Problem 50 - 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 \(f .\) $$f(x)=-1, g(x)=4$$

Step-by-Step Solution

Answer

The graph of \(f(x)=-1\) is a horizontal line at \(y=-1\) and the graph of \(g(x)=4\) is a horizontal line at \(y=4\). \(g(x)\) is a vertical shift 5 units up from \(f(x)\).

Step 1: Understanding the Functions

Firstly, it's essential to understand that \(f(x)=-1\) and \(g(x)=4\) are constant functions. This means regardless of the value of \(x\), the output of \(f(x)\) will always be \( -1\) and the output of \(g(x)\) will always be \(4\).

Step 2: Create the Graph

For \(x\) values ranging from -2 to 2, plot the points \((-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1)\) for function \(f(x)\) and \((-2, 4), (-1, 4), (0, 4), (1, 4), (2, 4)\) for function \(g(x)\). Also, draw horizontal lines through these points.

Step 3: Comparing the Graphs of \(f\) and \(g\)

The graphs of both functions are horizontal lines. The graph of \(g(x)\) is positioned above the graph of \(f(x)\) because the value output by \(g(x)\) is greater than the value output by \(f(x)\). Additionally, since the difference between the \(y\)-values of \(g\) and \(f\) is \(5\), we can say that \(g(x)\) is a vertical shift of \(5\) units up from \(f(x)\).

Key Concepts

Constant Functions

Constant functions are among the simplest types of functions you will encounter in algebra. A constant function is a function that always returns the same value regardless of what the input is. In mathematical terms, a constant function can be expressed as \( f(x) = c \) where \( c \) is a fixed number.

This means that for any value of \( x \), the output \( f(x) \) remains constant. It's neither increasing nor decreasing. For example, if \( f(x) = -1 \), as in the given problem, then for any \( x \) you choose, \( f(x) \) will always equal \(-1\). Similarly, if \( g(x) = 4 \), no matter what \( x \) is, \( g(x) \) will always equal \(4\).

• The graph of a constant function is a straight horizontal line.
• The line is parallel to the x-axis.
• All points on the line have the same y-coordinate.

Horizontal Lines

When graphing functions like \( f(x) = -1 \) or \( g(x) = 4 \), we see horizontal lines on a coordinate plane. A horizontal line is a line where every point on the line has the same \( y \)-value. This type of line remains flat and does not slope upwards or downwards.

Horizontal lines are characterized by having an equation of the form \( y = c \), where \( c \) is the constant value of \( y \) for all points on the line. In our case:

• The function \( f(x) = -1 \) is a horizontal line at \( y = -1 \).
• The function \( g(x) = 4 \) is a horizontal line at \( y = 4 \).

Identifying these lines on a graph is straightforward. You just plot several points that have the constant \( y \)-value and connect them to form a straight line. Each point lying on the line for \( f(x) \) is below each corresponding point on the line for \( g(x) \).

Horizontal lines simplify many graphing tasks, especially linear equations where the slope \( m \) equals zero.

Vertical Shift

A vertical shift moves a graph up or down in a coordinate plane. Specifically, a vertical shift involves adding a constant to each output \( y \)-value of a function. In this exercise, the function \( g(x) = 4 \) represents a vertical shift of \( f(x) = -1 \).

To understand how this works, consider that \( g(x) \) is simply \( f(x) \) shifted upward by \(5\) units. Here's why:

• The difference between the \( y \)-values of \( g(x) \) and \( f(x) \) is calculated as \( 4 - (-1) = 5 \).
• This means every point on \( g(x) \) is vertically 5 units higher than the corresponding point on \( f(x) \).

This shift alters only the position of the graph relative to the \( x \)-axis and not its shape because horizontal lines remain unchanged when shifted up or down. When graphing, simply note that the shift doesn't affect the compact form or equation of the line, making plotting variations straightforward.