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 plotting several points on the graph.
- For example, choose x-values of -4, -2, -1, 1, 2, and 4.
- Calculate the corresponding y-values by substituting these x-values into the equation: y = -10/x.
- Plot these points on the graph.
- Note that the y-values become very large as x approaches zero and as x moves away from zero in both positive and negative directions.
- Draw a smooth curve that passes through these points, keeping in mind the behavior around x = 0.
- The graph should approach the x-axis from the positive and negative side without actually crossing it.
- This behavior is due to the reciprocal relationship; as x gets very close to zero, the reciprocal (-10/x) approaches positive or negative infinity.
2. Graphing y = 1/(x+3) + 3:
- Start by plotting several points using the same method as above.
- For example, choose x-values of -4, -2, -1, 1, 2, and 4.
- Determine the corresponding y-values by substituting these x-values into the equation: y = 1/(x+3) + 3.
- Plot these points on the graph.
- Note that this function represents a vertical shift of the graph y = 1/x by 3 units upward and the graph becomes narrower around x = -3.
- Draw a smooth curve that passes through these points, considering the behavior around x = -3.
- The graph should not intersect the vertical line x = -3, as the denominator (x+3) approaches zero.
Domain and Range:
- The domain of a rational function is all the values that x can take on.
- For y = -10/x, the domain is all real numbers except x = 0 since division by zero is undefined.
- For y = 1/(x+3) + 3, the domain is all real numbers except for x = -3 since division by zero is undefined.
- The range of a rational function is all the resulting y-values.
- For y = -10/x, the range is all real numbers except y = 0 since the function does not intersect the x-axis.
- For y = 1/(x+3) + 3, the range is all real numbers except y = 3 since the function does not intersect the horizontal line y = 3.
Comparison with y = 1/x:
- The graph y = -10/x is similar to y = 1/x, but it is reflected in the y-axis and vertically stretched by a factor of 10. The asymptote is the x-axis.
- The graph y = 1/(x+3) + 3 is similar to y = 1/x, but it is shifted 3 units upward, and the graph becomes narrower around x = -3.