Question

How do you graph `f(x)=7/(x+4)` using holes, vertical and horizontal asymptotes, x and y intercepts?

Answer 1

Holes : None

Vertical asymptote : `x = -4`

Horizontal asymptote : `y = 0`

`x`-intercept : None

`y`-intercept : `(0, 7/4)`

Explanation:

Holes are literally what they mean—a hole in a graph. It's when both numerator and denominator of the equation have an exact same factor. For example, in the equation `f(x) = (3(x+1))/(x+1)`, there would be a hole at `-1`.

In our equation `f(x) = 7/(x+4)`, there are no factors on both numerator and denominator of the function, so there are no holes in the graph.

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Vertical asymptotes are vertical lines where the graph approaches and get closer to but NEVER touches.

To find whether there is/are vertical asymptote(s), we set the denominator of the function equal to `0`, like this:
Our denominator of the function is `x+4`, so we do this:
`x+4 = 0`
`x = -4`

As said earlier, vertical asymptotes are vertical lines, meaning they start with `x =`.

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Horizontal asymptotes are horizontal lines where the graph approaches; however, it can be crossed.

The following are the rules for finding horizontal asymptotes:
In the following,
Let m be the degree of the numerator.
Let n be the degree of the denominator.

1. if m > n , then there is no horizontal asymptote

2. if m = n , then the horizontal asymptote is dividing the coefficients of the numerator and denominator

3. if m < n , then the horizontal asymptote is y=0

From the equation `f(x) = 7/(x+4)`, we can see that the degree of the denominator is higher than the degree of the numerator (there's an `x` on the bottom but no `x` on top). This means that the horizontal asymptote is `y = 0`.

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X-intercepts are where the graph touches the x-axis. It's sort of similar to finding the vertical asymptote, but we look at the numerator instead.
We set the numerator equal to `0`:
`7 = 0` BUT THAT IS NEVER TRUE! `7 != 0`
This means that there are no x-intercepts.

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Y-intercepts are the values of `y` when you plug in `0` for `x` or where the graph touches the y-axis.
As said, let's plug in `0` for the `x` values:
`f(0) = 7/(x+4)`
`f(0) = 7/(0+4)`
`f(0) = 7/4`

So there is an y-intercept at `(0, 7/4)`.

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If you need more help or want to watch a video, feel free to watch this:

Sorry, I know it's kind of long!
Hope this helps!