Sketch a graph y = |x – 3| – 2 and describe the
translations.
To sketch the graph of y = |x – 3| – 2, let's first analyze the parent function y = |x|.
The parent function y = |x| produces a V-shaped graph with the vertex at (0, 0). For positive values of x, y = x, and for negative values of x, y = -x. Therefore, the graph is always positive.
Now let's analyze the given equation y = |x – 3| – 2. This equation modifies the parent function by applying two translation operations: a horizontal translation and a vertical translation.
1. Horizontal Translation:
The "-3" inside the absolute value function shifts the graph horizontally to the right by 3 units. This means the vertex, which was initially at (0, 0), would now be at (3, 0).
2. Vertical Translation:
The "-2" subtracted from the absolute value function shifts the graph vertically downwards by 2 units. This means the vertex, which was initially at (3, 0), would now be at (3, -2).
By using these translations, we can graph the equation y = |x – 3| – 2.
The graph would appear as follows:
(assuming x-axis is horizontal and y-axis is vertical)
|
|
|
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_/|___
| - - - - -
| - -
| - -
X--------------------
| - -
| - -
| - -
|______-
|
|
The vertex of the graph would be at (3, -2). The graph forms an upside-down V above the x-axis, centered at x = 3, with its tip at (3, -2).