sketch the graph of x square minus one over x minus x square
there are lots of online graphing web sites.
If you meant y = (x^2-1)/(x-x^2) then note that is just
(x+1)/(x-1) / x(1-x) = -(x+1)/x = -1 - 1/x
for x≠1
So it has a hole at x=1, and then also
a vertical asymptote at x=0
a horizontal asymptote at y = -1
To sketch the graph of the function f(x) = (x^2 - 1)/(x - x^2), we can follow these steps:
Step 1: Determine the domain of the function.
The function is defined for all values of x, except for those that make the denominator equal to zero. So, x - x^2 = 0 gives us x(1 - x) = 0 ⇒ x = 0 or x = 1. Therefore, the function is defined for all x except 0 and 1. The domain of the function is (-∞, 0) ∪ (0, 1) ∪ (1, ∞).
Step 2: Find the x-intercepts.
The x-intercepts occur when the numerator, x^2 - 1, equals zero. Solving the equation x^2 - 1 = 0 gives us x = -1 and x = 1 as the x-intercepts.
Step 3: Find the y-intercept.
The y-intercept is the value of the function when x = 0. Substituting x = 0 into the function, we get f(0) = (-1 - 0)/(0 - 0) = 0. So, the y-intercept is at (0, 0).
Step 4: Determine the behavior of the function as x approaches positive and negative infinity.
As x approaches positive or negative infinity, the function simplifies to x^2/x = x. So, the function approaches positive or negative infinity as x approaches positive or negative infinity, respectively.
Step 5: Find the vertical asymptotes.
Vertical asymptotes occur when the denominator is equal to zero. In this case, we already determined that x = 0 and x = 1 are not in the domain of the function. Therefore, there are no vertical asymptotes.
Step 6: Sketch the graph using the information obtained.
Based on the steps above, we have the following key points on the graph:
- x-intercepts: (-1, 0), (1, 0)
- y-intercept: (0, 0)
- Behavior as x approaches positive and negative infinity: Approaching positive and negative infinity, respectively.
Now, plot these points on the coordinate plane and sketch the curve accordingly.
To sketch the graph of the given expression, we can follow these steps:
1. Simplify the expression: x^2 - 1 / (x - x^2)
2. Factor the numerator and denominator, if possible: (x - 1)(x + 1) / x(1 - x)
3. Determine the vertical asymptotes: Set the denominator equal to zero and solve for x. In this case, x(1 - x) = 0. Thus, there are two vertical asymptotes at x = 0 and x = 1.
4. Determine the horizontal asymptote: As x approaches positive or negative infinity, the expression approaches zero. Therefore, the horizontal asymptote is y = 0.
5. Find the x-intercepts: Set the expression equal to zero and solve for x. This occurs when (x - 1)(x + 1) = 0, so the x-intercepts are x = -1 and x = 1.
6. Determine the behavior around the x-intercepts: To determine whether each x-intercept is a local maximum or minimum, we can use the first derivative test.
- To the left of x = -1, the derivative is negative, indicating a local maximum.
- Between x = -1 and x = 0, the derivative is positive, indicating a local minimum.
- Between x = 0 and x = 1, the derivative is negative, indicating a local maximum.
- To the right of x = 1, the derivative is positive, indicating a local minimum.
7. Draw the graph based on the information obtained:
- Draw vertical asymptotes at x = 0 and x = 1.
- Draw the horizontal asymptote at y = 0.
- Mark the x-intercepts at x = -1 and x = 1.
- Sketch the behavior of the function around the x-intercepts based on the first derivative test.
The final graph should show a function with two vertical asymptotes, two x-intercepts, and a horizontal asymptote. The function will have a local maximum to the left of x = -1, a local minimum between x = -1 and x = 0, a local maximum between x = 0 and x = 1, and a local minimum to the right of x = 1.