How do you sketch y=2^x-2^-x? Is there a some type of trick, without using a table of values.
To sketch the graph of the equation y = 2^x - 2^(-x) without using a table of values, you can follow these steps:
1. Analyze the behavior of the equation:
- As x approaches negative infinity, both terms in the equation approach zero. Therefore, their difference approaches zero as well.
- As x approaches positive infinity, the first term 2^x increases exponentially while the second term 2^(-x) approaches zero. Therefore, their difference becomes significantly larger.
2. Identify key points:
- The y-intercept occurs when x = 0. Substituting x = 0 into the equation, we get:
y = 2^0 - 2^(-0) = 1 - 1 = 0. So, the y-intercept is (0, 0).
- To find the x-intercepts, set y = 0:
2^x - 2^(-x) = 0
2^x = 2^(-x)
Taking the logarithm base 2 of both sides, we get:
x = -x
2x = 0
x = 0. So, the x-intercept is (0, 0).
3. Determine the symmetry:
- Since the equation contains only even powers of x, it exhibits symmetry about the y-axis.
4. Sketch the graph:
- Based on the y-intercept, x-intercept, and symmetry, we know that the graph starts at (0, 0) and it lies symmetrically about the y-axis.
- To determine the shape of the curve, we can examine the behavior of the equation for a small range of x-values. For example, you can choose two points: one on the left side of the y-axis and the other on the right side. These points could be x = -1 and x = 1.
- Substituting x = -1 into the equation, we get:
y = 2^(-1) - 2^1 = 1/2 - 2 = -3/2
So, the point (-1, -3/2) lies on the graph.
- Substituting x = 1 into the equation, we get:
y = 2^1 - 2^(-1) = 2 - 1/2 = 3/2
So, the point (1, 3/2) lies on the graph.
- Based on these points, you can draw a smooth curve pass through them symmetrically about the y-axis.
Remember, it's always a good idea to use additional points if available or use graphing software to verify your sketch.