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) = x², g(x) = x² - 2

Verified step by step guidance

1. Step 1: Begin by understanding the given functions. The first function, f(x) = x², represents a basic parabola that opens upwards with its vertex at the origin (0, 0). The second function, g(x) = x² - 2, is a transformation of f(x) where the entire graph is shifted downward by 2 units.
2. Step 2: Create a table of values for f(x) = x². Select integer values for x from -2 to 2. For each x-value, calculate f(x) by squaring x. For example, when x = -2, f(x) = (-2)² = 4. Repeat this for all x-values.
3. Step 3: Create a table of values for g(x) = x² - 2. Use the same x-values from -2 to 2. For each x-value, calculate g(x) by squaring x and then subtracting 2. For example, when x = -2, g(x) = (-2)² - 2 = 4 - 2 = 2. Repeat this for all x-values.
4. Step 4: Plot the points for both functions on the same rectangular coordinate system. For f(x), plot the points (x, f(x)) from the table created in Step 2. For g(x), plot the points (x, g(x)) from the table created in Step 3.
5. Step 5: Analyze the graphs. Notice that the graph of g(x) = x² - 2 is identical in shape to the graph of f(x) = x², but it is shifted downward by 2 units. This vertical shift is due to the '-2' in the equation g(x).

Key Concepts

Graphing Functions

Graphing functions involves plotting points on a coordinate system based on the function's output for given input values. For the functions f(x) = x² and g(x) = x² - 2, you will calculate the output for selected integer values of x, which helps visualize the shape and behavior of the functions. Understanding how to plot these points accurately is essential for comparing the two graphs.

Transformation of Functions

Transformation of functions refers to the changes made to the graph of a function based on modifications to its equation. In this case, g(x) = x² - 2 represents a vertical shift of the graph of f(x) = x² downward by 2 units. Recognizing these transformations allows for a better understanding of how the graphs relate to each other.

Vertex of a Parabola

The vertex of a parabola is the highest or lowest point on its graph, depending on its orientation. For the function f(x) = x², the vertex is at the origin (0,0), while for g(x) = x² - 2, the vertex shifts to (0,-2). Identifying the vertex is crucial for understanding the overall shape and position of the parabolas in relation to each other.