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Chapter 1: Problem 85

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

Short Answer

Expert verified
The graph of the function \(h(x) = |x+4|-2\) is the graph of the absolute value function \(f(x)=|x|\), shifted to the left by 4 units and down by 2 units. The vertex of the graph is at the point (-4,-2).

Step by step solution

01

Draw Base Function

Start by sketching the base function \(f(x)=|x|\) which is a V-shape centered at the origin. This means the vertex of the V is at (0,0), and the left and right arms of the v shape are lines that have slope of -1 and 1 respectively.
02

Apply horizontal shift

The expression \(x+4\) inside the absolute value brackets instructs a horizontal shift. The graph of the function will move to the left 4 units because of '+4'. Shift the vertex from (0,0) to (-4, 0) and the rest of the graph accordingly.
03

Apply vertical shift

The number \(-2\) outside the absolute value brackets suggests a vertical shift. This will move the graph downward by 2 units. Shift the vertex from (-4,0) to (-4,-2) and the rest of the graph accordingly.
04

Draw the Final Graph

After applying the transformations correctly, sketch the final graph. The graph should look like the original V-shaped graph but shifted 4 units to the left and 2 units down.

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Most popular questions from this chapter

Define a piecewise function on the intervals \((-\infty, 2],(2,5)\) and \([5, \infty)\) that does not "jump" at 2 or 5 such that one piece is a constant function, another piece is an increasing function, and the third piece is a decreasing function.

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