for function f(x)=(x)/x^2-4.
1. State the vertical and horizontal asymtopes.
2. Graph this function.
Assuming you meant f(x) = x/(x^2 - 4)
there are two vertical asymptotes, one at x = 2 and the other at x=-2
the HA is the x-axis
please solve this equation above
To find the vertical and horizontal asymptotes of the function f(x) = x / (x^2 - 4), we can follow these steps:
1. Vertical Asymptotes:
To find the vertical asymptotes, we need to determine the values of x for which the denominator of the function becomes zero. We set the denominator (x^2 - 4) equal to zero and solve for x:
x^2 - 4 = 0
By factoring the equation, we get:
(x - 2)(x + 2) = 0
So, x = 2 and x = -2 are the values where the denominator becomes zero. Therefore, there are vertical asymptotes at x = 2 and x = -2.
2. Horizontal Asymptotes:
To find the horizontal asymptotes, we need to examine the behavior of the function as x approaches positive or negative infinity. To do this, we compare the degrees of the numerator and denominator.
The degree of the numerator is 1 (since it's a linear function) and the degree of the denominator is 2 (since it's a quadratic function).
When the degree of the numerator is less than the degree of the denominator, the horizontal asymptote can be found using the following rules:
- If the degree of the numerator is one less than the degree of the denominator (as in this case), then the horizontal asymptote is y = 0.
Therefore, in this case, the function f(x) = x / (x^2 - 4) has a horizontal asymptote at y = 0.
Now, let's move on to graphing the function:
To graph the function, we can start by plotting some key points and then joining them to form a curve.
1. Find the x-intercepts:
To find the x-intercepts, we set the numerator equal to zero:
x = 0
So, the function intersects the x-axis at x = 0.
2. Find the y-intercept:
To find the y-intercept, we substitute x = 0 into the function:
f(0) = 0 / (0^2 - 4) = 0 / (-4) = 0
So, the function intersects the y-axis at y = 0.
3. Additional Points:
To get a sense of the overall shape of the function, you can find a few more points by substituting some values of x into the function. For example, you can choose x = -3, -1, 1, and 3, and calculate the corresponding y-values:
f(-3) = (-3) / ((-3)^2 - 4) = -3 / (9 - 4) = -3 / 5
f(-1) = (-1) / ((-1)^2 - 4) = -1 / (1 - 4) = 1 / 3
f(1) = 1 / ((1)^2 - 4) = 1 / (1 - 4) = -1
f(3) = 3 / ((3)^2 - 4) = 3 / (9 - 4) = 3 / 5
4. Plot the points:
Now that you have the x-intercept, y-intercept, and a few additional points, you can plot them on a graph.
5. Draw the curve:
Finally, connect the points with a smooth curve that approaches the vertical asymptotes at x = 2 and x = -2, and approaches the horizontal asymptote at y = 0.
Note: It's important to remember that graphing functions accurately may require more points and precision, but this procedure should give you a general idea of the shape of the graph.