Question

graph the given functions, f and g, in the same rectangular coordinate system. describe how the graph of g is related to the graph of f.

f(x) = -2x

g(x) = -2x - 3

...

use the graphing tool to graph the functions.

click to enlarge graph (image of coordinate system with a line)

Explanation:

Step1: Identify Function Types

Both \( f(x) = -2x \) and \( g(x) = -2x - 3 \) are linear functions (in the form \( y = mx + b \), where \( m \) is slope and \( b \) is y - intercept). For \( f(x) \), \( m=-2 \), \( b = 0 \); for \( g(x) \), \( m=-2 \), \( b=-3 \).

Step2: Analyze Transformations

Linear functions with the same slope (\( m=-2 \) for both) are parallel. The difference is in the y - intercept. The function \( g(x) \) can be seen as \( f(x)-3 \), which means we take the graph of \( f(x) \) and shift it down by 3 units (since we subtract 3 from the output of \( f(x) \)).

Step3: Graphing (Conceptual)

• For \( f(x)=-2x \), when \( x = 0 \), \( y = 0 \); when \( x = 1 \), \( y=-2 \). So two points are \( (0,0) \) and \( (1, - 2) \).
• For \( g(x)=-2x - 3 \), when \( x = 0 \), \( y=-3 \); when \( x = 1 \), \( y=-2 - 3=-5 \). So two points are \( (0,-3) \) and \( (1, - 5) \). Plotting these points, we see that the line of \( g(x) \) is parallel to \( f(x) \) and shifted down 3 units.

Answer:

The graph of \( g(x)=-2x - 3 \) is the graph of \( f(x)=-2x \) shifted vertically down by 3 units. The two lines are parallel (since they have the same slope \( m = - 2\)).