Answer:
The graph of y = (2/(x+5)) - 3 is a transformation of the graph of y = 1/x, which has a vertical asymptote at x = 0 and passes through the point (1,1), by a horizontal shift of 5 units to the left, a vertical stretch of 2 units, and a downward shift of 3 units.
Explanation:
The graph of y = (2/(x+5)) - 3 can be obtained by applying vertical and horizontal transformations to the graph of y = 1/x.
Specifically, we can describe the graph of y = 1/x as the parent function. It has a vertical asymptote at x = 0, a horizontal asymptote at y = 0, and it passes through the point (1,1).
The graph of y = (2/(x+5)) - 3 can be obtained by first shifting the graph of y = 1/x horizontally to the left by 5 units, then vertically stretching it by a factor of 2, and finally shifting it downward by 3 units.
This means that the vertical asymptote of y = 1/x at x = 0 is shifted to x = -5 for y = (2/(x+5)) - 3. The horizontal asymptote of y = 1/x at y = 0 remains the same for y = (2/(x+5)) - 3.
The point (1,1) on the graph of y = 1/x is shifted to (-4,-1) on the graph of y = (2/(x+5)) - 3.
Therefore, the graph of y = (2/(x+5)) - 3 is a transformation of the graph of y = 1/x, which has a vertical asymptote at x = 0 and passes through the point (1,1), by a horizontal shift of 5 units to the left, a vertical stretch of 2 units, and a downward shift of 3 units.
Visually, the graph of y = (2/(x+5)) - 3 will look like a reflected and vertically stretched version of the graph of y = 1/x, with a vertical asymptote at x = -5 and passing through the point (-4,-1).