Here are the equations of the transformed graphs for each problem:
19.) To translate the graph of \( y = 2^x \) up 13 units, add 13 to the output: \[ y = 2^x + 13 \]
20.) To reflect the graph of \( y = 2^x \) in the x-axis, negate the entire function, and then translate down 8 units: \[ y = -2^x - 8 \]
21.) To reflect the graph of \( y = \sqrt{x} \) in the y-axis, replace \( x \) with \( -x \), and then translate down 10 units: \[ y = -\sqrt{-x} - 10 \]
22.) To translate the graph of \( y = x \) left 9 units and down 1 unit, change \( x \) to \( x + 9 \) and subtract 1: \[ y = (x + 9) - 1 = x + 8 \]
23.) To reflect the graph of \( y = x^2 \) in the x-axis, negate the entire function, and then translate 3 units to the right: \[ y = -(x - 3)^2 \]
24.) To translate the graph of \( y = x^3 \) left 5 units and up 1 unit, replace \( x \) with \( x + 5 \) and add 1: \[ y = (x + 5)^3 + 1 \]
25.) To reflect the graph of \( y = |x| \) in the x-axis, negate the entire function, then translate right 4 units and down 6: \[ y = -|x - 4| - 6 \]
These equations represent the transformed graphs based on the descriptions provided.
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