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)= |x|, g(x) = |x| +1

Verified step by step guidance

1. Identify the parent function: The function \( f(x) = |x| \) is the absolute value function, which is V-shaped and symmetric about the y-axis.
2. Determine the transformation for \( g(x) = |x| + 1 \): This function is a vertical shift of \( f(x) \) upwards by 1 unit.
3. Create a table of values for \( f(x) = |x| \) using integers from -2 to 2: Calculate \( f(x) \) for each \( x \) value.
4. Create a table of values for \( g(x) = |x| + 1 \) using the same integers: Add 1 to each \( f(x) \) value to get \( g(x) \).
5. Graph both functions on the same coordinate system: Plot the points from both tables and draw the V-shaped graphs, noting the vertical shift of \( g(x) \) compared to \( f(x) \).

Key Concepts

Absolute Value Function

The absolute value function, denoted as f(x) = |x|, outputs the non-negative value of x. This means that for any input x, the function reflects negative values to positive ones, creating a V-shaped graph that opens upwards. Understanding this function is crucial for analyzing how it behaves across different intervals of x.

Vertical Shifts

A vertical shift occurs when a function is adjusted by adding or subtracting a constant from it. In the case of g(x) = |x| + 1, the '+1' indicates that the entire graph of f(x) = |x| is moved up by one unit. This concept is essential for understanding how the graph of g relates to that of f.

Graphing Functions

Graphing functions involves plotting points on a coordinate system based on the function's output for various input values. For the given functions, selecting integer values for x from -2 to 2 allows for a clear visualization of both f and g. This process helps in identifying transformations and relationships between the graphs.