10. Sketch a graph y = |x – 3| – 2 and describe the translations.
To sketch the graph of y = |x – 3| – 2, we can start with the parent function y = |x| and apply two translations.
1. "x – 3": This translation shifts the entire graph to the right by 3 units. Any point (x, y) on the graph is transformed to (x – 3, y).
2. "- 2": This translation shifts the graph downward by 2 units. Each y-coordinate is decreased by 2 units.
Combining both translations, the graph of y = |x – 3| – 2 will resemble the absolute value function shifted to the right by 3 units and then shifted downward by 2 units.
The vertex or lowest point of the graph occurs when x – 3 = 0, giving x = 3. Therefore, the vertex is at (3, -2).
The line y = -2 is the horizontal asymptote for this graph. The graph approaches this line but never touches or crosses it.
Below is a rough sketch of the graph of y = |x – 3| – 2:
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-2|___/
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-3 -2 -1 0 1 2 3 4 5 6 7
Note: The graph is "V" shaped with the vertex at (3, -2).