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|

Verified step by step guidance

Start by graphing the basic absolute value function, \( f(x) = |x| \), which is a V-shaped graph with its vertex at the origin (0,0) and opens upwards.

Identify the transformation needed for \( h(x) = -|x+3| \). Notice that \( |x+3| \) represents a horizontal shift of the graph of \( |x| \) to the left by 3 units.

Apply the horizontal shift to the graph of \( f(x) = |x| \). This means moving the vertex from (0,0) to (-3,0).

Next, apply the vertical reflection indicated by the negative sign in front of the absolute value. This reflection flips the graph over the x-axis, so the V-shape now opens downwards.

Combine these transformations to graph \( h(x) = -|x+3| \). The vertex of the graph is at (-3,0), and it opens downwards, forming an upside-down V-shape.

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 is V-shaped, with its vertex at the origin (0,0), and it reflects any negative input to positive. Understanding this function is crucial as it serves as the foundation for graphing transformations.

Transformations of Functions

Transformations involve shifting, reflecting, stretching, or compressing the graph of a function. In this case, the function h(x) = -|x+3| involves a horizontal shift to the left by 3 units and a vertical reflection across the x-axis. Mastery of these transformations allows for the manipulation of the base graph to achieve the desired function.

Graphing Techniques

Graphing techniques include plotting key points, identifying transformations, and understanding the overall shape of the function. For h(x) = -|x+3|, one would start with the graph of f(x) = |x|, apply the transformations, and then accurately sketch the new graph. Proficiency in these techniques is essential for visualizing and interpreting functions.

Watch next

Master Graphs of Common Functions with a bite sized video explanation from Patrick

Start learning