Begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. g(x) = -|x + 4| +2

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Start with the base graph of the absolute value function, f(x) = |x|. This graph is a V-shaped graph with its vertex at the origin (0, 0), opening upwards.

Identify the transformation inside the absolute value: x + 4. This represents a horizontal shift. Specifically, the graph of |x| is shifted 4 units to the left because of the +4 inside the absolute value.

Next, consider the negative sign in front of the absolute value, -|x + 4|. This reflects the graph of |x + 4| across the x-axis, causing the V-shape to open downwards instead of upwards.

Now, account for the +2 outside the absolute value. This represents a vertical shift. The entire graph of -|x + 4| is shifted 2 units upward.

Combine all the transformations: Start with the base graph of |x|, shift it 4 units to the left, reflect it across the x-axis, and finally shift it 2 units upward. Plot the resulting graph to visualize g(x) = -|x + 4| + 2.

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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 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) = -|x + 4| + 2, the transformations include a horizontal shift left by 4 units, a reflection across the x-axis, and a vertical shift up by 2 units.

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 the changes and accurately represent the new function on a coordinate plane.