( f(x) \) by 2.

• The vertex of \( g(x) = |x| - 2 \) shifts from \((0, 0)\) to \((0, -2)\).
• Every point on \( f(x) \) simply moves 2 units lower to form the graph of \( g(x) \).

Vertical shifts are a straightforward way to reposition the graph of a function without altering its inherent geometry or symmetry characteristics.

Symmetry

Symmetry in graphs helps to make predictions about a function's behavior and appearance. The symmetry of the absolute value function \( f(x) = |x| \) is a prime example. The function is symmetric with respect to the y-axis, known as y-axis symmetry.

You can determine this symmetry because when you substitute \(-x\) for \(x\), the absolute value function gives the same result: \( |x| = |-x| \). This implies that whenever you reflect the graph across the y-axis, the result remains unchanged.

• Positive and negative values of \( x \) give the same distance from the y-axis, producing a mirror-like quality.
• This symmetry aspect holds even after vertical transformations, like shifting down by 2 in the equation \( g(x) = |x| - 2 \).

Symmetry in such functions helps in easing the verification of the graph and in establishing the relationship between different parts of the function.

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a two-dimensional plane for graphing functions and equations. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).

This system allows us to plot points and graph functions by using ordered pairs \((x, y)\). The x-axis represents input values, while the y-axis represents output values of a function.

• The point \( (0,0) \), where the axes intersect, is the origin.
• Positive values are plotted to the right (for x) and above (for y) from the origin, while negative values are to the left and below.

When graphing functions like \( f(x) = |x| \) and \( g(x) = |x|-2 \), we use this system to visually represent and analyze their behavior. The clarity of this coordinate system makes it easier to interpret transformations and compare different functions at a glance.