Sketch the curve calculus. Example 3 Sketch the graph of - 4/x^2-1
Wrong again.
If y = - 4/(x^2-1) then y(0) = -4/-1 = 4
So, since the y-intercept is at (0,4) and there are no x-intercepts, the graph opens upward, not downward.
Apologies for the confusion in my previous response. You are correct. Let's revise the graph based on the corrected information.
- The y-intercept is at (0, 4).
- There are no x-intercepts.
- The function has vertical asymptotes at x = -1 and x = 1.
- The function is symmetric about the y-axis.
Please find the revised graph below:
```
| . . . . . . . |
| : : |
| : : |
|------- -------|
+ ∞ . . + ∞
| : : |
| : : |
| . . . . . . . |
| : : |
| : : |
|--------o---------o-------- |
- ∞ -1 0 1 + ∞
|--------o--------o-------- |
| : : |
| : : |
| . . . . . . . |
| : : |
| : : |
- ∞
```
The revised graph accurately represents the behavior and key features of the function f(x) = -4/(x^2 - 1), with the correct y-intercept at (0, 4) and upward opening.
To sketch the graph of the function f(x) = -4/(x^2-1), we will follow these steps:
Step 1: Determine the vertical asymptotes.
To find the vertical asymptotes, set the denominator equal to 0 and solve for x:
x^2 - 1 = 0
(x-1)(x+1) = 0
x = 1 or x = -1
So, the vertical asymptotes are x = 1 and x = -1.
Step 2: Identify the horizontal asymptote.
To determine the horizontal asymptote, we examine the behavior of the function as x approaches positive infinity and negative infinity. Let's look at the end behavior:
As x approaches positive infinity, the term 4/x^2 becomes closer to 0, and the function approaches 0. So, the horizontal asymptote is y = 0.
Step 3: Find the x-intercepts.
To find the x-intercepts, we set the numerator equal to 0 and solve for x:
-4 = 0
Since there is no value of x that satisfies this equation, there are no x-intercepts.
Step 4: Determine the y-intercept.
To find the y-intercept, we evaluate the function at x = 0:
f(0) = -4/(0^2 - 1) = -4/-1 = 4
So, the y-intercept is 4.
Step 5: Plot the points and sketch the curve.
We can now plot the vertical asymptotes at x = 1 and x = -1. Then, we plot the y-intercept at (0, 4). Finally, we can sketch the curve, considering the behavior of the function between the asymptotes and its approach towards the horizontal asymptote.
The resulting sketch of the graph of f(x) = -4/(x^2-1) should look something like this:
```
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----------------------------------
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| /
```
Note: The graph is not symmetric with respect to the y-axis because of the negative sign in the function.
To sketch the graph of the function f(x) = -4/(x^2 - 1), we can start by analyzing its behavior at certain points.
First, we look at the vertical asymptotes. The function has vertical asymptotes when the denominator is equal to zero, so x^2 - 1 = 0. Solving this equation, we find that x = -1 and x = 1. These values can be seen as vertical asymptotes since the function approaches positive or negative infinity as x approaches these values.
Next, we examine the behavior near the x-intercepts. To find the x-intercepts, we set the numerator equal to zero, yielding -4 = 0. However, this equation has no solution, so the function has no x-intercepts.
Additionally, we check the behavior near the y-intercept. To find the y-intercept, we substitute x = 0 into the equation, giving f(0) = -4/(0^2 - 1) = -4. Therefore, the y-intercept is (0, -4).
We can also analyze the symmetry of the function. By replacing x with -x in the equation, we get f(-x) = -4/((-x)^2 - 1) = -4/(x^2 - 1), which means the function is even and symmetric about the y-axis.
With this information, we can now sketch the graph:
- As x approaches -1 from the left, the function approaches negative infinity. As x approaches -1 from the right, the function approaches positive infinity.
- As x approaches 1 from the left, the function approaches positive infinity. As x approaches 1 from the right, the function approaches negative infinity.
- The function has one vertical asymptote at x = -1 and another at x = 1.
- There are no x-intercepts.
- The y-intercept is (0, -4).
- The function is symmetric about the y-axis.
Based on these properties, we can sketch the graph of f(x) = -4/(x^2 - 1) as shown below:
```
| . . . . . . . |
| : : |
| : : |
|------- -------|
+ ∞ . . + ∞
| : : |
| : : |
| . . . . . . . |
| : : |
| : : |
|--------o---------o-------- |
- ∞ -1 0 1 + ∞
|--------o--------o-------- |
| : : |
| : : |
| . . . . . . . |
| : : |
| : : |
+ ∞ . . + ∞
|-------- ------ |
| : : |
| : : |
| . . . . . . . |
| : : |
| : : |
- ∞
```
This graph accurately represents the behavior and key features of the function f(x) = -4/(x^2 - 1).