Graph the function using transformations. Show your work and state
a) the domain
b) the range
c) the asymptotes
f(x) = 3 - 1/(x + 4)^2
if g(x) = 1/x^2
1/(x+4)^2 is g(x) shifted left 4 units
-1/(x+4)^2 flips it over the y-ais
3 - 1/(x+4)^2 shifts it up 3 units
so, f(x) is 1/x^2 shifted left, reflected, and then shifted up.
Same with the asymptotes, which I am sure you can figure out for 1/x^2.
domain is all reals except x = -4
range is all reals < 3
To graph the function f(x) = 3 - 1/(x + 4)^2 using transformations, we first need to identify the transformations involved. Let's break it down step by step:
1. Vertical Translation:
The function starts with f(x) = 3, which means it is a horizontal line passing through the y-coordinate 3. In terms of transformations, this is a vertical translation of 3 units upward.
2. Horizontal Translation:
The next part is 1/(x + 4)^2. This expression includes a horizontal translation, shifting the graph 4 units to the left.
Now let's find the key points and plot the function step by step:
1. Identify the vertical asymptote:
The denominator (x + 4)^2 equals zero when x = -4. Therefore, the vertical asymptote is x = -4.
2. Find the x-intercept:
To find the x-intercept, set f(x) = 0 and solve for x:
0 = 3 - 1/(x + 4)^2
1/(x + 4)^2 = 3
(x + 4)^2 = 1/3
Take the square root:
x + 4 = ± √(1/3)
x = -4 ± √(1/3)
So the x-intercepts are x = -4 - √(1/3) and x = -4 + √(1/3).
3. Determine the domain:
The domain includes all real numbers except -4 (to avoid division by zero). So the domain is (-∞, -4) U (-4, ∞).
4. Determine the range:
Since the function is a vertical line translated 3 units upward, the range is (3, ∞).
5. Graph the function:
Using the information we gathered, we can now plot our graph.
- Mark the vertical asymptote at x = -4.
- Plot the x-intercepts at x = -4 - √(1/3) and x = -4 + √(1/3).
- Since the range is (3, ∞), draw a horizontal line passing through the y-coordinate 3.
Here is a rough sketch of the graph:
```
|
4 +------------------------------
|
3 +----
|
2 +
|
1 +
|
0 +
|
+--------------------------
-6 -4 -2 0 2 4 6
```
This represents the graph of the function f(x) = 3 - 1/(x + 4)^2, showing the domain, range, and asymptotes.