Graph each rational function;identify the domain and ranges;and compare the graph with the graph y=1/x.
1.y=-10/x
2.y=1/(x+3)+3
To graph each rational function and identify the domain and range, we can follow these steps:
1. **Graphing y = -10/x:**
- Start by identifying any vertical asymptotes. In this case, since the denominator is x, the function has a vertical asymptote at x = 0.
- Next, identify any horizontal asymptotes. Since the degree of the numerator is less than the degree of the denominator (-10 is just a constant), the function has a horizontal asymptote at y = 0.
- Plot a few points on both sides of the vertical asymptote. For example, when x = -1, y = -10/(-1) = 10; when x = 1, y = -10/(1) = -10.
- Draw the graph passing through these points, getting closer and closer to the asymptotes, and confirming the asymptotes.
- The domain of this function is all real numbers except x = 0 (to avoid division by zero).
- The range of this function is all real numbers except y = 0.
2. **Graphing y = 1/(x + 3) + 3:**
- Start by identifying any vertical asymptotes. In this case, the function has a vertical asymptote at x = -3 (since the denominator is x + 3).
- Next, identify any horizontal asymptotes. Since the degree of the numerator is less than the degree of the denominator (1 in the numerator, 1 in the denominator), the function has a horizontal asymptote at y = 0.
- Plot a few points on both sides of the vertical asymptote. For example, when x = -4, y = 1/(-4 + 3) + 3 = -2 + 3 = 1; when x = -2, y = 1/(-2 + 3) + 3 = 2 + 3 = 5.
- Draw the graph passing through these points, getting closer and closer to the asymptote, and confirming the asymptotes.
- The domain of this function is all real numbers except x = -3.
- The range of this function is all real numbers.
Now, let's compare these graphs with the graph of y = 1/x:
- The graph of y = -10/x is a reflection of y = 1/x across the x-axis and vertically stretched by a factor of 10. It has the same vertical asymptote (x = 0) and horizontal asymptote (y = 0).
- The graph of y = 1/(x + 3) + 3 is a translation of y = 1/x to the left by 3 units and vertically shifted upwards by 3 units. It has a vertical asymptote at x = -3 and a horizontal asymptote at y = 0.
These comparisons help us see the similarities and differences between the graphs.