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. h(x) = |x + 3| - 2

Verified step by step guidance

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.

Identify the transformations needed to graph \( h(x) = |x + 3| - 2 \). The expression \( |x + 3| \) indicates a horizontal shift.

The term \( x + 3 \) inside the absolute value function represents a horizontal shift to the left by 3 units. This means the vertex of the graph will move from (0,0) to (-3,0).

The \( -2 \) outside the absolute value function represents a vertical shift downward by 2 units. This means the vertex will move from (-3,0) to (-3,-2).

Combine these transformations: shift the graph of \( f(x) = |x| \) left by 3 units and down by 2 units to obtain the graph of \( h(x) = |x + 3| - 2 \). The new vertex is at (-3,-2), and the graph maintains its V-shape, opening upwards.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 graphing transformations.

Graph Transformations

Graph transformations involve shifting, reflecting, stretching, or compressing the graph of a function. In the case of h(x) = |x + 3| - 2, the graph of f(x) = |x| is shifted left by 3 units and down by 2 units. Recognizing these transformations allows for the accurate graphing of modified functions.

Vertex of a Function

The vertex of a function is the point where the graph changes direction, often representing a minimum or maximum value. For the absolute value function, the vertex is at (0,0). In the transformed function h(x), the vertex shifts to (-3, -2), which is essential for accurately plotting the new graph.

Watch next

Master Intro to Transformations with a bite sized video explanation from Patrick

Start learning