In Exercises 81–94, begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. g(x) = -2|x+3|+2

Verified step by step guidance

1. Start by graphing the basic absolute value function, \( f(x) = |x| \). This graph is a V-shaped graph with its vertex at the origin (0,0) and opens upwards.
2. Identify the transformations needed to graph \( g(x) = -2|x+3|+2 \) based on the basic graph of \( f(x) = |x| \).
3. Apply a horizontal shift to the graph of \( f(x) = |x| \) by moving it 3 units to the left. This is due to the \( x+3 \) inside the absolute value, resulting in \( |x+3| \).
4. Apply a vertical stretch and reflection. The coefficient \(-2\) indicates a vertical stretch by a factor of 2 and a reflection over the x-axis. This changes the graph to open downwards.
5. Finally, apply a vertical shift by moving the graph 2 units up. This is due to the \(+2\) outside the absolute value, resulting in the final graph of \( g(x) = -2|x+3|+2 \).

Key Concepts

Absolute Value Function

The absolute value function, denoted as f(x) = |x|, outputs the non-negative value of x. This function has a V-shaped graph that opens upwards, with its vertex at the origin (0,0). Understanding this function is crucial as it serves as the foundation for applying transformations to graph other functions.

Transformations of Functions

Transformations involve shifting, reflecting, stretching, or compressing the graph of a function. For example, adding a constant inside the absolute value affects horizontal shifts, while adding outside affects vertical shifts. In the function g(x) = -2|x+3|+2, the transformations include a horizontal shift left by 3 units, a vertical stretch by a factor of 2, and a reflection across the x-axis.

Graphing Techniques

Graphing techniques involve plotting points and understanding how transformations affect the shape and position of a graph. For the function g(x), one must first graph f(x) = |x|, then apply the identified transformations systematically. This process helps visualize how the original graph is altered, leading to an accurate representation of the new function.