In Exercises 39–50, 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) = -2x, g(x) = -2x-1

Verified step by step guidance

1. Step 1: Start by understanding the given functions. The first function is f(x) = -2x, which is a linear function with a slope of -2 and a y-intercept of 0. The second function is g(x) = -2x - 1, which is also a linear function with the same slope of -2 but a y-intercept of -1.
2. Step 2: Create a table of values for f(x) = -2x. Select integer values for x from -2 to 2. For each x-value, substitute it into f(x) to calculate the corresponding y-value. For example, when x = -2, f(x) = -2(-2) = 4. Repeat this for x = -1, 0, 1, and 2.
3. Step 3: Similarly, create a table of values for g(x) = -2x - 1. Again, substitute integer values for x from -2 to 2 into g(x) to calculate the corresponding y-values. For example, when x = -2, g(x) = -2(-2) - 1 = 3. Repeat this for x = -1, 0, 1, and 2.
4. Step 4: Plot the points from both tables on the same rectangular coordinate system. For f(x), plot the points (x, f(x)) and for g(x), plot the points (x, g(x)). Connect the points for each function to form straight lines, as both are linear functions.
5. Step 5: Analyze the relationship between the graphs of f(x) and g(x). Notice that the graph of g(x) is a vertical shift of the graph of f(x) by -1 unit. This means that every point on the graph of g(x) is 1 unit lower than the corresponding point on the graph of f(x).

Key Concepts

Linear Functions

Linear functions are mathematical expressions that create a straight line when graphed. They can be represented in the form f(x) = mx + b, where m is the slope and b is the y-intercept. Understanding the characteristics of linear functions, such as slope and intercepts, is essential for analyzing their graphs and relationships.

Graphing Techniques

Graphing techniques involve plotting points on a coordinate system to visualize mathematical functions. For linear functions, selecting integer values for x allows for easy calculation of corresponding y values. This process helps in accurately representing the function's behavior and identifying key features such as intersections and shifts.

Transformations of Functions

Transformations of functions refer to changes made to the graph of a function, such as translations, reflections, and dilations. In this case, the function g(x) = -2x - 1 is a vertical translation of f(x) = -2x, shifted down by one unit. Understanding these transformations is crucial for comparing and relating the graphs of different functions.