graph the function using transformations, show and state , a) the domain, B) range c) the asymptotes
F(x)= 1/x^2-4
please show work!
I see no use for transformations if F(x) = 1/(x^2-4), so I'll assume you meant what you wrote:
F(x) = (1/x^2) - 4
That is just the graph of 1/x^2, translated 4 units downward.
So, since 1/x^2 has asymptotes x=0 and y=0, F has asymptotes x=0 and y = -4.
Domain is all reals except x=0
Range is all reals except y=-4
To graph the function f(x) = 1/(x^2 - 4), we can use several transformations. Here's how you can do it:
1. Start with the parent function f(x) = 1/x^2, which is a basic quadratic function with an asymptote at y = 0 and is symmetric about the y-axis.
2. Apply a horizontal shift by 2 units to the right. To do this, replace x in the parent function with (x - 2). Now, we have g(x) = 1/(x - 2)^2.
3. Now, apply a vertical shift upward by 4 units. To do this, add 4 to the function g(x). Our transformed function will be h(x) = 1/(x - 2)^2 + 4.
Now, let's determine the domain, range, and asymptotes for the function h(x):
a) Domain: The domain consists of all real numbers except the values that make the denominator zero. In this case, the denominator is (x - 2)^2, so we need to exclude x = 2. Therefore, the domain is (-∞, 2) U (2, ∞).
b) Range: To find the range, we need to analyze the behavior of the function. Since the denominator is a squared term, it is always positive (except at x = 2). As a result, the value of h(x) approaches 0 as x approaches positive or negative infinity. Plus, the constant term 4 added to the function gives a vertical shift upward by 4 units. Therefore, the range is (4, ∞).
c) Asymptotes: The function h(x) = 1/(x - 2)^2 + 4 has the following asymptotes:
- Horizontal asymptote: As x approaches positive or negative infinity, h(x) approaches 4, resulting in a horizontal asymptote at y = 4.
- Vertical asymptote: Since the denominator is (x - 2)^2, there is a vertical asymptote at x = 2.
Now, using this information, you can plot the graph of f(x) = 1/(x^2 - 4) by first plotting the parent function f(x) = 1/x^2, then applying the transformations mentioned above.